Explanation:
Okay, so I need to find the modulo 7 remainder of the sum of all the odd numbers from 1 up to 199. Let me think about how to approach this.

First, let me recall that the sum of the first n odd numbers is n squared. Wait, is that right? Let me check with a small example. The first odd number is 1, sum is 1, which is 1². The first two odd numbers: 1 + 3 = 4, which is 2². Yeah, that seems correct. So, if I can figure out how many terms there are in this sequence from 1 to 199, then I can square that number to get the sum, and then take modulo 7 of that result.

But wait, let me confirm if 199 is indeed the last term. The problem says "1+3+5+7+9+...+195+197+199." So yes, the last term is 199. So I need to find how many terms are there in this sequence.

Since these are odd numbers, they form an arithmetic sequence where the first term a₁ = 1, the common difference d = 2, and the last term aₙ = 199. The formula for the nth term of an arithmetic sequence is aₙ = a₁ + (n - 1)d. Let me plug in the values:

199 = 1 + (n - 1)*2

Subtract 1 from both sides:

198 = (n - 1)*2

Divide both sides by 2:

99 = n - 1

Add 1 to both sides:

n = 100

So there are 100 terms in this sequence. Therefore, the sum S is n² = 100² = 10,000. Wait, so the sum is 10,000? Let me check that again. If there are 100 terms, then the sum is 100² = 10,000. That seems correct.

But let me verify with another approach. The formula for the sum of an arithmetic sequence is S = n/2 * (a₁ + aₙ). Let me compute that:

n = 100, a₁ = 1, aₙ = 199.

So S = 100/2 * (1 + 199) = 50 * 200 = 10,000. Yep, same result. Okay, so the sum is indeed 10,000.

Now, I need to find 10,000 mod 7. Let me compute that.

First, I can divide 10,000 by 7 and find the remainder. Alternatively, I can break down 10,000 into smaller parts that are easier to compute modulo 7. Let me try both methods.

First method: Let me divide 10,000 by 7.

Well, 7*1428 = 9996. Because 7*1400 = 9800, 7*28 = 196, so 9800 + 196 = 9996. Then 10,000 - 9996 = 4. Therefore, 10,000 mod 7 is 4.

Wait, so the remainder is 4? Let me check that again.

Alternatively, maybe I can compute 10,000 mod 7 by breaking it down. Let me recall that 10 mod 7 is 3, so 10 ≡ 3 mod 7. Therefore, 10^k mod 7 can be calculated using powers of 3.

But 10,000 is 10^4. So 10^1 mod 7 = 3, 10^2 mod 7 = 3*3 = 9 mod 7 = 2, 10^3 mod 7 = 2*3 = 6 mod 7, 10^4 mod 7 = 6*3 = 18 mod 7 = 4. So that also gives 4. So that's consistent. Therefore, 10,000 mod 7 is 4.

Therefore, the answer is 4? Wait, but let me make sure that I didn't make a mistake in calculating the sum. Because if the sum is indeed 10,000, then mod 7 is 4. But maybe I made a mistake in calculating the number of terms?

Let me check again. The sequence is 1, 3, 5,...,199. So the nth term is 1 + (n-1)*2 = 2n - 1. So setting 2n - 1 = 199, so 2n = 200, so n = 100. So that's correct. Therefore, the sum is 100² = 10,000.

Alternatively, maybe I can compute the sum modulo 7 directly, without calculating the entire sum? Maybe that would be easier? Let me think.

Because sometimes when dealing with modulo operations, it's easier to take modulo at each step. But since the sum is 10,000, which is manageable, but maybe there's a smarter way.

Alternatively, since we know that the sum is n² where n = 100, so we can compute 100² mod 7.

But 100 mod 7: Let me compute 7*14 = 98, so 100 - 98 = 2. Therefore, 100 ≡ 2 mod 7. Therefore, 100² ≡ 2² = 4 mod 7. So that's the same result.

Therefore, regardless of the method, the remainder is 4.

But let me try another approach just to be thorough. Let me compute the sum modulo 7 by considering each term modulo 7 and adding them up.

But since there are 100 terms, that might take a while, but maybe there's a pattern.

First, note that the sequence of odd numbers from 1 to 199. Let me think about their residues modulo 7.

Each odd number can be written as 2k + 1, where k ranges from 0 to 99 (since the last term is 199 = 2*99 + 1). Wait, but maybe that's complicating.

Alternatively, since we have 100 terms, and modulo 7 cycles every 7 numbers. So perhaps we can find how many complete cycles of 7 there are in the sequence, and then the remainder.

But the problem is that the numbers aren't consecutive integers, they are odd numbers. So the residues modulo 7 will repeat every 14 numbers? Wait, because adding 7 to an odd number gives an even number, so the residues modulo 7 for odd numbers would cycle every 7 numbers, but since we are only taking odd numbers, maybe every 14 numbers? Wait, let me think.

Let me consider the residues of odd numbers modulo 7. The odd numbers modulo 7 can be 1, 3, 5, 7≡0, 9≡2, 11≡4, 13≡6, 15≡1, etc. So the residues cycle every 7 numbers, but since we are only taking odd numbers, the residues cycle every 7 numbers? Wait, let me check:

Starting from 1:

1 mod7 = 1

3 mod7 = 3

5 mod7 = 5

7 mod7 = 0

9 mod7 = 2

11 mod7 = 4

13 mod7 = 6

15 mod7 = 1

17 mod7 = 3

... and so on. So the residues cycle every 7 terms: 1,3,5,0,2,4,6, then repeats 1,3,5,0,2,4,6,...

Therefore, the residues of the odd numbers modulo 7 cycle every 7 terms with the pattern [1,3,5,0,2,4,6].

Therefore, each block of 7 terms contributes a sum of 1+3+5+0+2+4+6 = 21.

So if we can find how many complete blocks of 7 terms there are in 100 terms, and then the remaining terms.

So 100 divided by 7 is 14 with a remainder of 2. Because 14*7 = 98, so 100 - 98 = 2.

Therefore, there are 14 complete blocks and 2 remaining terms.

Each block contributes 21 mod7. Let me compute 21 mod7: 21 is 3*7, so 21 mod7 = 0. Therefore, each block contributes 0 mod7.

Therefore, the total contribution from the 14 blocks is 14*0 = 0 mod7.

Now, the remaining 2 terms. What are those terms?

Since each block is 7 terms, the first 98 terms are 14 blocks. The 99th term is the first term of the next block, which is 1 mod7, and the 100th term is the second term of the block, which is 3 mod7.

Wait, let me check. Let me think about the sequence of residues. The first block (terms 1-7) has residues [1,3,5,0,2,4,6]. The second block (terms 8-14) would be [1,3,5,0,2,4,6], and so on. Therefore, term number 7k + 1 has residue 1, term 7k + 2 has residue 3, etc.

Therefore, term 99: Let me compute 99 divided by 7. 7*14 = 98, so term 98 is the last term of the 14th block. Then term 99 is the first term of the next block, which is residue 1. Term 100 is the second term of the block, residue 3.

Therefore, the remaining two terms have residues 1 and 3.

Therefore, the total sum mod7 is 0 + 1 + 3 = 4 mod7.

Which matches the previous result.

Therefore, regardless of the method, the remainder is 4.

But just to be absolutely thorough, let me check with another approach. Let me compute the sum S = 10,000. Let me divide 10,000 by 7.

As I did before, 7*1428 = 9996. Then 10,000 - 9996 = 4. Therefore, remainder 4.

Alternatively, maybe I can compute 10,000 divided by 7 step by step.

But since 7*1000 = 7000. Then 10,000 - 7000 = 3000.

Then 7*400 = 2800. 3000 - 2800 = 200.

Then 7*28 = 196. 200 - 196 = 4.

So total is 1000 + 400 + 28 = 1428, with remainder 4. Yep, same result.

Therefore, all methods lead to the conclusion that the remainder is 4.

But just to make sure that I didn't make a mistake in the number of terms. Let me check again.

The sequence is 1, 3, 5,...,199.

Number of terms: since it's an arithmetic sequence with first term 1, last term 199, common difference 2.

Number of terms n = ((last term - first term)/common difference) + 1 = ((199 - 1)/2) + 1 = (198/2) + 1 = 99 + 1 = 100. Correct.

Therefore, sum is 100² = 10,000.

Therefore, 10,000 mod7 = 4.

Alternatively, maybe I can compute 100 mod7 first, then square it.

As before, 100 divided by 7 is 14*7=98, remainder 2. So 100 ≡ 2 mod7. Therefore, 100² ≡ 2² = 4 mod7.

Therefore, same result.

Therefore, I think I can be confident that the answer is 4.

But just to make sure, let me try calculating the sum modulo 7 step by step for a smaller case.

For example, let's take the sum of the first 7 odd numbers: 1+3+5+7+9+11+13.

Sum is 1+3=4, +5=9, +7=16, +9=25, +11=36, +13=49.

49 mod7 is 0. Which matches our previous calculation, since each block of 7 terms sums to 21, which is 0 mod7.

Similarly, if I take the first 8 terms: sum is 49 + 15 = 64. 64 mod7: 7*9=63, so 64 mod7=1. Which would be the same as adding the next term (15) which is 1 mod7.

Similarly, first 9 terms: 64 + 17 = 81. 81 mod7: 7*11=77, 81-77=4. Which is 1 + 3 = 4 mod7. Which matches our previous calculation.

Therefore, this seems to check out.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{4}

To find the modulo 7 remainder of the sum of all odd numbers from 1 to 199, we begin by analyzing the structure of the sequence.

---

### Step 1: Identify the Sequence

The sequence is:

$$
1 + 3 + 5 + 7 + \dots + 199
$$

This is an arithmetic sequence of **odd numbers**, with:

- First term $ a = 1 $
- Common difference $ d = 2 $
- Last term $ l = 199 $

We can find the number of terms $ n $ using the formula for the $ n $-th term of an arithmetic sequence:

$$
l = a + (n - 1)d
$$

$$
199 = 1 + (n - 1) \cdot 2 \Rightarrow 198 = 2(n - 1) \Rightarrow n = 100
$$

So, there are **100 terms** in the sequence.

---

### Step 2: Compute the Sum

The sum $ S $ of the first $ n $ odd numbers is given by the formula:

$$
S = n^2
$$

$$
S = 100^2 = 10,000
$$

---

### Step 3: Compute the Modulo 7 Remainder

We now compute:

$$
10,000 \mod 7
$$

We can simplify this by first computing $ 100 \mod 7 $:

$$
100 \div 7 = 14 \text{ remainder } 2 \Rightarrow 100 \equiv 2 \mod 7
$$

Then:

$$
100^2 \equiv 2^2 = 4 \mod 7
$$

---

### Step 4: Verification via Residue Cycles

Alternatively, we can consider the residues of the odd numbers modulo 7. The residues of odd numbers modulo 7 cycle every 7 terms as:

$$
[1, 3, 5, 0, 2, 4, 6]
$$

Each block of 7 terms sums to:

$$
1 + 3 + 5 + 0 + 2 + 4 + 6 = 21 \equiv 0 \mod 7
$$

There are $ \left\lfloor \frac{100}{7} \right\rfloor = 14 $ full blocks, contributing $ 14 \times 0 = 0 \mod 7 $

The remaining 2 terms are the first two terms of the next block: $ 1 $ and $ 3 $, which sum to:

$$
1 + 3 = 4 \mod 7
$$

So again, the total is:

$$
0 + 4 = 4 \mod 7
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the modulo 7 remainder of the sum of all the odd numbers from 1 up to 199. Let me think about how to approach this.

First, let me recall that the sum of the first n odd numbers is n squared. Wait, is that right? Let me check with a small example. The first odd number is 1, sum is 1, which is 1². The first two odd numbers: 1 + 3 = 4, which is 2². Yeah, that seems correct. So, if I can figure out how many terms there are in this sequence from 1 to 199, then I can square that number to get the sum, and then take modulo 7 of that result.

But wait, let me confirm if 199 is indeed the last term. The problem says "1+3+5+7+9+...+195+197+199." So yes, the last term is 199. So I need to find how many terms are there in this sequence.

Since these are odd numbers, they form an arithmetic sequence where the first term a₁ = 1, the common difference d = 2, and the last term aₙ = 199. The formula for the nth term of an arithmetic sequence is aₙ = a₁ + (n - 1)d. Let me plug in the values:

199 = 1 + (n - 1)*2

Subtract 1 from both sides:

198 = (n - 1)*2

Divide both sides by 2:

99 = n - 1

Add 1 to both sides:

n = 100

So there are 100 terms in this sequence. Therefore, the sum S is n² = 100² = 10,000. Wait, so the sum is 10,000? Let me check that again. If there are 100 terms, then the sum is 100² = 10,000. That seems correct.

But let me verify with another approach. The formula for the sum of an arithmetic sequence is S = n/2 * (a₁ + aₙ). Let me compute that:

n = 100, a₁ = 1, aₙ = 199.

So S = 100/2 * (1 + 199) = 50 * 200 = 10,000. Yep, same result. Okay, so the sum is indeed 10,000.

