Explanation:
Okay, so I need to find the remainder when the sum of numbers from 1 to 10 is divided by 9. Let me think about how to approach this.

First, I remember that there's a formula for the sum of the first n natural numbers. Is it n(n + 1)/2? Let me check with a small number. If n is 3, then the sum is 1 + 2 + 3 = 6. Using the formula: 3(3 + 1)/2 = 12/2 = 6. Yep, that works. So for n = 10, the sum should be 10(10 + 1)/2. Let me calculate that.

10 times 11 is 110, divided by 2 is 55. So the total sum is 55. Now, I need to find the remainder when 55 is divided by 9.

Hmm, how do I do that? Well, division with remainder... So 9 times 6 is 54, right? Because 9*6 = 54. Then 55 minus 54 is 1. So the remainder is 1? Wait, is that correct? Let me verify.

Alternatively, maybe I can do this without calculating the entire sum. Maybe there's a way to find the remainder by considering each term modulo 9 and then adding them up? Let me try that approach to cross-check.

So, the numbers from 1 to 10. Let me list them and find each modulo 9:

1 mod 9 = 1

2 mod 9 = 2

3 mod 9 = 3

4 mod 9 = 4

5 mod 9 = 5

6 mod 9 = 6

7 mod 9 = 7

8 mod 9 = 8

9 mod 9 = 0 (since 9 is divisible by 9)

10 mod 9 = 1 (because 10 divided by 9 is 1 with a remainder of 1)

So now, adding up all these remainders: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1. Let me compute that step by step.

Starting from the beginning:

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 0 = 36

36 + 1 = 37

So the total sum of the remainders is 37. Now, I need to find the remainder when 37 is divided by 9. Let me do that.

9 times 4 is 36. So 37 minus 36 is 1. Therefore, the remainder is 1.

Wait, so both methods give me the same answer, 1. That seems consistent. But let me check once more to be sure.

Alternatively, maybe I can use properties of modular arithmetic. For example, the sum from 1 to n modulo m can be calculated by summing each term modulo m and then taking modulo m again. Which is exactly what I did in the second approach.

Alternatively, since I know the total sum is 55, and 55 divided by 9 is 6 with a remainder of 1, as 9*6=54, 55-54=1. So that's straightforward.

But just to make sure I didn't make a mistake in calculating the sum. Let me add the numbers from 1 to 10 manually.

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 9 = 45

45 + 10 = 55.

Yes, that's correct. So the total is indeed 55. Then 55 divided by 9. Let me do the division again.

9*5=45, 9*6=54, 9*7=63. So 54 is less than 55, so 6*9=54, remainder 1. So yes, remainder 1.

Alternatively, maybe I can think of 55 mod 9. Since 9*6=54, 55-54=1. So 55 ≡ 1 mod 9.

Alternatively, maybe there's another way. For example, using the fact that 1 + 2 + ... + 9 is 45, which is divisible by 9, since 9*5=45. Then adding 10, which is 1 mod 9. So 45 + 10 = 55. Since 45 is 0 mod 9, then 55 mod 9 is 10 mod 9, which is 1.

That's another way to think about it. So breaking it down into the sum up to 9 and then adding 10.

So that also gives me 1.

Therefore, all methods lead to the same answer. So I think the remainder is 1.

But just to make sure, let me check with another approach. Let me compute 55 divided by 9.

9*6 = 54. 55 - 54 = 1. So quotient is 6, remainder 1.

Alternatively, if I use the division algorithm: for integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a = 55, b = 9. So 55 = 9*6 + 1. Therefore, remainder r is 1.

Therefore, regardless of the method, the remainder is 1.

I think that's solid. So the answer is 1.

**Final Answer**
The remainder is \boxed{1}.

To find the remainder when the sum $1 + 2 + 3 + \dots + 10$ is divided by 9, we can approach the problem in a few different ways, all of which lead to the same conclusion.

---

### Step 1: Use the Formula for the Sum of the First $n$ Natural Numbers

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

$$
S = \frac{n(n + 1)}{2}
$$

For $n = 10$:

$$
S = \frac{10 \times 11}{2} = \frac{110}{2} = 55
$$

So, the total sum is 55.