Now, I need to find 10,000 mod 7. Let me compute that.

First, I can divide 10,000 by 7 and find the remainder. Alternatively, I can break down 10,000 into smaller parts that are easier to compute modulo 7. Let me try both methods.

First method: Let me divide 10,000 by 7.

Well, 7*1428 = 9996. Because 7*1400 = 9800, 7*28 = 196, so 9800 + 196 = 9996. Then 10,000 - 9996 = 4. Therefore, 10,000 mod 7 is 4.

Wait, so the remainder is 4? Let me check that again.

Alternatively, maybe I can compute 10,000 mod 7 by breaking it down. Let me recall that 10 mod 7 is 3, so 10 ≡ 3 mod 7. Therefore, 10^k mod 7 can be calculated using powers of 3.

But 10,000 is 10^4. So 10^1 mod 7 = 3, 10^2 mod 7 = 3*3 = 9 mod 7 = 2, 10^3 mod 7 = 2*3 = 6 mod 7, 10^4 mod 7 = 6*3 = 18 mod 7 = 4. So that also gives 4. So that's consistent. Therefore, 10,000 mod 7 is 4.

Therefore, the answer is 4? Wait, but let me make sure that I didn't make a mistake in calculating the sum. Because if the sum is indeed 10,000, then mod 7 is 4. But maybe I made a mistake in calculating the number of terms?

Let me check again. The sequence is 1, 3, 5,...,199. So the nth term is 1 + (n-1)*2 = 2n - 1. So setting 2n - 1 = 199, so 2n = 200, so n = 100. So that's correct. Therefore, the sum is 100² = 10,000.

Alternatively, maybe I can compute the sum modulo 7 directly, without calculating the entire sum? Maybe that would be easier? Let me think.

Because sometimes when dealing with modulo operations, it's easier to take modulo at each step. But since the sum is 10,000, which is manageable, but maybe there's a smarter way.

Alternatively, since we know that the sum is n² where n = 100, so we can compute 100² mod 7.

But 100 mod 7: Let me compute 7*14 = 98, so 100 - 98 = 2. Therefore, 100 ≡ 2 mod 7. Therefore, 100² ≡ 2² = 4 mod 7. So that's the same result.

Therefore, regardless of the method, the remainder is 4.

But let me try another approach just to be thorough. Let me compute the sum modulo 7 by considering each term modulo 7 and adding them up.

But since there are 100 terms, that might take a while, but maybe there's a pattern.

First, note that the sequence of odd numbers from 1 to 199. Let me think about their residues modulo 7.

Each odd number can be written as 2k + 1, where k ranges from 0 to 99 (since the last term is 199 = 2*99 + 1). Wait, but maybe that's complicating.

Alternatively, since we have 100 terms, and modulo 7 cycles every 7 numbers. So perhaps we can find how many complete cycles of 7 there are in the sequence, and then the remainder.

But the problem is that the numbers aren't consecutive integers, they are odd numbers. So the residues modulo 7 will repeat every 14 numbers? Wait, because adding 7 to an odd number gives an even number, so the residues modulo 7 for odd numbers would cycle every 7 numbers, but since we are only taking odd numbers, maybe every 14 numbers? Wait, let me think.

Let me consider the residues of odd numbers modulo 7. The odd numbers modulo 7 can be 1, 3, 5, 7≡0, 9≡2, 11≡4, 13≡6, 15≡1, etc. So the residues cycle every 7 numbers, but since we are only taking odd numbers, the residues cycle every 7 numbers? Wait, let me check:

Starting from 1:

1 mod7 = 1

3 mod7 = 3

5 mod7 = 5

7 mod7 = 0

9 mod7 = 2

11 mod7 = 4

13 mod7 = 6

15 mod7 = 1

17 mod7 = 3

... and so on. So the residues cycle every 7 terms: 1,3,5,0,2,4,6, then repeats 1,3,5,0,2,4,6,...

Therefore, the residues of the odd numbers modulo 7 cycle every 7 terms with the pattern [1,3,5,0,2,4,6].

Therefore, each block of 7 terms contributes a sum of 1+3+5+0+2+4+6 = 21.

So if we can find how many complete blocks of 7 terms there are in 100 terms, and then the remaining terms.

So 100 divided by 7 is 14 with a remainder of 2. Because 14*7 = 98, so 100 - 98 = 2.

Therefore, there are 14 complete blocks and 2 remaining terms.

Each block contributes 21 mod7. Let me compute 21 mod7: 21 is 3*7, so 21 mod7 = 0. Therefore, each block contributes 0 mod7.

Therefore, the total contribution from the 14 blocks is 14*0 = 0 mod7.

Now, the remaining 2 terms. What are those terms?

Since each block is 7 terms, the first 98 terms are 14 blocks. The 99th term is the first term of the next block, which is 1 mod7, and the 100th term is the second term of the block, which is 3 mod7.

Wait, let me check. Let me think about the sequence of residues. The first block (terms 1-7) has residues [1,3,5,0,2,4,6]. The second block (terms 8-14) would be [1,3,5,0,2,4,6], and so on. Therefore, term number 7k + 1 has residue 1, term 7k + 2 has residue 3, etc.

Therefore, term 99: Let me compute 99 divided by 7. 7*14 = 98, so term 98 is the last term of the 14th block. Then term 99 is the first term of the next block, which is residue 1. Term 100 is the second term of the block, residue 3.

Therefore, the remaining two terms have residues 1 and 3.

Therefore, the total sum mod7 is 0 + 1 + 3 = 4 mod7.

Which matches the previous result.

Therefore, regardless of the method, the remainder is 4.

But just to be absolutely thorough, let me check with another approach. Let me compute the sum S = 10,000. Let me divide 10,000 by 7.

As I did before, 7*1428 = 9996. Then 10,000 - 9996 = 4. Therefore, remainder 4.

Alternatively, maybe I can compute 10,000 divided by 7 step by step.

But since 7*1000 = 7000. Then 10,000 - 7000 = 3000.

Then 7*400 = 2800. 3000 - 2800 = 200.

Then 7*28 = 196. 200 - 196 = 4.

So total is 1000 + 400 + 28 = 1428, with remainder 4. Yep, same result.

Therefore, all methods lead to the conclusion that the remainder is 4.

But just to make sure that I didn't make a mistake in the number of terms. Let me check again.

The sequence is 1, 3, 5,...,199.

Number of terms: since it's an arithmetic sequence with first term 1, last term 199, common difference 2.

Number of terms n = ((last term - first term)/common difference) + 1 = ((199 - 1)/2) + 1 = (198/2) + 1 = 99 + 1 = 100. Correct.

Therefore, sum is 100² = 10,000.

Therefore, 10,000 mod7 = 4.

Alternatively, maybe I can compute 100 mod7 first, then square it.

As before, 100 divided by 7 is 14*7=98, remainder 2. So 100 ≡ 2 mod7. Therefore, 100² ≡ 2² = 4 mod7.

Therefore, same result.

Therefore, I think I can be confident that the answer is 4.

But just to make sure, let me try calculating the sum modulo 7 step by step for a smaller case.

For example, let's take the sum of the first 7 odd numbers: 1+3+5+7+9+11+13.

Sum is 1+3=4, +5=9, +7=16, +9=25, +11=36, +13=49.

49 mod7 is 0. Which matches our previous calculation, since each block of 7 terms sums to 21, which is 0 mod7.

Similarly, if I take the first 8 terms: sum is 49 + 15 = 64. 64 mod7: 7*9=63, so 64 mod7=1. Which would be the same as adding the next term (15) which is 1 mod7.

Similarly, first 9 terms: 64 + 17 = 81. 81 mod7: 7*11=77, 81-77=4. Which is 1 + 3 = 4 mod7. Which matches our previous calculation.

Therefore, this seems to check out.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{4}

To find the modulo 7 remainder of the sum of all odd numbers from 1 to 199, we begin by analyzing the structure of the sequence.

---

### Step 1: Identify the Sequence

The sequence is:

$$
1 + 3 + 5 + 7 + \dots + 199
$$

This is an arithmetic sequence of **odd numbers**, with:

- First term $ a = 1 $
- Common difference $ d = 2 $
- Last term $ l = 199 $

We can find the number of terms $ n $ using the formula for the $ n $-th term of an arithmetic sequence:

$$
l = a + (n - 1)d
$$

$$
199 = 1 + (n - 1) \cdot 2 \Rightarrow 198 = 2(n - 1) \Rightarrow n = 100
$$

So, there are **100 terms** in the sequence.

---

### Step 2: Compute the Sum

The sum $ S $ of the first $ n $ odd numbers is given by the formula:

$$
S = n^2
$$

$$
S = 100^2 = 10,000
$$

---

### Step 3: Compute the Modulo 7 Remainder

We now compute:

$$
10,000 \mod 7
$$

We can simplify this by first computing $ 100 \mod 7 $:

$$
100 \div 7 = 14 \text{ remainder } 2 \Rightarrow 100 \equiv 2 \mod 7
$$

Then:

$$
100^2 \equiv 2^2 = 4 \mod 7
$$

---

### Step 4: Verification via Residue Cycles

Alternatively, we can consider the residues of the odd numbers modulo 7. The residues of odd numbers modulo 7 cycle every 7 terms as:

$$
[1, 3, 5, 0, 2, 4, 6]
$$

Each block of 7 terms sums to:

$$
1 + 3 + 5 + 0 + 2 + 4 + 6 = 21 \equiv 0 \mod 7
$$

There are $ \left\lfloor \frac{100}{7} \right\rfloor = 14 $ full blocks, contributing $ 14 \times 0 = 0 \mod 7 $

The remaining 2 terms are the first two terms of the next block: $ 1 $ and $ 3 $, which sum to:

$$
1 + 3 = 4 \mod 7
$$

So again, the total is:

$$
0 + 4 = 4 \mod 7
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution:
Instead of adding up the sum and finding the residue, we can find the residue of each number to make computation easier.

Each group of 7 numbers would have the sum of residues $1+3+5+0+2+4+6 \equiv 21 \equiv 0 \pmod7$. Since we only have odd numbers in the sum, every $7$ odd numbers is $14$ integers. Because every group has a residue of $7$, we can ignore them.

There are $\left\lfloor \frac{199}{14}\right\rfloor=14$ sets of $14$ integers, which is equivalent to $7$ odd numbers in our sum. This leaves $197$ and $199$, which have residues $1+3 \equiv \boxed{4} \pmod7$.


Explanation:
Okay, so I need to find $ f^{-1}(f^{-1}(6)) $. Let me start by recalling what an inverse function does. The inverse function $ f^{-1} $ essentially undoes what $ f $ does. So if $ f(a) = b $, then $ f^{-1}(b) = a $.

Given the function values: $ f(1) = 2 $, $ f(2) = 6 $, and $ f(3) = 5 $. Let me write those down to keep track:

- $ f(1) = 2 $
- $ f(2) = 6 $
- $ f(3) = 5 $

So, if I want to find $ f^{-1}(6) $, I need to figure out what input to $ f $ gives me 6. Looking at the given values, $ f(2) = 6 $, so that means $ f^{-1}(6) = 2 $.

Wait, let me make sure. If $ f(2) = 6 $, then applying $ f^{-1} $ to both sides gives $ f^{-1}(f(2)) = f^{-1}(6) $, which simplifies to $ 2 = f^{-1}(6) $. Yep, that seems right. So $ f^{-1}(6) = 2 $.

Now, the problem asks for $ f^{-1}(f^{-1}(6)) $, which is $ f^{-1}(2) $. So now I need to find $ f^{-1}(2) $. Again, using the same logic, $ f^{-1}(2) $ is the value of $ x $ such that $ f(x) = 2 $.

Looking back at the given function values, $ f(1) = 2 $. Therefore, $ f^{-1}(2) = 1 $.

So putting it all together: $ f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1 $.

Wait, let me check again to make sure I didn't mix up any steps.

First step: $ f^{-1}(6) $. Since $ f(2) = 6 $, inverse function would take 6 back to 2. Then, taking the inverse function of that result, which is 2, so $ f^{-1}(2) $. Since $ f(1) = 2 $, inverse function takes 2 back to 1. So yes, the answer is 1.

But just to be thorough, let me make sure there's no other information I need to consider. The problem didn't specify the domain or codomain of the function, but since they gave three specific values, maybe there's more to the function? But since we're only dealing with the inverse function at specific points, and the inverse function is only defined for the outputs of the original function.

Given that $ f(1) = 2 $, $ f(2) = 6 $, $ f(3) = 5 $, so the outputs are 2, 6, 5. Therefore, the inverse function $ f^{-1} $ is defined for 2, 5, 6. So when we take $ f^{-1}(6) $, that's 2, and then $ f^{-1}(2) $ is 1.

Is there a possibility that the function isn't one-to-one? Wait, if the function isn't one-to-one, then the inverse wouldn't exist. But the problem says that $ f^{-1} $ is the inverse of $ f $, so that implies that $ f $ is bijective (both injective and surjective) on its domain. However, since we are only given three points, maybe the function is only defined on those three points? Or maybe it's defined on more points, but we only know these three.

But regardless, since we are only dealing with the values given, and the inverse function is defined for the outputs of $ f $, which are 2, 5, 6. So for the purposes of this problem, we can safely assume that $ f $ is invertible on the given domain, and that the inverse function is defined for those outputs.

Therefore, my conclusion is that $ f^{-1}(f^{-1}(6)) = 1 $.

But let me try to write it step by step again:

1. Find $ f^{-1}(6) $:
   - Since $ f(2) = 6 $, then $ f^{-1}(6) = 2 $.

2. Now find $ f^{-1}(f^{-1}(6)) = f^{-1}(2) $:
   - Since $ f(1) = 2 $, then $ f^{-1}(2) = 1 $.

Therefore, the final answer is 1.

I think that's solid. Let me check if there's another way to approach this. Maybe by constructing the inverse function table?

Original function:

Input | Output
--- | ---
1 | 2
2 | 6
3 | 5

Inverse function would swap inputs and outputs:

Input | Output
--- | ---
2 | 1
6 | 2
5 | 3

So, $ f^{-1}(2) = 1 $, $ f^{-1}(6) = 2 $, $ f^{-1}(5) = 3 $.

Therefore, $ f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1 $. Yep, same result.

So I think that's correct. I don't see any mistakes in my reasoning. The key was to remember that the inverse function swaps the input and output of the original function. So applying the inverse twice here is like going back two steps in the original function.

Another way to think about it: $ f^{-1}(6) $ is the input that gives 6 when put into f, which is 2. Then $ f^{-1}(2) $ is the input that gives 2 when put into f, which is 1. So the answer is 1.

I think that's thorough. I can't think of any other angles to check. Maybe if there was more to the function? But since the problem doesn't give more information, we can't assume anything else. So with the given data, 1 is the correct answer.

**Final Answer**
\boxed{1}

To solve the problem, we are given a function $ f $ and its inverse $ f^{-1} $, along with specific values:

- $ f(1) = 2 $
- $ f(2) = 6 $
- $ f(3) = 5 $

We are asked to compute $ f^{-1}(f^{-1}(6)) $.

---

### Step 1: Understand the Inverse Function

The inverse function $ f^{-1} $ satisfies the property:

$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
$$

This means that if $ f(a) = b $, then $ f^{-1}(b) = a $.

---

### Step 2: Compute $ f^{-1}(6) $

From the given values, we know:

$$
f(2) = 6
$$

So, by the definition of the inverse function:

$$
f^{-1}(6) = 2
$$

---

### Step 3: Compute $ f^{-1}(f^{-1}(6)) $

We now compute:

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2)
$$

From the given values, we know:

$$
f(1) = 2
$$

So:

$$
f^{-1}(2) = 1
$$

---

### Final Answer

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1
$$

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to find $ f^{-1}(f^{-1}(6)) $. Let me start by recalling what an inverse function does. The inverse function $ f^{-1} $ essentially undoes what $ f $ does. So if $ f(a) = b $, then $ f^{-1}(b) = a $.

Given the function values: $ f(1) = 2 $, $ f(2) = 6 $, and $ f(3) = 5 $. Let me write those down to keep track:

- $ f(1) = 2 $
- $ f(2) = 6 $
- $ f(3) = 5 $

So, if I want to find $ f^{-1}(6) $, I need to figure out what input to $ f $ gives me 6. Looking at the given values, $ f(2) = 6 $, so that means $ f^{-1}(6) = 2 $.

Wait, let me make sure. If $ f(2) = 6 $, then applying $ f^{-1} $ to both sides gives $ f^{-1}(f(2)) = f^{-1}(6) $, which simplifies to $ 2 = f^{-1}(6) $. Yep, that seems right. So $ f^{-1}(6) = 2 $.

Now, the problem asks for $ f^{-1}(f^{-1}(6)) $, which is $ f^{-1}(2) $. So now I need to find $ f^{-1}(2) $. Again, using the same logic, $ f^{-1}(2) $ is the value of $ x $ such that $ f(x) = 2 $.

Looking back at the given function values, $ f(1) = 2 $. Therefore, $ f^{-1}(2) = 1 $.

So putting it all together: $ f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1 $.

Wait, let me check again to make sure I didn't mix up any steps.

First step: $ f^{-1}(6) $. Since $ f(2) = 6 $, inverse function would take 6 back to 2. Then, taking the inverse function of that result, which is 2, so $ f^{-1}(2) $. Since $ f(1) = 2 $, inverse function takes 2 back to 1. So yes, the answer is 1.

But just to be thorough, let me make sure there's no other information I need to consider. The problem didn't specify the domain or codomain of the function, but since they gave three specific values, maybe there's more to the function? But since we're only dealing with the inverse function at specific points, and the inverse function is only defined for the outputs of the original function.

Given that $ f(1) = 2 $, $ f(2) = 6 $, $ f(3) = 5 $, so the outputs are 2, 6, 5. Therefore, the inverse function $ f^{-1} $ is defined for 2, 5, 6. So when we take $ f^{-1}(6) $, that's 2, and then $ f^{-1}(2) $ is 1.

Is there a possibility that the function isn't one-to-one? Wait, if the function isn't one-to-one, then the inverse wouldn't exist. But the problem says that $ f^{-1} $ is the inverse of $ f $, so that implies that $ f $ is bijective (both injective and surjective) on its domain. However, since we are only given three points, maybe the function is only defined on those three points? Or maybe it's defined on more points, but we only know these three.

But regardless, since we are only dealing with the values given, and the inverse function is defined for the outputs of $ f $, which are 2, 5, 6. So for the purposes of this problem, we can safely assume that $ f $ is invertible on the given domain, and that the inverse function is defined for those outputs.

Therefore, my conclusion is that $ f^{-1}(f^{-1}(6)) = 1 $.

But let me try to write it step by step again:

1. Find $ f^{-1}(6) $:
   - Since $ f(2) = 6 $, then $ f^{-1}(6) = 2 $.

2. Now find $ f^{-1}(f^{-1}(6)) = f^{-1}(2) $:
   - Since $ f(1) = 2 $, then $ f^{-1}(2) = 1 $.

Therefore, the final answer is 1.

I think that's solid. Let me check if there's another way to approach this. Maybe by constructing the inverse function table?

Original function:

Input | Output
--- | ---
1 | 2
2 | 6
3 | 5

Inverse function would swap inputs and outputs:

Input | Output
--- | ---
2 | 1
6 | 2
5 | 3

So, $ f^{-1}(2) = 1 $, $ f^{-1}(6) = 2 $, $ f^{-1}(5) = 3 $.

Therefore, $ f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1 $. Yep, same result.

So I think that's correct. I don't see any mistakes in my reasoning. The key was to remember that the inverse function swaps the input and output of the original function. So applying the inverse twice here is like going back two steps in the original function.

Another way to think about it: $ f^{-1}(6) $ is the input that gives 6 when put into f, which is 2. Then $ f^{-1}(2) $ is the input that gives 2 when put into f, which is 1. So the answer is 1.

I think that's thorough. I can't think of any other angles to check. Maybe if there was more to the function? But since the problem doesn't give more information, we can't assume anything else. So with the given data, 1 is the correct answer.

**Final Answer**
\boxed{1}

To solve the problem, we are given a function $ f $ and its inverse $ f^{-1} $, along with specific values:

- $ f(1) = 2 $
- $ f(2) = 6 $
- $ f(3) = 5 $

We are asked to compute $ f^{-1}(f^{-1}(6)) $.

---

### Step 1: Understand the Inverse Function

The inverse function $ f^{-1} $ satisfies the property:

$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
$$

This means that if $ f(a) = b $, then $ f^{-1}(b) = a $.

---

### Step 2: Compute $ f^{-1}(6) $

From the given values, we know:

$$
f(2) = 6
$$

So, by the definition of the inverse function:

$$
f^{-1}(6) = 2
$$

---

### Step 3: Compute $ f^{-1}(f^{-1}(6)) $

We now compute:

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2)
$$

From the given values, we know:

$$
f(1) = 2
$$

So:

$$
f^{-1}(2) = 1
$$

---

### Final Answer

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1
$$

$$
\boxed{1}
$$

Solution: Since $f(2) = 6$, we have $f^{-1}(6)=2$. (Note that the hypothesis that $f$ has an inverse implies that there are no other values of $x$ with $f(x) = 6$.)  Similarly, $f(1) =2$ implies $f^{-1}(2)=1$.   So $f^{-1}(f^{-1}(6))=f^{-1}(2)=\boxed{1}$.


Explanation:
Okay, so I need to find the coordinates of point P on the line y = -x + 6 such that P is equidistant from points A(10, -10) and O(0, 0). That means PA = PO. Let me think about how to approach this.

First, since P lies on the line y = -x + 6, I can represent P as (x, -x + 6). That way, any point on the line can be expressed in terms of x. Then, I need to set up the distance formulas for PA and PO and set them equal to each other.

Let me recall the distance formula between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. So, PA would be the distance between P(x, -x + 6) and A(10, -10), and PO would be the distance between P(x, -x + 6) and O(0, 0).

Let me write those out:

PA^2 = (x - 10)^2 + (-x + 6 - (-10))^2
PO^2 = (x - 0)^2 + (-x + 6 - 0)^2

Since PA = PO, their squares should also be equal. So, I can set PA^2 = PO^2 and solve for x. That might be easier than dealing with square roots.

Let me compute PA^2 first:

PA^2 = (x - 10)^2 + (-x + 6 + 10)^2
Simplify the second term: -x + 6 + 10 = -x + 16
So PA^2 = (x - 10)^2 + (-x + 16)^2

Similarly, PO^2 = x^2 + (-x + 6)^2

So, set them equal:

(x - 10)^2 + (-x + 16)^2 = x^2 + (-x + 6)^2

Now, let me expand each term step by step.

First, expand (x - 10)^2:

(x - 10)^2 = x^2 - 20x + 100

Then, (-x + 16)^2. Let me note that (-x + 16) is the same as (16 - x), so squaring that:

(16 - x)^2 = x^2 - 32x + 256

So PA^2 becomes:

x^2 - 20x + 100 + x^2 - 32x + 256

Combine like terms:

2x^2 - 52x + 356

Now, let me compute PO^2:

x^2 + (-x + 6)^2

First, (-x + 6)^2 is same as (x - 6)^2:

(x - 6)^2 = x^2 - 12x + 36

Therefore, PO^2 is:

x^2 + x^2 - 12x + 36 = 2x^2 - 12x + 36

So now, set PA^2 = PO^2:

2x^2 - 52x + 356 = 2x^2 - 12x + 36

Hmm, let me subtract 2x^2 from both sides:

-52x + 356 = -12x + 36

Now, let me bring all terms to one side. Let me add 52x to both sides:

356 = 40x + 36

Then subtract 36 from both sides:

356 - 36 = 40x

Which is 320 = 40x

Divide both sides by 40:

x = 320 / 40 = 8

So x = 8. Then, since P is on the line y = -x + 6, substitute x = 8:

y = -8 + 6 = -2

Therefore, the coordinates of P are (8, -2). Let me check if this is correct.

First, check if PA = PO.

Compute PA: distance between (8, -2) and A(10, -10):

Difference in x: 10 - 8 = 2

Difference in y: -10 - (-2) = -8

PA^2 = 2^2 + (-8)^2 = 4 + 64 = 68

PO: distance between (8, -2) and O(0, 0):

Difference in x: 8 - 0 = 8

Difference in y: -2 - 0 = -2

PO^2 = 8^2 + (-2)^2 = 64 + 4 = 68

Yes, PA^2 = PO^2 = 68, so PA = PO. That checks out.

Wait, but let me think again. Is there another way to approach this problem? Maybe using the concept of perpendicular bisectors or something?

Alternatively, since P is equidistant from A and O, it lies on the perpendicular bisector of segment AO. Maybe I can find the perpendicular bisector of AO and find its intersection with the line y = -x + 6. That might be another method.

Let me try that approach to verify.

First, find the midpoint of AO. A is (10, -10), O is (0, 0). Midpoint M has coordinates ((10 + 0)/2, (-10 + 0)/2) = (5, -5).

Then, the slope of AO is ( -10 - 0 ) / (10 - 0 ) = (-10)/10 = -1. Therefore, the perpendicular bisector will have a slope that is the negative reciprocal of -1, which is 1.

So the equation of the perpendicular bisector is y - (-5) = 1*(x - 5), which simplifies to y + 5 = x - 5, so y = x - 10.

Now, the point P lies on both the perpendicular bisector of AO and the line y = -x + 6. Therefore, we can solve the system:

y = x - 10

y = -x + 6

Set them equal:

x - 10 = -x + 6

Add x to both sides:

2x - 10 = 6

Add 10 to both sides:

2x = 16

x = 8

Then y = -8 + 6 = -2. So P is (8, -2). Same result as before. So that confirms the answer is correct.

Therefore, the coordinates of P are (8, -2). So I think that's the answer.

But let me just check once more. Maybe there's a mistake in my calculations?

First method: solving equations, got x=8, y=-2. Second method: perpendicular bisector, same result. Also checked distances, they are equal. So seems correct.