---

### Step 2: Find the Remainder When 55 is Divided by 9

We now compute the remainder of $55 \div 9$:

$$
55 \div 9 = 6 \text{ remainder } 1
$$

This is because:

$$
9 \times 6 = 54 \quad \text{and} \quad 55 - 54 = 1
$$

So, the remainder is:

$$
\boxed{1}
$$

---

### Step 3: Cross-Verification Using Modular Arithmetic

We can also verify this by computing the sum modulo 9 directly. That is, we compute each number from 1 to 10 modulo 9 and then sum the results:

$$
\begin{align*}
1 \mod 9 &= 1 \\
2 \mod 9 &= 2 \\
3 \mod 9 &= 3 \\
4 \mod 9 &= 4 \\
5 \mod 9 &= 5 \\
6 \mod 9 &= 6 \\
7 \mod 9 &= 7 \\
8 \mod 9 &= 8 \\
9 \mod 9 &= 0 \\
10 \mod 9 &= 1 \\
\end{align*}
$$

Now sum these remainders:

$$
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1 = 37
$$

Now compute $37 \mod 9$:

$$
37 \div 9 = 4 \text{ remainder } 1
$$

So again, the remainder is:

$$
\boxed{1}
$$

---

### Final Answer

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to find the remainder when the sum of numbers from 1 to 10 is divided by 9. Let me think about how to approach this.

First, I remember that there's a formula for the sum of the first n natural numbers. Is it n(n + 1)/2? Let me check with a small number. If n is 3, then the sum is 1 + 2 + 3 = 6. Using the formula: 3(3 + 1)/2 = 12/2 = 6. Yep, that works. So for n = 10, the sum should be 10(10 + 1)/2. Let me calculate that.

10 times 11 is 110, divided by 2 is 55. So the total sum is 55. Now, I need to find the remainder when 55 is divided by 9.

Hmm, how do I do that? Well, division with remainder... So 9 times 6 is 54, right? Because 9*6 = 54. Then 55 minus 54 is 1. So the remainder is 1? Wait, is that correct? Let me verify.

Alternatively, maybe I can do this without calculating the entire sum. Maybe there's a way to find the remainder by considering each term modulo 9 and then adding them up? Let me try that approach to cross-check.

So, the numbers from 1 to 10. Let me list them and find each modulo 9:

1 mod 9 = 1

2 mod 9 = 2

3 mod 9 = 3

4 mod 9 = 4

5 mod 9 = 5

6 mod 9 = 6

7 mod 9 = 7

8 mod 9 = 8

9 mod 9 = 0 (since 9 is divisible by 9)

10 mod 9 = 1 (because 10 divided by 9 is 1 with a remainder of 1)

So now, adding up all these remainders: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1. Let me compute that step by step.

Starting from the beginning:

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 0 = 36

36 + 1 = 37

So the total sum of the remainders is 37. Now, I need to find the remainder when 37 is divided by 9. Let me do that.

9 times 4 is 36. So 37 minus 36 is 1. Therefore, the remainder is 1.

Wait, so both methods give me the same answer, 1. That seems consistent. But let me check once more to be sure.

Alternatively, maybe I can use properties of modular arithmetic. For example, the sum from 1 to n modulo m can be calculated by summing each term modulo m and then taking modulo m again. Which is exactly what I did in the second approach.

Alternatively, since I know the total sum is 55, and 55 divided by 9 is 6 with a remainder of 1, as 9*6=54, 55-54=1. So that's straightforward.

But just to make sure I didn't make a mistake in calculating the sum. Let me add the numbers from 1 to 10 manually.

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 9 = 45

45 + 10 = 55.

Yes, that's correct. So the total is indeed 55. Then 55 divided by 9. Let me do the division again.

9*5=45, 9*6=54, 9*7=63. So 54 is less than 55, so 6*9=54, remainder 1. So yes, remainder 1.

Alternatively, maybe I can think of 55 mod 9. Since 9*6=54, 55-54=1. So 55 ≡ 1 mod 9.

Alternatively, maybe there's another way. For example, using the fact that 1 + 2 + ... + 9 is 45, which is divisible by 9, since 9*5=45. Then adding 10, which is 1 mod 9. So 45 + 10 = 55. Since 45 is 0 mod 9, then 55 mod 9 is 10 mod 9, which is 1.

That's another way to think about it. So breaking it down into the sum up to 9 and then adding 10.

So that also gives me 1.

Therefore, all methods lead to the same answer. So I think the remainder is 1.

But just to make sure, let me check with another approach. Let me compute 55 divided by 9.