Alternatively, maybe I can visualize. The line y = -x + 6 is a straight line with slope -1, crossing y-axis at 6 and x-axis at 6. The point (8, -2) is on this line. Let me see, when x=8, y=-8+6=-2. Correct.

Midpoint of AO is (5, -5). The perpendicular bisector is y = x - 10. Let me check if (5, -5) lies on that line: y = 5 - 10 = -5. Yes, correct. So the perpendicular bisector is correct.

Intersection of y = x - 10 and y = -x + 6 is indeed at x=8, y=-2. All steps check out.

Therefore, I think the answer is (8, -2). So I can write that as the final answer.

**Final Answer**
The coordinates of point $ P $ are \boxed{(8, -2)}.

To find the coordinates of point $ P $ on the line $ y = -x + 6 $ such that $ P $ is equidistant from points $ A(10, -10) $ and $ O(0, 0) $, we proceed as follows:

---

### Step 1: Represent Point $ P $ on the Line

Since $ P $ lies on the line $ y = -x + 6 $, we can express the coordinates of $ P $ as:
$$
P = (x, -x + 6)
$$

---

### Step 2: Use the Distance Formula

We are given that $ PA = PO $, so we can equate the squared distances to avoid dealing with square roots:

- Distance from $ P $ to $ A(10, -10) $:
$$
PA^2 = (x - 10)^2 + (-x + 6 + 10)^2 = (x - 10)^2 + (-x + 16)^2
$$

- Distance from $ P $ to $ O(0, 0) $:
$$
PO^2 = x^2 + (-x + 6)^2
$$

Set the two expressions equal:
$$
(x - 10)^2 + (-x + 16)^2 = x^2 + (-x + 6)^2
$$

---

### Step 3: Expand and Simplify

**Left-hand side:**
$$
(x - 10)^2 = x^2 - 20x + 100 \\
(-x + 16)^2 = x^2 - 32x + 256 \\
\Rightarrow \text{Total} = 2x^2 - 52x + 356
$$

**Right-hand side:**
$$
x^2 + (-x + 6)^2 = x^2 + (x^2 - 12x + 36) = 2x^2 - 12x + 36
$$

Set both sides equal:
$$
2x^2 - 52x + 356 = 2x^2 - 12x + 36
$$

Subtract $ 2x^2 $ from both sides:
$$
-52x + 356 = -12x + 36
$$

Solve for $ x $:
$$
-52x + 356 = -12x + 36 \\
-40x = -320 \\
x = 8
$$

---

### Step 4: Find the Corresponding $ y $-Coordinate

Substitute $ x = 8 $ into the line equation $ y = -x + 6 $:
$$
y = -8 + 6 = -2
$$

So, the coordinates of point $ P $ are:
$$
P = (8, -2)
$$

---

### Step 5: Verification

- Distance from $ P(8, -2) $ to $ A(10, -10) $:
$$
PA^2 = (8 - 10)^2 + (-2 + 10)^2 = (-2)^2 + (8)^2 = 4 + 64 = 68
$$

- Distance from $ P(8, -2) $ to $ O(0, 0) $:
$$
PO^2 = (8 - 0)^2 + (-2 - 0)^2 = 64 + 4 = 68
$$

Since $ PA^2 = PO^2 $, the distances are equal, confirming the solution.

---

### Final Answer

$$
\boxed{(8, -2)}
$$

[Assistant]
Explanation:
Okay, so I need to find the coordinates of point P on the line y = -x + 6 such that P is equidistant from points A(10, -10) and O(0, 0). That means PA = PO. Let me think about how to approach this.

First, since P lies on the line y = -x + 6, I can represent P as (x, -x + 6). That way, any point on the line can be expressed in terms of x. Then, I need to set up the distance formulas for PA and PO and set them equal to each other.

Let me recall the distance formula between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. So, PA would be the distance between P(x, -x + 6) and A(10, -10), and PO would be the distance between P(x, -x + 6) and O(0, 0).

Let me write those out:

PA^2 = (x - 10)^2 + (-x + 6 - (-10))^2
PO^2 = (x - 0)^2 + (-x + 6 - 0)^2

Since PA = PO, their squares should also be equal. So, I can set PA^2 = PO^2 and solve for x. That might be easier than dealing with square roots.

Let me compute PA^2 first:

PA^2 = (x - 10)^2 + (-x + 6 + 10)^2
Simplify the second term: -x + 6 + 10 = -x + 16
So PA^2 = (x - 10)^2 + (-x + 16)^2

Similarly, PO^2 = x^2 + (-x + 6)^2

So, set them equal:

(x - 10)^2 + (-x + 16)^2 = x^2 + (-x + 6)^2

Now, let me expand each term step by step.

First, expand (x - 10)^2:

(x - 10)^2 = x^2 - 20x + 100

Then, (-x + 16)^2. Let me note that (-x + 16) is the same as (16 - x), so squaring that:

(16 - x)^2 = x^2 - 32x + 256

So PA^2 becomes:

x^2 - 20x + 100 + x^2 - 32x + 256

Combine like terms:

2x^2 - 52x + 356

Now, let me compute PO^2:

x^2 + (-x + 6)^2

First, (-x + 6)^2 is same as (x - 6)^2:

(x - 6)^2 = x^2 - 12x + 36

Therefore, PO^2 is:

x^2 + x^2 - 12x + 36 = 2x^2 - 12x + 36

So now, set PA^2 = PO^2:

2x^2 - 52x + 356 = 2x^2 - 12x + 36

Hmm, let me subtract 2x^2 from both sides:

-52x + 356 = -12x + 36

Now, let me bring all terms to one side. Let me add 52x to both sides:

356 = 40x + 36

Then subtract 36 from both sides:

356 - 36 = 40x

Which is 320 = 40x

Divide both sides by 40:

x = 320 / 40 = 8

So x = 8. Then, since P is on the line y = -x + 6, substitute x = 8:

y = -8 + 6 = -2

Therefore, the coordinates of P are (8, -2). Let me check if this is correct.

First, check if PA = PO.

Compute PA: distance between (8, -2) and A(10, -10):

Difference in x: 10 - 8 = 2

Difference in y: -10 - (-2) = -8

PA^2 = 2^2 + (-8)^2 = 4 + 64 = 68

PO: distance between (8, -2) and O(0, 0):

Difference in x: 8 - 0 = 8

Difference in y: -2 - 0 = -2

PO^2 = 8^2 + (-2)^2 = 64 + 4 = 68

Yes, PA^2 = PO^2 = 68, so PA = PO. That checks out.

Wait, but let me think again. Is there another way to approach this problem? Maybe using the concept of perpendicular bisectors or something?

Alternatively, since P is equidistant from A and O, it lies on the perpendicular bisector of segment AO. Maybe I can find the perpendicular bisector of AO and find its intersection with the line y = -x + 6. That might be another method.

Let me try that approach to verify.

First, find the midpoint of AO. A is (10, -10), O is (0, 0). Midpoint M has coordinates ((10 + 0)/2, (-10 + 0)/2) = (5, -5).

Then, the slope of AO is ( -10 - 0 ) / (10 - 0 ) = (-10)/10 = -1. Therefore, the perpendicular bisector will have a slope that is the negative reciprocal of -1, which is 1.

So the equation of the perpendicular bisector is y - (-5) = 1*(x - 5), which simplifies to y + 5 = x - 5, so y = x - 10.

Now, the point P lies on both the perpendicular bisector of AO and the line y = -x + 6. Therefore, we can solve the system:

y = x - 10

y = -x + 6

Set them equal:

x - 10 = -x + 6

Add x to both sides:

2x - 10 = 6

Add 10 to both sides:

2x = 16

x = 8

Then y = -8 + 6 = -2. So P is (8, -2). Same result as before. So that confirms the answer is correct.

Therefore, the coordinates of P are (8, -2). So I think that's the answer.

But let me just check once more. Maybe there's a mistake in my calculations?

First method: solving equations, got x=8, y=-2. Second method: perpendicular bisector, same result. Also checked distances, they are equal. So seems correct.

Alternatively, maybe I can visualize. The line y = -x + 6 is a straight line with slope -1, crossing y-axis at 6 and x-axis at 6. The point (8, -2) is on this line. Let me see, when x=8, y=-8+6=-2. Correct.

Midpoint of AO is (5, -5). The perpendicular bisector is y = x - 10. Let me check if (5, -5) lies on that line: y = 5 - 10 = -5. Yes, correct. So the perpendicular bisector is correct.

Intersection of y = x - 10 and y = -x + 6 is indeed at x=8, y=-2. All steps check out.

Therefore, I think the answer is (8, -2). So I can write that as the final answer.

**Final Answer**
The coordinates of point $ P $ are \boxed{(8, -2)}.

To find the coordinates of point $ P $ on the line $ y = -x + 6 $ such that $ P $ is equidistant from points $ A(10, -10) $ and $ O(0, 0) $, we proceed as follows:

---

### Step 1: Represent Point $ P $ on the Line

Since $ P $ lies on the line $ y = -x + 6 $, we can express the coordinates of $ P $ as:
$$
P = (x, -x + 6)
$$

---

### Step 2: Use the Distance Formula

We are given that $ PA = PO $, so we can equate the squared distances to avoid dealing with square roots:

- Distance from $ P $ to $ A(10, -10) $:
$$
PA^2 = (x - 10)^2 + (-x + 6 + 10)^2 = (x - 10)^2 + (-x + 16)^2
$$

- Distance from $ P $ to $ O(0, 0) $:
$$
PO^2 = x^2 + (-x + 6)^2
$$

Set the two expressions equal:
$$
(x - 10)^2 + (-x + 16)^2 = x^2 + (-x + 6)^2
$$

---

### Step 3: Expand and Simplify

**Left-hand side:**
$$
(x - 10)^2 = x^2 - 20x + 100 \\
(-x + 16)^2 = x^2 - 32x + 256 \\
\Rightarrow \text{Total} = 2x^2 - 52x + 356
$$

**Right-hand side:**
$$
x^2 + (-x + 6)^2 = x^2 + (x^2 - 12x + 36) = 2x^2 - 12x + 36
$$

Set both sides equal:
$$
2x^2 - 52x + 356 = 2x^2 - 12x + 36
$$

Subtract $ 2x^2 $ from both sides:
$$
-52x + 356 = -12x + 36
$$

Solve for $ x $:
$$
-52x + 356 = -12x + 36 \\
-40x = -320 \\
x = 8
$$

---

### Step 4: Find the Corresponding $ y $-Coordinate

Substitute $ x = 8 $ into the line equation $ y = -x + 6 $:
$$
y = -8 + 6 = -2
$$

So, the coordinates of point $ P $ are:
$$
P = (8, -2)
$$

---

### Step 5: Verification

- Distance from $ P(8, -2) $ to $ A(10, -10) $:
$$
PA^2 = (8 - 10)^2 + (-2 + 10)^2 = (-2)^2 + (8)^2 = 4 + 64 = 68
$$

- Distance from $ P(8, -2) $ to $ O(0, 0) $:
$$
PO^2 = (8 - 0)^2 + (-2 - 0)^2 = 64 + 4 = 68
$$

Since $ PA^2 = PO^2 $, the distances are equal, confirming the solution.

---

### Final Answer

$$
\boxed{(8, -2)}
$$

Solution:
If $P$ is equidistant from $A$ and $O$, it must lie on the perpendicular bisector of $AO$. Since $A$ has coordinates $(10,-10)$ and $O$ has coordinates $(0,0)$, $AO$ has slope $\frac{-10-0}{10-0}=-1$.  The perpendicular bisector of $AO$ must have slope $-\frac{1}{-1}=1$, and must also pass through the midpoint of $AO$, which is $(5,-5)$. Therefore, the perpendicular bisector has equation $y-(-5)=x-5$ or $y=x-10$.

$P$ is the point of intersection of the lines $y=x-10$ and the line $y=-x+6$.  Setting these equations equal and solving for $x$ yields $-x+6=x-10 \Rightarrow x=8$.  It follows that $y=-8+6=-2$ and $P=(x,y)=\boxed{(8,-2)}$.


Explanation:
Okay, so I need to figure out for which grade the number of male bus riders is closest to 135% of the number of female bus riders. The table gives percentages for males and females in each grade, and there are 300 males and 300 females surveyed in each grade. Let me break this down step by step.

First, let me understand the question. For each grade, I need to calculate the actual number of male bus riders and the actual number of female bus riders. Then, I need to check what 135% of the female riders would be for that grade, and see which grade has the male riders closest to that value.

Let me start by recalling that percentages can be converted to actual numbers by multiplying the percentage by the total number of people surveyed. Since there are 300 males and 300 females in each grade, the total number of students surveyed per grade is 600, but maybe I don't need that. Wait, actually, the question is about male and female riders separately. So for each grade, the number of male bus riders is 300 times the male percentage, and similarly for females.

So let me write that down for each grade:

For each grade:
- Number of male bus riders = 300 * (male percentage)
- Number of female bus riders = 300 * (female percentage)

Then, 135% of the number of female bus riders would be 1.35 * (number of female bus riders).

So, for each grade, I need to compute 1.35 * (female riders) and compare it to the number of male riders. Then, find which grade has the male riders closest to that value.

Alternatively, maybe I can compute the difference between the male riders and 1.35 times female riders, and see which difference is the smallest.

Let me try this approach for each grade.

Starting with 9th grade:

9th grade:
- Male percentage: 41.1%
- Female percentage: 39.4%

Calculating number of male riders: 300 * 41.1% = 300 * 0.411 = let me compute that. 300 * 0.4 is 120, 300 * 0.011 is 3.3, so total is 120 + 3.3 = 123.3. Wait, but percentages are per 100, so 41.1% of 300 is 0.411 * 300. Let me check that again. 0.411 * 300. 0.4 * 300 = 120, 0.01 * 300 = 3, 0.001 * 300 = 0.3. So 0.411 * 300 = 120 + 3 + 0.3 = 123.3. So 123.3 male riders.

Similarly, female riders: 300 * 39.4% = 300 * 0.394. Let me compute that. 0.3 * 300 = 90, 0.09 * 300 = 27, 0.004 * 300 = 1.2. So adding up: 90 + 27 = 117 + 1.2 = 118.2. So 118.2 female riders.

Now, 135% of female riders is 1.35 * 118.2. Let me calculate that. 1 * 118.2 = 118.2, 0.35 * 118.2. Let me compute 0.3 * 118.2 = 35.46, 0.05 * 118.2 = 5.91. So total 35.46 + 5.91 = 41.37. Therefore, 1.35 * 118.2 = 118.2 + 41.37 = 159.57.

So, for 9th grade, male riders are 123.3, and 135% of female riders is 159.57. The difference between male riders and 135% female riders is 123.3 - 159.57 = -36.27. But since we are looking for the closest, maybe we take absolute value? Or maybe compare the absolute difference. So the absolute difference is 36.27.

Now moving on to 10th grade:

10th grade:
- Male percentage: 34.4%
- Female percentage: 33.1%

Male riders: 300 * 0.344. Let me compute that. 0.3 * 300 = 90, 0.04 * 300 = 12, 0.004 * 300 = 1.2. So total is 90 + 12 + 1.2 = 103.2.

Female riders: 300 * 0.331. Let me compute that. 0.3 * 300 = 90, 0.03 * 300 = 9, 0.001 * 300 = 0.3. So total is 90 + 9 + 0.3 = 99.3.

Then, 135% of female riders is 1.35 * 99.3. Let me calculate that. 1 * 99.3 = 99.3, 0.35 * 99.3. Let me compute 0.3 * 99.3 = 29.79, 0.05 * 99.3 = 4.965. Adding those gives 29.79 + 4.965 = 34.755. Therefore, total is 99.3 + 34.755 = 134.055.

So, male riders are 103.2, and 135% of female riders is 134.055. The difference is 103.2 - 134.055 = -30.855. Absolute difference is 30.855.

Comparing this to the previous grade's difference of 36.27, this is smaller. So 10th grade is closer so far.

Now moving on to 11th grade:

11th grade:
- Male percentage: 20.6%
- Female percentage: 13.8%

Male riders: 300 * 0.206. Let me compute that. 0.2 * 300 = 60, 0.006 * 300 = 1.8. So total is 60 + 1.8 = 61.8.

Female riders: 300 * 0.138. Let me compute that. 0.1 * 300 = 30, 0.03 * 300 = 9, 0.008 * 300 = 2.4. So total is 30 + 9 + 2.4 = 41.4.

Then, 135% of female riders is 1.35 * 41.4. Let me calculate that. 1 * 41.4 = 41.4, 0.35 * 41.4. Let me compute 0.3 * 41.4 = 12.42, 0.05 * 41.4 = 2.07. Adding those gives 12.42 + 2.07 = 14.49. Therefore, total is 41.4 + 14.49 = 55.89.

So, male riders are 61.8, and 135% of female riders is 55.89. The difference is 61.8 - 55.89 = 5.91. Absolute difference is 5.91.

That's much smaller than the previous differences. So 11th grade is closer now.

Now moving on to 12th grade:

12th grade:
- Male percentage: 11.6%
- Female percentage: 8.6%

Male riders: 300 * 0.116. Let me compute that. 0.1 * 300 = 30, 0.01 * 300 = 3, 0.006 * 300 = 1.8. So total is 30 + 3 + 1.8 = 34.8.

Female riders: 300 * 0.086. Let me compute that. 0.08 * 300 = 24, 0.006 * 300 = 1.8. So total is 24 + 1.8 = 25.8.

Then, 135% of female riders is 1.35 * 25.8. Let me calculate that. 1 * 25.8 = 25.8, 0.35 * 25.8. Let me compute 0.3 * 25.8 = 7.74, 0.05 * 25.8 = 1.29. Adding those gives 7.74 + 1.29 = 9.03. Therefore, total is 25.8 + 9.03 = 34.83.

So, male riders are 34.8, and 135% of female riders is 34.83. The difference is 34.8 - 34.83 = -0.03. Absolute difference is 0.03.

Wow, that's extremely close. So the difference here is only 0.03, which is much smaller than the previous differences. So 12th grade is the closest so far.

Wait, so comparing all the absolute differences:

- 9th grade: 36.27
- 10th grade: 30.855
- 11th grade: 5.91
- 12th grade: 0.03

So the smallest difference is for 12th grade. Therefore, the answer should be 12th grade?

But let me check my calculations again because the difference for 12th grade is so small. Let me verify the numbers again.

Starting with 12th grade:

Male percentage: 11.6%, so male riders: 300 * 0.116. Let me compute 0.116 * 300. 0.1 * 300 = 30, 0.01 * 300 = 3, 0.006 * 300 = 1.8. So 30 + 3 + 1.8 = 34.8. Correct.

Female percentage: 8.6%, so female riders: 300 * 0.086. Let me compute 0.086 * 300. 0.08 * 300 = 24, 0.006 * 300 = 1.8. So 24 + 1.8 = 25.8. Correct.

Then 135% of female riders: 1.35 * 25.8. Let me compute that again. 25.8 * 1.35. Let me do 25.8 * 1.35.

Alternatively, 25.8 * 1.35 = 25.8 * (1 + 0.3 + 0.05) = 25.8 + 7.74 + 1.29 = 25.8 + 7.74 = 33.54 + 1.29 = 34.83. Correct.

So male riders are 34.8, and 135% of female riders is 34.83. The difference is 34.8 - 34.83 = -0.03. So absolute difference is 0.03. That's very close.

But let me check if I did the 11th grade correctly.

For 11th grade:

Male riders: 300 * 20.6% = 300 * 0.206. Let me compute 0.2 * 300 = 60, 0.006 * 300 = 1.8. So 60 + 1.8 = 61.8. Correct.

Female riders: 300 * 13.8% = 300 * 0.138. Let me compute 0.1 * 300 = 30, 0.03 * 300 = 9, 0.008 * 300 = 2.4. So 30 + 9 + 2.4 = 41.4. Correct.

Then 135% of female riders: 1.35 * 41.4. Let me compute that again. 41.4 * 1.35. Let me do 41.4 * 1.35.

Alternatively, 41.4 * 1.35 = 41.4 * (1 + 0.3 + 0.05) = 41.4 + 12.42 + 2.07 = 41.4 + 12.42 = 53.82 + 2.07 = 55.89. Correct.

So male riders are 61.8, and 135% of female riders is 55.89. The difference is 61.8 - 55.89 = 5.91. Correct.

So 12th grade has a difference of 0.03, which is way smaller. So that seems like the closest.

But let me check if there's any mistake in interpreting the question. The question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, for each grade, compute male riders vs 1.35 * female riders. The difference between them. The one with the smallest difference is the answer.

So for 12th grade, male riders are 34.8, and 1.35 * female riders is 34.83. So the difference is 0.03. That's very close.

But let me check if there's a possible error in calculation. For example, maybe I should have done something else? Let me check the 12th grade again.

Wait, 135% of female riders is 1.35 * female riders. So if female riders are 25.8, then 1.35 * 25.8 is indeed 34.83. Male riders are 34.8. So the difference is 34.8 vs 34.83. So the difference is 0.03.

But since we are dealing with actual numbers of people, which should be whole numbers. Wait, but the percentages are given with one decimal place, so when we multiply by 300, we get numbers with one decimal place. For example, 34.8 is 34.8, which is 34.8 people? Wait, but you can't have 0.8 of a person. However, since the percentages are given as exact numbers, maybe we are supposed to treat them as exact, even though in reality you can't have fractions of people. But since the problem is presented with percentages, and the answer is likely expecting us to use the exact decimal values, even if they are not whole numbers.

Therefore, proceeding with the decimal values is correct here.

So, given that, the difference for 12th grade is 0.03, which is the smallest. Therefore, the answer is 12th grade.

But let me check again for 11th grade. The difference was 5.91, which is bigger than 0.03. For 10th grade, 30.855, which is bigger. For 9th grade, 36.27. So 12th grade is the closest.

But wait, the problem says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, maybe I need to check if 135% of female riders is actually closer to male riders.

Alternatively, maybe I should compute the absolute difference between male riders and 1.35 * female riders. Which is exactly what I did.

So, for 12th grade, the difference is 0.03, which is very small.

But let me check if there's a possible miscalculation here. Let me recompute 1.35 * 25.8.

25.8 * 1.35. Let me do this multiplication step by step.

First, 25.8 * 1 = 25.8

25.8 * 0.3 = 7.74

25.8 * 0.05 = 1.29

Adding them up: 25.8 + 7.74 = 33.54 + 1.29 = 34.83. Correct.

Male riders: 34.8. So difference is 34.8 - 34.83 = -0.03. Absolute value is 0.03.

So that's correct.

But let me check if the question is asking for the closest in terms of percentage difference or absolute difference. The question says "closest to 135% of the number of its female bus riders". So, it's about the actual number of male riders being closest to 1.35 times the number of female riders. So, the absolute difference between male riders and 1.35 * female riders.

Therefore, 12th grade is the closest.

But let me check if there's a possible mistake in the problem interpretation. For example, maybe the question is asking for the male riders to be closest to 135% of the female riders, meaning that 135% of female riders is the target, and male riders are compared to that.

Alternatively, maybe the question is phrased as "the number of its male bus riders closest to 135% of the number of its female bus riders", so the male riders are being compared to 135% of female riders. So, for each grade, compute male riders and 1.35 * female riders, and see which is closest.

Yes, that's exactly what I did.

Alternatively, maybe the question is asking for the male percentage to be closest to 135% of the female percentage? But that would be different. But the question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So it's about the actual numbers, not percentages.

Therefore, my conclusion is that 12th grade is the answer.

But let me check once more for 12th grade.

Male riders: 34.8

135% of female riders: 34.83

So, 34.8 is very close to 34.83. The difference is 0.03.

But let me check if there's another grade with a smaller difference. For example, maybe I made a mistake with 11th grade?

For 11th grade, male riders: 61.8, 1.35 * female riders: 55.89. The difference is 5.91.

So, 5.91 is bigger than 0.03.

So, 12th grade is the closest.

But let me check if the problem says "closest to 135% of the number of its female bus riders". So, if the male riders are 34.8 and 135% of female is 34.83, then the male riders are just slightly less than 135% of female riders.

But since the question is about which grade is closest, and 0.03 is the smallest difference, then 12th grade is the answer.

But let me check if there's a possible error in calculation for 12th grade.

Wait, female riders: 8.6% of 300. Let me compute 8.6% of 300.

8.6% is 0.086. 0.086 * 300.

Well, 10% of 300 is 30, so 8% is 24, 0.6% is 1.8, so total is 24 + 1.8 = 25.8. Correct.

Then 1.35 * 25.8.

Alternatively, 25.8 * 1.35. Let me compute 25.8 * 1.35.

Let me convert 25.8 to 258/10, and 1.35 to 135/100.

So 258/10 * 135/100 = (258 * 135) / 1000.

But maybe that's too complicated. Alternatively, 25.8 * 1.35.

Alternatively, think of 25.8 * 1.35 = 25.8 * (1 + 0.3 + 0.05) = 25.8 + 7.74 + 1.29 = 34.83. Correct.

So, yes, that's correct.

Therefore, the answer is 12th grade.

But wait, the problem says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, if the male riders are 34.8 and 135% of female is 34.83, then the male riders are just 0.03 less than 135% of female riders.

But maybe the question is expecting us to round to the nearest whole number? Because you can't have 0.8 of a person.

Wait, but the problem says "the number of its male bus riders", which would be a whole number. However, the percentages are given with one decimal place, so when we calculate the number of riders, we get decimal numbers. But maybe we need to round them?

Wait, the problem says "the data in the table below shows the percent of bus riders in a survey of Central H.S. students; 300 males and 300 females in each grade were surveyed." So, the percentages are given, and the actual number of riders would be calculated by multiplying the percentage by 300. However, since percentages can result in decimal numbers, but in reality, the number of riders must be whole numbers. However, since the problem is presented with percentages, and the answer is likely expecting us to use the exact decimal values, even though they are not whole numbers.