9*6 = 54. 55 - 54 = 1. So quotient is 6, remainder 1.

Alternatively, if I use the division algorithm: for integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a = 55, b = 9. So 55 = 9*6 + 1. Therefore, remainder r is 1.

Therefore, regardless of the method, the remainder is 1.

I think that's solid. So the answer is 1.

**Final Answer**
The remainder is \boxed{1}.

To find the remainder when the sum $1 + 2 + 3 + \dots + 10$ is divided by 9, we can approach the problem in a few different ways, all of which lead to the same conclusion.

---

### Step 1: Use the Formula for the Sum of the First $n$ Natural Numbers

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

$$
S = \frac{n(n + 1)}{2}
$$

For $n = 10$:

$$
S = \frac{10 \times 11}{2} = \frac{110}{2} = 55
$$

So, the total sum is 55.

---

### Step 2: Find the Remainder When 55 is Divided by 9

We now compute the remainder of $55 \div 9$:

$$
55 \div 9 = 6 \text{ remainder } 1
$$

This is because:

$$
9 \times 6 = 54 \quad \text{and} \quad 55 - 54 = 1
$$

So, the remainder is:

$$
\boxed{1}
$$

---

### Step 3: Cross-Verification Using Modular Arithmetic

We can also verify this by computing the sum modulo 9 directly. That is, we compute each number from 1 to 10 modulo 9 and then sum the results:

$$
\begin{align*}
1 \mod 9 &= 1 \\
2 \mod 9 &= 2 \\
3 \mod 9 &= 3 \\
4 \mod 9 &= 4 \\
5 \mod 9 &= 5 \\
6 \mod 9 &= 6 \\
7 \mod 9 &= 7 \\
8 \mod 9 &= 8 \\
9 \mod 9 &= 0 \\
10 \mod 9 &= 1 \\
\end{align*}
$$

Now sum these remainders:

$$
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1 = 37
$$

Now compute $37 \mod 9$:

$$
37 \div 9 = 4 \text{ remainder } 1
$$

So again, the remainder is:

$$
\boxed{1}
$$

---

### Final Answer

$$
\boxed{1}
$$

Solution: Looking at our sum, we can see that the numbers $1$ through $8$ can be paired off to form $9,$ so we may eliminate them. That is, $1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 9.$ Therefore, the only remaining terms are $9$ and $10,$ and $9$ is obviously also divisible by $9,$ hence we only need to find the remainder of $10$ when divided by $9,$ which is $\boxed{1}.$


Explanation:
Okay, let me try to figure out this problem. So, Susan and Pam are buying tickets to a show, and we need to find out how many more dollars Pam pays than Susan. Let me start by understanding the problem step by step.

First, the full price of one ticket is $20. Susan buys 4 tickets with a 25% discount coupon, and Pam buys 5 tickets with a 30% discount coupon. The question is asking for the difference in the amount they each pay, specifically how much more Pam pays compared to Susan.

Alright, so I need to calculate the total amount Susan pays after applying her discount, then do the same for Pam, and then subtract Susan's total from Pam's total to find the difference. Let me write that down.

Starting with Susan. She buys 4 tickets. The original price per ticket is $20, so without any discount, she would pay 4 times $20. But she has a 25% discount. So, first, maybe I should calculate the discount amount per ticket and then subtract that from the original price? Or maybe calculate the discounted price per ticket first?

Hmm, I think it's better to calculate the discounted price per ticket first. Let me recall that a 25% discount means she pays 75% of the original price. Because 100% - 25% = 75%. So, if each ticket is $20, then the discounted price per ticket is 25% off, so 75% of $20. Let me compute that.

So, 25% of $20 is 0.25 * 20 = $5. So, the discount per ticket is $5. Therefore, the discounted price per ticket is $20 - $5 = $15. Alternatively, 75% of $20 is 0.75 * 20 = $15. Either way, same result. So, Susan pays $15 per ticket. Since she buys 4 tickets, her total cost is 4 * $15. Let me calculate that. 4 * 15 is 60. So, Susan pays $60 in total.

Now, moving on to Pam. She buys 5 tickets with a 30% discount. Let me do the same process. First, the original price per ticket is $20. A 30% discount means she pays 70% of the original price. Let me check that. 100% - 30% = 70%. So, the discounted price per ticket is 70% of $20. Let me compute that. 0.70 * 20. Hmm, 0.7 * 20. Well, 0.7 * 20 is 14. So, each ticket costs her $14 after the discount. Therefore, for 5 tickets, her total cost is 5 * $14. Let me calculate that. 5 * 14 is 70. So, Pam pays $70 in total.