But maybe the problem expects us to round the number of riders to the nearest whole number? Let me check.

For example, in 12th grade, male riders would be 34.8, which is 35 when rounded. Female riders are 25.8, which is 26 when rounded. Then, 135% of 26 is 1.35 * 26 = 35.1. So, male riders rounded is 35, which is very close to 35.1. The difference is 0.1.

But if we don't round, the difference is 0.03.

But the problem says "the number of its male bus riders", which is an actual count. So, if we have to consider that the number of riders must be whole numbers, then we need to calculate the exact number of riders, which would be 34.8 for males and 25.8 for females. But since you can't have 0.8 of a person, maybe the actual numbers are rounded.

But the problem doesn't specify whether to round or not. However, since the percentages are given with one decimal place, and the number of people is 300, which is a multiple of 10, maybe the actual number of riders is a whole number. Let me check for 12th grade.

Wait, 11.6% of 300. Let me compute 11.6% of 300.

11.6% is 11.6/100 = 0.116. 0.116 * 300 = 34.8.

But 34.8 is not a whole number. However, maybe the survey results are rounded, so the actual number could be 35 or 34. Similarly for females: 8.6% of 300 is 25.8, which could be 26 or 25.

But since the problem gives the percentages as exact numbers, maybe we are supposed to use the exact decimal values.

Alternatively, maybe the problem expects us to use the exact numbers, even if they are not whole numbers, because the percentages are given with decimals.

In that case, the difference of 0.03 is acceptable.

Alternatively, if we consider that the number of riders must be whole numbers, then we need to check if the percentages are rounded. For example, if the actual percentage for males is 11.6%, which could be rounded from something like 11.55% or 11.65%, leading to 34.65 or 34.95, which would round to 35. Similarly for females.

But since the problem doesn't specify, I think we should proceed with the exact decimal values as calculated.

Therefore, the answer is 12th grade.

But let me check once more for 12th grade.

Male riders: 34.8

135% of female riders: 34.83

Difference: 0.03.

So, that's the closest.

But let me check if there's a possible mistake in the question. For example, maybe the question is asking for the male percentage to be closest to 135% of the female percentage? Let me check that.

For example, for 12th grade, male percentage is 11.6%, female percentage is 8.6%. 135% of female percentage is 1.35 * 8.6% = 11.59%. Which is very close to male percentage of 11.6%.

Ah! Wait, that's another way to interpret the question. Maybe the question is asking for the male percentage to be closest to 135% of the female percentage.

But the original question says: "the number of its male bus riders closest to 135% of the number of its female bus riders".

So, "number of male bus riders" vs "number of female bus riders". So, it's about the actual counts, not percentages.

But if we consider the percentages, then for 12th grade, male percentage is 11.6%, and 135% of female percentage is 1.35 * 8.6% = 11.59%, which is extremely close.

But the question is about numbers, not percentages. However, maybe the problem is ambiguously worded?

But given the exact wording, it's about numbers.

But let me check both interpretations.

First interpretation: numbers. As above, 12th grade is the answer.

Second interpretation: percentages. For each grade, compute male percentage vs 1.35 * female percentage.

Let me check that.

For 9th grade: male percentage 41.1%, 1.35 * female percentage = 1.35 * 39.4% = let me compute that. 1.35 * 39.4.

Let me compute 1 * 39.4 = 39.4, 0.35 * 39.4.

0.3 * 39.4 = 11.82, 0.05 * 39.4 = 1.97. Total is 11.82 + 1.97 = 13.79. So total is 39.4 + 13.79 = 53.19%.

So male percentage is 41.1% vs 53.19%. Difference is 12.09%.

For 10th grade: male percentage 34.4%, 1.35 * 33.1% = 1.35 * 33.1.

Compute 1 * 33.1 = 33.1, 0.35 * 33.1.

0.3 * 33.1 = 9.93, 0.05 * 33.1 = 1.655. Total is 9.93 + 1.655 = 11.585. Total is 33.1 + 11.585 = 44.685%.

Difference between male percentage (34.4%) and 44.685% is 10.285%.

For 11th grade: male percentage 20.6%, 1.35 * 13.8% = 1.35 * 13.8.

Compute 1 * 13.8 = 13.8, 0.35 * 13.8.

0.3 * 13.8 = 4.14, 0.05 * 13.8 = 0.69. Total is 4.14 + 0.69 = 4.83. Total is 13.8 + 4.83 = 18.63%.

Difference between male percentage (20.6%) and 18.63% is 1.97%.

For 12th grade: male percentage 11.6%, 1.35 * 8.6% = 1.35 * 8.6.

Compute 1 * 8.6 = 8.6, 0.35 * 8.6.

0.3 * 8.6 = 2.58, 0.05 * 8.6 = 0.43. Total is 2.58 + 0.43 = 3.01. Total is 8.6 + 3.01 = 11.61%.

Difference between male percentage (11.6%) and 11.61% is 0.01%.

So, if the question was about percentages, then 12th grade is also the closest, with a difference of 0.01%.

But the original question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, it's about numbers, not percentages. However, in both interpretations, 12th grade is the answer.

But let me check if there's a different answer if we consider the numbers.

But according to the first interpretation, the answer is 12th grade.

But let me check if there's a possible mistake in the problem. For example, maybe the question is asking for the male percentage to be closest to 135% of the female percentage, which would be 12th grade.

But since the question says "number of its male bus riders", I think it's about the actual numbers.

But given that both interpretations lead to 12th grade, maybe that's the answer.

But let me check once more for 12th grade with numbers.

Male riders: 34.8

135% of female riders: 34.83

Difference: 0.03.

But if we consider that the number of riders must be whole numbers, then maybe we need to round.

For example, if male riders are 34.8, which is approximately 35, and female riders are 25.8, which is approximately 26. Then, 135% of 26 is 35.1, which is approximately 35. So, male riders are 35, which is very close to 35.1. The difference is 0.1.

But if we don't round, the difference is 0.03.

But since the problem didn't specify rounding, I think we should go with the exact decimal values.

Therefore, the answer is 12th grade.

But let me check if the answer is different if we consider that the question is asking for the male percentage to be closest to 135% of the female percentage.

In that case, as we saw, 12th grade has a difference of 0.01%, which is extremely close.

But again, the question says "number of its male bus riders", not "percentage".

But given that both interpretations lead to 12th grade, and the problem is from a math competition or something similar, maybe the answer is 12th grade.

Alternatively, maybe I made a mistake in calculation for another grade. Let me check 11th grade again.

For 11th grade, male riders: 61.8, 135% of female riders: 55.89. Difference is 5.91.

If we round to whole numbers, male riders would be 62, female riders 41. Then, 135% of 41 is 55.35, so difference is 62 - 55.35 = 6.65.

But even with rounding, the difference is still larger than 0.03.

Therefore, regardless of rounding, 12th grade is the closest.

Therefore, I think the answer is 12th grade.

But let me check once more for 12th grade.

Male riders: 34.8

Female riders: 25.8

135% of female riders: 1.35 * 25.8 = 34.83

Difference: 34.8 - 34.83 = -0.03.

So, the male riders are 0.03 less than 135% of female riders.

But since the question is about "closest to", the absolute difference is 0.03, which is the smallest.

Therefore, the answer is 12th grade.

But let me check if the answer is different if we consider that the question is asking for the male riders to be 135% of the female riders, meaning that the male riders are 1.35 times the female riders.

In that case, we would compute for each grade, the ratio of male riders to female riders, and see which is closest to 1.35.

Let me try that approach.

For each grade:

Ratio = (male riders) / (female riders)

Compute this ratio and see which is closest to 1.35.

Let me compute that for each grade.

Starting with 9th grade:

Male riders: 123.3, female riders: 118.2. Ratio = 123.3 / 118.2 ≈ 1.043.

Difference from 1.35: 1.35 - 1.043 = 0.307.

10th grade:

Male riders: 103.2, female riders: 99.3. Ratio = 103.2 / 99.3 ≈ 1.039.

Difference from 1.35: 1.35 - 1.039 = 0.311.

11th grade:

Male riders: 61.8, female riders: 41.4. Ratio = 61.8 / 41.4 ≈ 1.493.

Difference from 1.35: 1.493 - 1.35 = 0.143.

12th grade:

Male riders: 34.8, female riders: 25.8. Ratio = 34.8 / 25.8 ≈ 1.348.

Difference from 1.35: 1.35 - 1.348 = 0.002.

So, the difference here is 0.002, which is extremely small.

So, if the question is asking for the ratio of male to female riders closest to 1.35, then 12th grade is the answer.

But the original question says: "the number of its male bus riders closest to 135% of the number of its female bus riders".

Which is equivalent to saying that male riders ≈ 1.35 * female riders, which is equivalent to male / female ≈ 1.35.

Therefore, this approach is equivalent.

So, in this case, the ratio for 12th grade is 1.348, which is 0.002 less than 1.35.

For 11th grade, the ratio is 1.493, which is 0.143 more than 1.35.

So, 12th grade is the closest.

Therefore, regardless of whether we compute the absolute difference between male riders and 1.35 * female riders, or compute the ratio and see how close it is to 1.35, we get that 12th grade is the answer.

Therefore, I think the answer is 12th grade.

But let me check once more for 12th grade ratio.

Male riders: 34.8, female riders: 25.8.

34.8 / 25.8 = let me compute that.

Divide numerator and denominator by 6: 34.8 / 6 = 5.8, 25.8 / 6 = 4.3. So 5.8 / 4.3 ≈ 1.3488.

Which is approximately 1.3488, which is 1.35 - 0.0012.

So, very close to 1.35.

Therefore, the answer is 12th grade.

But let me check if the problem is from an official source, maybe there's a trick. But since I don't have that context, I have to rely on my calculations.

Therefore, I think the answer is 12th grade.

**Final Answer**
\boxed{12}

To determine which grade has the number of **male bus riders closest to 135% of the number of female bus riders**, we follow a structured approach:

---

### **Step 1: Understand the Data**

We are given the percentage of male and female bus riders in each grade, and we know that **300 males and 300 females** were surveyed in each grade. This means:

- **Number of male riders** = $ 300 \times \text{male percentage} $
- **Number of female riders** = $ 300 \times \text{female percentage} $

We are to compare the **actual number of male riders** to **135% of the number of female riders** for each grade and find the grade with the **smallest absolute difference**.

---

### **Step 2: Compute for Each Grade**

Let’s compute the number of male and female riders, and then compute 135% of the female riders for each grade.

#### **9th Grade**
- Male riders: $ 300 \times 0.411 = 123.3 $
- Female riders: $ 300 \times 0.394 = 118.2 $
- 135% of female riders: $ 1.35 \times 118.2 = 159.57 $
- Difference: $ |123.3 - 159.57| = 36.27 $

#### **10th Grade**
- Male riders: $ 300 \times 0.344 = 103.2 $
- Female riders: $ 300 \times 0.331 = 99.3 $
- 135% of female riders: $ 1.35 \times 99.3 = 134.055 $
- Difference: $ |103.2 - 134.055| = 30.855 $

#### **11th Grade**
- Male riders: $ 300 \times 0.206 = 61.8 $
- Female riders: $ 300 \times 0.138 = 41.4 $
- 135% of female riders: $ 1.35 \times 41.4 = 55.89 $
- Difference: $ |61.8 - 55.89| = 5.91 $

#### **12th Grade**
- Male riders: $ 300 \times 0.116 = 34.8 $
- Female riders: $ 300 \times 0.086 = 25.8 $
- 135% of female riders: $ 1.35 \times 25.8 = 34.83 $
- Difference: $ |34.8 - 34.83| = 0.03 $

---

### **Step 3: Compare the Differences**

| Grade       | Absolute Difference |
|-------------|---------------------|
| 9th Grade   | 36.27               |
| 10th Grade  | 30.855              |
| 11th Grade  | 5.91                |
| 12th Grade  | **0.03**            |

---

### **Conclusion**

The **12th grade** has the **smallest absolute difference** between the number of male riders and 135% of the number of female riders. Therefore, the number of male bus riders in the **12th grade** is **closest to 135% of the number of female bus riders**.

---

### **Final Answer**

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for which grade the number of male bus riders is closest to 135% of the number of female bus riders. The table gives percentages for males and females in each grade, and there are 300 males and 300 females surveyed in each grade. Let me break this down step by step.

First, let me understand the question. For each grade, I need to calculate the actual number of male bus riders and the actual number of female bus riders. Then, I need to check what 135% of the female riders would be for that grade, and see which grade has the male riders closest to that value.

Let me start by recalling that percentages can be converted to actual numbers by multiplying the percentage by the total number of people surveyed. Since there are 300 males and 300 females in each grade, the total number of students surveyed per grade is 600, but maybe I don't need that. Wait, actually, the question is about male and female riders separately. So for each grade, the number of male bus riders is 300 times the male percentage, and similarly for females.

So let me write that down for each grade:

For each grade:
- Number of male bus riders = 300 * (male percentage)
- Number of female bus riders = 300 * (female percentage)

Then, 135% of the number of female bus riders would be 1.35 * (number of female bus riders).

So, for each grade, I need to compute 1.35 * (female riders) and compare it to the number of male riders. Then, find which grade has the male riders closest to that value.

Alternatively, maybe I can compute the difference between the male riders and 1.35 times female riders, and see which difference is the smallest.

Let me try this approach for each grade.

Starting with 9th grade:

9th grade:
- Male percentage: 41.1%
- Female percentage: 39.4%

Calculating number of male riders: 300 * 41.1% = 300 * 0.411 = let me compute that. 300 * 0.4 is 120, 300 * 0.011 is 3.3, so total is 120 + 3.3 = 123.3. Wait, but percentages are per 100, so 41.1% of 300 is 0.411 * 300. Let me check that again. 0.411 * 300. 0.4 * 300 = 120, 0.01 * 300 = 3, 0.001 * 300 = 0.3. So 0.411 * 300 = 120 + 3 + 0.3 = 123.3. So 123.3 male riders.

Similarly, female riders: 300 * 39.4% = 300 * 0.394. Let me compute that. 0.3 * 300 = 90, 0.09 * 300 = 27, 0.004 * 300 = 1.2. So adding up: 90 + 27 = 117 + 1.2 = 118.2. So 118.2 female riders.

Now, 135% of female riders is 1.35 * 118.2. Let me calculate that. 1 * 118.2 = 118.2, 0.35 * 118.2. Let me compute 0.3 * 118.2 = 35.46, 0.05 * 118.2 = 5.91. So total 35.46 + 5.91 = 41.37. Therefore, 1.35 * 118.2 = 118.2 + 41.37 = 159.57.

So, for 9th grade, male riders are 123.3, and 135% of female riders is 159.57. The difference between male riders and 135% female riders is 123.3 - 159.57 = -36.27. But since we are looking for the closest, maybe we take absolute value? Or maybe compare the absolute difference. So the absolute difference is 36.27.

Now moving on to 10th grade:

10th grade:
- Male percentage: 34.4%
- Female percentage: 33.1%

Male riders: 300 * 0.344. Let me compute that. 0.3 * 300 = 90, 0.04 * 300 = 12, 0.004 * 300 = 1.2. So total is 90 + 12 + 1.2 = 103.2.

Female riders: 300 * 0.331. Let me compute that. 0.3 * 300 = 90, 0.03 * 300 = 9, 0.001 * 300 = 0.3. So total is 90 + 9 + 0.3 = 99.3.

Then, 135% of female riders is 1.35 * 99.3. Let me calculate that. 1 * 99.3 = 99.3, 0.35 * 99.3. Let me compute 0.3 * 99.3 = 29.79, 0.05 * 99.3 = 4.965. Adding those gives 29.79 + 4.965 = 34.755. Therefore, total is 99.3 + 34.755 = 134.055.

So, male riders are 103.2, and 135% of female riders is 134.055. The difference is 103.2 - 134.055 = -30.855. Absolute difference is 30.855.

Comparing this to the previous grade's difference of 36.27, this is smaller. So 10th grade is closer so far.

Now moving on to 11th grade:

11th grade:
- Male percentage: 20.6%
- Female percentage: 13.8%

Male riders: 300 * 0.206. Let me compute that. 0.2 * 300 = 60, 0.006 * 300 = 1.8. So total is 60 + 1.8 = 61.8.

Female riders: 300 * 0.138. Let me compute that. 0.1 * 300 = 30, 0.03 * 300 = 9, 0.008 * 300 = 2.4. So total is 30 + 9 + 2.4 = 41.4.

Then, 135% of female riders is 1.35 * 41.4. Let me calculate that. 1 * 41.4 = 41.4, 0.35 * 41.4. Let me compute 0.3 * 41.4 = 12.42, 0.05 * 41.4 = 2.07. Adding those gives 12.42 + 2.07 = 14.49. Therefore, total is 41.4 + 14.49 = 55.89.

So, male riders are 61.8, and 135% of female riders is 55.89. The difference is 61.8 - 55.89 = 5.91. Absolute difference is 5.91.

That's much smaller than the previous differences. So 11th grade is closer now.

Now moving on to 12th grade:

12th grade:
- Male percentage: 11.6%
- Female percentage: 8.6%

Male riders: 300 * 0.116. Let me compute that. 0.1 * 300 = 30, 0.01 * 300 = 3, 0.006 * 300 = 1.8. So total is 30 + 3 + 1.8 = 34.8.

Female riders: 300 * 0.086. Let me compute that. 0.08 * 300 = 24, 0.006 * 300 = 1.8. So total is 24 + 1.8 = 25.8.

Then, 135% of female riders is 1.35 * 25.8. Let me calculate that. 1 * 25.8 = 25.8, 0.35 * 25.8. Let me compute 0.3 * 25.8 = 7.74, 0.05 * 25.8 = 1.29. Adding those gives 7.74 + 1.29 = 9.03. Therefore, total is 25.8 + 9.03 = 34.83.

So, male riders are 34.8, and 135% of female riders is 34.83. The difference is 34.8 - 34.83 = -0.03. Absolute difference is 0.03.

Wow, that's extremely close. So the difference here is only 0.03, which is much smaller than the previous differences. So 12th grade is the closest so far.

Wait, so comparing all the absolute differences:

- 9th grade: 36.27
- 10th grade: 30.855
- 11th grade: 5.91
- 12th grade: 0.03

So the smallest difference is for 12th grade. Therefore, the answer should be 12th grade?

But let me check my calculations again because the difference for 12th grade is so small. Let me verify the numbers again.

Starting with 12th grade:

Male percentage: 11.6%, so male riders: 300 * 0.116. Let me compute 0.116 * 300. 0.1 * 300 = 30, 0.01 * 300 = 3, 0.006 * 300 = 1.8. So 30 + 3 + 1.8 = 34.8. Correct.

Female percentage: 8.6%, so female riders: 300 * 0.086. Let me compute 0.086 * 300. 0.08 * 300 = 24, 0.006 * 300 = 1.8. So 24 + 1.8 = 25.8. Correct.

Then 135% of female riders: 1.35 * 25.8. Let me compute that again. 25.8 * 1.35. Let me do 25.8 * 1.35.

Alternatively, 25.8 * 1.35 = 25.8 * (1 + 0.3 + 0.05) = 25.8 + 7.74 + 1.29 = 25.8 + 7.74 = 33.54 + 1.29 = 34.83. Correct.

So male riders are 34.8, and 135% of female riders is 34.83. The difference is 34.8 - 34.83 = -0.03. So absolute difference is 0.03. That's very close.

But let me check if I did the 11th grade correctly.

For 11th grade:

Male riders: 300 * 20.6% = 300 * 0.206. Let me compute 0.2 * 300 = 60, 0.006 * 300 = 1.8. So 60 + 1.8 = 61.8. Correct.

Female riders: 300 * 13.8% = 300 * 0.138. Let me compute 0.1 * 300 = 30, 0.03 * 300 = 9, 0.008 * 300 = 2.4. So 30 + 9 + 2.4 = 41.4. Correct.

Then 135% of female riders: 1.35 * 41.4. Let me compute that again. 41.4 * 1.35. Let me do 41.4 * 1.35.

Alternatively, 41.4 * 1.35 = 41.4 * (1 + 0.3 + 0.05) = 41.4 + 12.42 + 2.07 = 41.4 + 12.42 = 53.82 + 2.07 = 55.89. Correct.

So male riders are 61.8, and 135% of female riders is 55.89. The difference is 61.8 - 55.89 = 5.91. Correct.

So 12th grade has a difference of 0.03, which is way smaller. So that seems like the closest.

But let me check if there's any mistake in interpreting the question. The question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, for each grade, compute male riders vs 1.35 * female riders. The difference between them. The one with the smallest difference is the answer.

So for 12th grade, male riders are 34.8, and 1.35 * female riders is 34.83. So the difference is 0.03. That's very close.

But let me check if there's a possible error in calculation. For example, maybe I should have done something else? Let me check the 12th grade again.

Wait, 135% of female riders is 1.35 * female riders. So if female riders are 25.8, then 1.35 * 25.8 is indeed 34.83. Male riders are 34.8. So the difference is 34.8 vs 34.83. So the difference is 0.03.

But since we are dealing with actual numbers of people, which should be whole numbers. Wait, but the percentages are given with one decimal place, so when we multiply by 300, we get numbers with one decimal place. For example, 34.8 is 34.8, which is 34.8 people? Wait, but you can't have 0.8 of a person. However, since the percentages are given as exact numbers, maybe we are supposed to treat them as exact, even though in reality you can't have fractions of people. But since the problem is presented with percentages, and the answer is likely expecting us to use the exact decimal values, even if they are not whole numbers.

Therefore, proceeding with the decimal values is correct here.

So, given that, the difference for 12th grade is 0.03, which is the smallest. Therefore, the answer is 12th grade.

But let me check again for 11th grade. The difference was 5.91, which is bigger than 0.03. For 10th grade, 30.855, which is bigger. For 9th grade, 36.27. So 12th grade is the closest.

But wait, the problem says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, maybe I need to check if 135% of female riders is actually closer to male riders.

Alternatively, maybe I should compute the absolute difference between male riders and 1.35 * female riders. Which is exactly what I did.

So, for 12th grade, the difference is 0.03, which is very small.

But let me check if there's a possible miscalculation here. Let me recompute 1.35 * 25.8.

25.8 * 1.35. Let me do this multiplication step by step.

First, 25.8 * 1 = 25.8

25.8 * 0.3 = 7.74

25.8 * 0.05 = 1.29

Adding them up: 25.8 + 7.74 = 33.54 + 1.29 = 34.83. Correct.

Male riders: 34.8. So difference is 34.8 - 34.83 = -0.03. Absolute value is 0.03.

So that's correct.

But let me check if the question is asking for the closest in terms of percentage difference or absolute difference. The question says "closest to 135% of the number of its female bus riders". So, it's about the actual number of male riders being closest to 1.35 times the number of female riders. So, the absolute difference between male riders and 1.35 * female riders.

Therefore, 12th grade is the closest.

But let me check if there's a possible mistake in the problem interpretation. For example, maybe the question is asking for the male riders to be closest to 135% of the female riders, meaning that 135% of female riders is the target, and male riders are compared to that.

Alternatively, maybe the question is phrased as "the number of its male bus riders closest to 135% of the number of its female bus riders", so the male riders are being compared to 135% of female riders. So, for each grade, compute male riders and 1.35 * female riders, and see which is closest.

Yes, that's exactly what I did.

Alternatively, maybe the question is asking for the male percentage to be closest to 135% of the female percentage? But that would be different. But the question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So it's about the actual numbers, not percentages.

Therefore, my conclusion is that 12th grade is the answer.

But let me check once more for 12th grade.

Male riders: 34.8

135% of female riders: 34.83

So, 34.8 is very close to 34.83. The difference is 0.03.

But let me check if there's another grade with a smaller difference. For example, maybe I made a mistake with 11th grade?

For 11th grade, male riders: 61.8, 1.35 * female riders: 55.89. The difference is 5.91.

So, 5.91 is bigger than 0.03.

So, 12th grade is the closest.

But let me check if the problem says "closest to 135% of the number of its female bus riders". So, if the male riders are 34.8 and 135% of female is 34.83, then the male riders are just slightly less than 135% of female riders.

But since the question is about which grade is closest, and 0.03 is the smallest difference, then 12th grade is the answer.

But let me check if there's a possible error in calculation for 12th grade.

Wait, female riders: 8.6% of 300. Let me compute 8.6% of 300.

8.6% is 0.086. 0.086 * 300.

Well, 10% of 300 is 30, so 8% is 24, 0.6% is 1.8, so total is 24 + 1.8 = 25.8. Correct.

Then 1.35 * 25.8.

Alternatively, 25.8 * 1.35. Let me compute 25.8 * 1.35.

Let me convert 25.8 to 258/10, and 1.35 to 135/100.

So 258/10 * 135/100 = (258 * 135) / 1000.

But maybe that's too complicated. Alternatively, 25.8 * 1.35.

Alternatively, think of 25.8 * 1.35 = 25.8 * (1 + 0.3 + 0.05) = 25.8 + 7.74 + 1.29 = 34.83. Correct.

So, yes, that's correct.

Therefore, the answer is 12th grade.

But wait, the problem says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, if the male riders are 34.8 and 135% of female is 34.83, then the male riders are just 0.03 less than 135% of female riders.

But maybe the question is expecting us to round to the nearest whole number? Because you can't have 0.8 of a person.

Wait, but the problem says "the number of its male bus riders", which would be a whole number. However, the percentages are given with one decimal place, so when we calculate the number of riders, we get decimal numbers. But maybe we need to round them?

Wait, the problem says "the data in the table below shows the percent of bus riders in a survey of Central H.S. students; 300 males and 300 females in each grade were surveyed." So, the percentages are given, and the actual number of riders would be calculated by multiplying the percentage by 300. However, since percentages can result in decimal numbers, but in reality, the number of riders must be whole numbers. However, since the problem is presented with percentages, and the answer is likely expecting us to use the exact decimal values, even though they are not whole numbers.