Wait, hold on. Let me verify that again. If the discount is 30%, then the amount she saves per ticket is 30% of $20, which is 0.30 * 20 = $6. Therefore, the discounted price per ticket is $20 - $6 = $14. Yep, same as before. So, 5 tickets would be 5 * 14 = $70. So, Pam pays $70.

So, Susan pays $60, Pam pays $70. The question is asking how many more dollars Pam pays than Susan. So, that would be Pam's total minus Susan's total. So, $70 - $60 = $10. Therefore, Pam pays $10 more than Susan. Is that the answer? Wait, let me check again to make sure I didn't make any calculation errors.

Let me recalculate Susan's total. 4 tickets at 25% off. Original total without discount: 4 * 20 = $80. Then, 25% discount on $80. Wait, hold on, maybe I should check if the discount is applied per ticket or on the total? The problem says Susan buys 4 tickets using a coupon that gives her a 25% discount. So, the coupon is probably applied per ticket, but sometimes discounts can be applied to the total. But in most cases, unless specified otherwise, a discount coupon would be applied per item. However, let me check both ways to be thorough.

First, if the discount is per ticket: as I did before, each ticket is $15, so 4 tickets is $60. Alternatively, if the discount is applied to the total amount. So, total original price is 4 * 20 = $80. Then, 25% discount on $80 is 0.25 * 80 = $20 discount. Therefore, total paid is $80 - $20 = $60. Same result. So, either way, Susan pays $60. So that's correct.

Similarly for Pam. Original total for 5 tickets is 5 * 20 = $100. 30% discount on $100 is 0.30 * 100 = $30 discount. Therefore, total paid is $100 - $30 = $70. Alternatively, per ticket discount: 30% off each ticket, so each ticket is 70% of $20, which is $14, so 5 tickets is 5 * 14 = $70. Same result. So, Pam pays $70. Therefore, the difference is $70 - $60 = $10. So, Pam pays $10 more than Susan.

Wait, but let me think again. The problem says "how many more dollars does Pam pay than Susan?" So, if Pam pays $70 and Susan pays $60, then the answer is $10. But let me check once more if I didn't mix up the numbers.

Wait, Susan's discount is 25%, so 25% off 4 tickets. Let me compute 25% of 4 tickets. Wait, no, the discount is per ticket. So, 25% off each ticket. So, each ticket is 20 * 0.75 = 15. 4 tickets is 15 * 4 = 60. Correct. Pam's discount is 30% off each ticket. So, 20 * 0.70 = 14 per ticket. 5 tickets is 14 * 5 = 70. Correct. So, difference is 70 - 60 = 10. So, answer is 10. So, \boxed{10}?

Wait, but let me check if there's any trick here. Sometimes problems can have tricks, like maybe the discount is applied differently. For example, maybe the coupon is for a certain amount, but the problem says "a coupon that gives her a 25% discount", so that's a percentage discount. So, that should be straightforward. So, I think my calculation is correct.

Alternatively, maybe I need to check if the question is asking for how many more dollars Pam pays than Susan, so if Pam pays more, then the answer is positive. Since 70 - 60 is 10, so yes, 10 dollars. So, I think that's correct.

But let me think again. Maybe I made a mistake in calculating the percentages? Let me check:

For Susan: 25% discount on each ticket. So, 25% of 20 is 5, so 20 - 5 = 15. 4 tickets: 15 * 4 = 60. Correct.

For Pam: 30% discount on each ticket. 30% of 20 is 6, so 20 - 6 = 14. 5 tickets: 14 * 5 = 70. Correct. 70 - 60 = 10. So, answer is 10. So, I think that's correct.

Alternatively, maybe the problem is in the interpretation of the discount. For example, if the coupon is a 25% discount on the total, but that's the same as 25% off each ticket. So, same result. So, no difference. So, I think my answer is correct.

Therefore, the answer is 10 dollars. So, Pam pays $10 more than Susan.

**Final Answer**
\boxed{10}

To determine how many more dollars Pam pays than Susan, we need to calculate the total cost each person pays after applying their respective discounts and then find the difference between these amounts.