But maybe the problem expects us to round the number of riders to the nearest whole number? Let me check.

For example, in 12th grade, male riders would be 34.8, which is 35 when rounded. Female riders are 25.8, which is 26 when rounded. Then, 135% of 26 is 1.35 * 26 = 35.1. So, male riders rounded is 35, which is very close to 35.1. The difference is 0.1.

But if we don't round, the difference is 0.03.

But the problem says "the number of its male bus riders", which is an actual count. So, if we have to consider that the number of riders must be whole numbers, then we need to calculate the exact number of riders, which would be 34.8 for males and 25.8 for females. But since you can't have 0.8 of a person, maybe the actual numbers are rounded.

But the problem doesn't specify whether to round or not. However, since the percentages are given with one decimal place, and the number of people is 300, which is a multiple of 10, maybe the actual number of riders is a whole number. Let me check for 12th grade.

Wait, 11.6% of 300. Let me compute 11.6% of 300.

11.6% is 11.6/100 = 0.116. 0.116 * 300 = 34.8.

But 34.8 is not a whole number. However, maybe the survey results are rounded, so the actual number could be 35 or 34. Similarly for females: 8.6% of 300 is 25.8, which could be 26 or 25.

But since the problem gives the percentages as exact numbers, maybe we are supposed to use the exact decimal values.

Alternatively, maybe the problem expects us to use the exact numbers, even if they are not whole numbers, because the percentages are given with decimals.

In that case, the difference of 0.03 is acceptable.

Alternatively, if we consider that the number of riders must be whole numbers, then we need to check if the percentages are rounded. For example, if the actual percentage for males is 11.6%, which could be rounded from something like 11.55% or 11.65%, leading to 34.65 or 34.95, which would round to 35. Similarly for females.

But since the problem doesn't specify, I think we should proceed with the exact decimal values as calculated.

Therefore, the answer is 12th grade.

But let me check once more for 12th grade.

Male riders: 34.8

135% of female riders: 34.83

Difference: 0.03.

So, that's the closest.

But let me check if there's a possible mistake in the question. For example, maybe the question is asking for the male percentage to be closest to 135% of the female percentage? Let me check that.

For example, for 12th grade, male percentage is 11.6%, female percentage is 8.6%. 135% of female percentage is 1.35 * 8.6% = 11.59%. Which is very close to male percentage of 11.6%.

Ah! Wait, that's another way to interpret the question. Maybe the question is asking for the male percentage to be closest to 135% of the female percentage.

But the original question says: "the number of its male bus riders closest to 135% of the number of its female bus riders".

So, "number of male bus riders" vs "number of female bus riders". So, it's about the actual counts, not percentages.

But if we consider the percentages, then for 12th grade, male percentage is 11.6%, and 135% of female percentage is 1.35 * 8.6% = 11.59%, which is extremely close.

But the question is about numbers, not percentages. However, maybe the problem is ambiguously worded?

But given the exact wording, it's about numbers.

But let me check both interpretations.

First interpretation: numbers. As above, 12th grade is the answer.

Second interpretation: percentages. For each grade, compute male percentage vs 1.35 * female percentage.

Let me check that.

For 9th grade: male percentage 41.1%, 1.35 * female percentage = 1.35 * 39.4% = let me compute that. 1.35 * 39.4.

Let me compute 1 * 39.4 = 39.4, 0.35 * 39.4.

0.3 * 39.4 = 11.82, 0.05 * 39.4 = 1.97. Total is 11.82 + 1.97 = 13.79. So total is 39.4 + 13.79 = 53.19%.

So male percentage is 41.1% vs 53.19%. Difference is 12.09%.

For 10th grade: male percentage 34.4%, 1.35 * 33.1% = 1.35 * 33.1.

Compute 1 * 33.1 = 33.1, 0.35 * 33.1.

0.3 * 33.1 = 9.93, 0.05 * 33.1 = 1.655. Total is 9.93 + 1.655 = 11.585. Total is 33.1 + 11.585 = 44.685%.

Difference between male percentage (34.4%) and 44.685% is 10.285%.

For 11th grade: male percentage 20.6%, 1.35 * 13.8% = 1.35 * 13.8.

Compute 1 * 13.8 = 13.8, 0.35 * 13.8.

0.3 * 13.8 = 4.14, 0.05 * 13.8 = 0.69. Total is 4.14 + 0.69 = 4.83. Total is 13.8 + 4.83 = 18.63%.

Difference between male percentage (20.6%) and 18.63% is 1.97%.

For 12th grade: male percentage 11.6%, 1.35 * 8.6% = 1.35 * 8.6.

Compute 1 * 8.6 = 8.6, 0.35 * 8.6.

0.3 * 8.6 = 2.58, 0.05 * 8.6 = 0.43. Total is 2.58 + 0.43 = 3.01. Total is 8.6 + 3.01 = 11.61%.

Difference between male percentage (11.6%) and 11.61% is 0.01%.

So, if the question was about percentages, then 12th grade is also the closest, with a difference of 0.01%.

But the original question says "the number of its male bus riders closest to 135% of the number of its female bus riders". So, it's about numbers, not percentages. However, in both interpretations, 12th grade is the answer.

But let me check if there's a different answer if we consider the numbers.

But according to the first interpretation, the answer is 12th grade.

But let me check if there's a possible mistake in the problem. For example, maybe the question is asking for the male percentage to be closest to 135% of the female percentage, which would be 12th grade.

But since the question says "number of its male bus riders", I think it's about the actual numbers.

But given that both interpretations lead to 12th grade, maybe that's the answer.

But let me check once more for 12th grade with numbers.

Male riders: 34.8

135% of female riders: 34.83

Difference: 0.03.

But if we consider that the number of riders must be whole numbers, then maybe we need to round.

For example, if male riders are 34.8, which is approximately 35, and female riders are 25.8, which is approximately 26. Then, 135% of 26 is 35.1, which is approximately 35. So, male riders are 35, which is very close to 35.1. The difference is 0.1.

But if we don't round, the difference is 0.03.

But since the problem didn't specify rounding, I think we should go with the exact decimal values.

Therefore, the answer is 12th grade.

But let me check if the answer is different if we consider that the question is asking for the male percentage to be closest to 135% of the female percentage.

In that case, as we saw, 12th grade has a difference of 0.01%, which is extremely close.

But again, the question says "number of its male bus riders", not "percentage".

But given that both interpretations lead to 12th grade, and the problem is from a math competition or something similar, maybe the answer is 12th grade.

Alternatively, maybe I made a mistake in calculation for another grade. Let me check 11th grade again.

For 11th grade, male riders: 61.8, 135% of female riders: 55.89. Difference is 5.91.

If we round to whole numbers, male riders would be 62, female riders 41. Then, 135% of 41 is 55.35, so difference is 62 - 55.35 = 6.65.

But even with rounding, the difference is still larger than 0.03.

Therefore, regardless of rounding, 12th grade is the closest.

Therefore, I think the answer is 12th grade.

But let me check once more for 12th grade.

Male riders: 34.8

Female riders: 25.8

135% of female riders: 1.35 * 25.8 = 34.83

Difference: 34.8 - 34.83 = -0.03.

So, the male riders are 0.03 less than 135% of female riders.

But since the question is about "closest to", the absolute difference is 0.03, which is the smallest.

Therefore, the answer is 12th grade.

But let me check if the answer is different if we consider that the question is asking for the male riders to be 135% of the female riders, meaning that the male riders are 1.35 times the female riders.

In that case, we would compute for each grade, the ratio of male riders to female riders, and see which is closest to 1.35.

Let me try that approach.

For each grade:

Ratio = (male riders) / (female riders)

Compute this ratio and see which is closest to 1.35.

Let me compute that for each grade.

Starting with 9th grade:

Male riders: 123.3, female riders: 118.2. Ratio = 123.3 / 118.2 ≈ 1.043.

Difference from 1.35: 1.35 - 1.043 = 0.307.

10th grade:

Male riders: 103.2, female riders: 99.3. Ratio = 103.2 / 99.3 ≈ 1.039.

Difference from 1.35: 1.35 - 1.039 = 0.311.

11th grade:

Male riders: 61.8, female riders: 41.4. Ratio = 61.8 / 41.4 ≈ 1.493.

Difference from 1.35: 1.493 - 1.35 = 0.143.

12th grade:

Male riders: 34.8, female riders: 25.8. Ratio = 34.8 / 25.8 ≈ 1.348.

Difference from 1.35: 1.35 - 1.348 = 0.002.

So, the difference here is 0.002, which is extremely small.

So, if the question is asking for the ratio of male to female riders closest to 1.35, then 12th grade is the answer.

But the original question says: "the number of its male bus riders closest to 135% of the number of its female bus riders".

Which is equivalent to saying that male riders ≈ 1.35 * female riders, which is equivalent to male / female ≈ 1.35.

Therefore, this approach is equivalent.

So, in this case, the ratio for 12th grade is 1.348, which is 0.002 less than 1.35.

For 11th grade, the ratio is 1.493, which is 0.143 more than 1.35.

So, 12th grade is the closest.

Therefore, regardless of whether we compute the absolute difference between male riders and 1.35 * female riders, or compute the ratio and see how close it is to 1.35, we get that 12th grade is the answer.

Therefore, I think the answer is 12th grade.

But let me check once more for 12th grade ratio.

Male riders: 34.8, female riders: 25.8.

34.8 / 25.8 = let me compute that.

Divide numerator and denominator by 6: 34.8 / 6 = 5.8, 25.8 / 6 = 4.3. So 5.8 / 4.3 ≈ 1.3488.

Which is approximately 1.3488, which is 1.35 - 0.0012.

So, very close to 1.35.

Therefore, the answer is 12th grade.

But let me check if the problem is from an official source, maybe there's a trick. But since I don't have that context, I have to rely on my calculations.

Therefore, I think the answer is 12th grade.

**Final Answer**
\boxed{12}

To determine which grade has the number of **male bus riders closest to 135% of the number of female bus riders**, we follow a structured approach:

---

### **Step 1: Understand the Data**

We are given the percentage of male and female bus riders in each grade, and we know that **300 males and 300 females** were surveyed in each grade. This means:

- **Number of male riders** = $ 300 \times \text{male percentage} $
- **Number of female riders** = $ 300 \times \text{female percentage} $

We are to compare the **actual number of male riders** to **135% of the number of female riders** for each grade and find the grade with the **smallest absolute difference**.

---

### **Step 2: Compute for Each Grade**

Let’s compute the number of male and female riders, and then compute 135% of the female riders for each grade.

#### **9th Grade**
- Male riders: $ 300 \times 0.411 = 123.3 $
- Female riders: $ 300 \times 0.394 = 118.2 $
- 135% of female riders: $ 1.35 \times 118.2 = 159.57 $
- Difference: $ |123.3 - 159.57| = 36.27 $

#### **10th Grade**
- Male riders: $ 300 \times 0.344 = 103.2 $
- Female riders: $ 300 \times 0.331 = 99.3 $
- 135% of female riders: $ 1.35 \times 99.3 = 134.055 $
- Difference: $ |103.2 - 134.055| = 30.855 $

#### **11th Grade**
- Male riders: $ 300 \times 0.206 = 61.8 $
- Female riders: $ 300 \times 0.138 = 41.4 $
- 135% of female riders: $ 1.35 \times 41.4 = 55.89 $
- Difference: $ |61.8 - 55.89| = 5.91 $

#### **12th Grade**
- Male riders: $ 300 \times 0.116 = 34.8 $
- Female riders: $ 300 \times 0.086 = 25.8 $
- 135% of female riders: $ 1.35 \times 25.8 = 34.83 $
- Difference: $ |34.8 - 34.83| = 0.03 $

---

### **Step 3: Compare the Differences**

| Grade       | Absolute Difference |
|-------------|---------------------|
| 9th Grade   | 36.27               |
| 10th Grade  | 30.855              |
| 11th Grade  | 5.91                |
| 12th Grade  | **0.03**            |

---

### **Conclusion**

The **12th grade** has the **smallest absolute difference** between the number of male riders and 135% of the number of female riders. Therefore, the number of male bus riders in the **12th grade** is **closest to 135% of the number of female bus riders**.

---

### **Final Answer**

$$
\boxed{12}
$$

Solution: The grade level in which the number of male bus riders is closest to $135\%$ of the number of female bus riders is the same grade level in which the percentage of male bus riders is closest to $135\%$ of the percentage of female bus riders. To find what percentage is $135\%$ of the percentage of female bus riders, we multiply each percentage by $1.35.$ In ninth grade, the percentage is $39.4\cdot1.35=53.19$ percent. In tenth grade, the percentage is $33.1\cdot1.35=44.685$ percent. In eleventh grade, the percentage is $13.8\cdot1.35=18.63$ percent. In twelfth grade, the percentage is $8.6\cdot1.35=11.61$ percent. From inspection, we see that $135\%$ of the percentage of female bus riders in twelfth grade is closest to the percentage of male bus riders in twelfth grade. So the answer is $\boxed{12}.$