---

### Step 1: Calculate Susan's Total Cost

- **Full price per ticket**: \$20
- **Discount**: 25% off
- **Discounted price per ticket**:
  $$
  20 \times (1 - 0.25) = 20 \times 0.75 = 15
  $$
- **Number of tickets**: 4
- **Total cost for Susan**:
  $$
  4 \times 15 = 60
  $$

---

### Step 2: Calculate Pam's Total Cost

- **Full price per ticket**: \$20
- **Discount**: 30% off
- **Discounted price per ticket**:
  $$
  20 \times (1 - 0.30) = 20 \times 0.70 = 14
  $$
- **Number of tickets**: 5
- **Total cost for Pam**:
  $$
  5 \times 14 = 70
  $$

---

### Step 3: Find the Difference

- **Pam's total**: \$70
- **Susan's total**: \$60
- **Difference**:
  $$
  70 - 60 = 10
  $$

---

### Final Answer

$$
\boxed{10}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, Susan and Pam are buying tickets to a show, and we need to find out how many more dollars Pam pays than Susan. Let me start by understanding the problem step by step.

First, the full price of one ticket is $20. Susan buys 4 tickets with a 25% discount coupon, and Pam buys 5 tickets with a 30% discount coupon. The question is asking for the difference in the amount they each pay, specifically how much more Pam pays compared to Susan.

Alright, so I need to calculate the total amount Susan pays after applying her discount, then do the same for Pam, and then subtract Susan's total from Pam's total to find the difference. Let me write that down.

Starting with Susan. She buys 4 tickets. The original price per ticket is $20, so without any discount, she would pay 4 times $20. But she has a 25% discount. So, first, maybe I should calculate the discount amount per ticket and then subtract that from the original price? Or maybe calculate the discounted price per ticket first?

Hmm, I think it's better to calculate the discounted price per ticket first. Let me recall that a 25% discount means she pays 75% of the original price. Because 100% - 25% = 75%. So, if each ticket is $20, then the discounted price per ticket is 25% off, so 75% of $20. Let me compute that.

So, 25% of $20 is 0.25 * 20 = $5. So, the discount per ticket is $5. Therefore, the discounted price per ticket is $20 - $5 = $15. Alternatively, 75% of $20 is 0.75 * 20 = $15. Either way, same result. So, Susan pays $15 per ticket. Since she buys 4 tickets, her total cost is 4 * $15. Let me calculate that. 4 * 15 is 60. So, Susan pays $60 in total.

Now, moving on to Pam. She buys 5 tickets with a 30% discount. Let me do the same process. First, the original price per ticket is $20. A 30% discount means she pays 70% of the original price. Let me check that. 100% - 30% = 70%. So, the discounted price per ticket is 70% of $20. Let me compute that. 0.70 * 20. Hmm, 0.7 * 20. Well, 0.7 * 20 is 14. So, each ticket costs her $14 after the discount. Therefore, for 5 tickets, her total cost is 5 * $14. Let me calculate that. 5 * 14 is 70. So, Pam pays $70 in total.

Wait, hold on. Let me verify that again. If the discount is 30%, then the amount she saves per ticket is 30% of $20, which is 0.30 * 20 = $6. Therefore, the discounted price per ticket is $20 - $6 = $14. Yep, same as before. So, 5 tickets would be 5 * 14 = $70. So, Pam pays $70.

So, Susan pays $60, Pam pays $70. The question is asking how many more dollars Pam pays than Susan. So, that would be Pam's total minus Susan's total. So, $70 - $60 = $10. Therefore, Pam pays $10 more than Susan. Is that the answer? Wait, let me check again to make sure I didn't make any calculation errors.

Let me recalculate Susan's total. 4 tickets at 25% off. Original total without discount: 4 * 20 = $80. Then, 25% discount on $80. Wait, hold on, maybe I should check if the discount is applied per ticket or on the total? The problem says Susan buys 4 tickets using a coupon that gives her a 25% discount. So, the coupon is probably applied per ticket, but sometimes discounts can be applied to the total. But in most cases, unless specified otherwise, a discount coupon would be applied per item. However, let me check both ways to be thorough.

First, if the discount is per ticket: as I did before, each ticket is $15, so 4 tickets is $60. Alternatively, if the discount is applied to the total amount. So, total original price is 4 * 20 = $80. Then, 25% discount on $80 is 0.25 * 80 = $20 discount. Therefore, total paid is $80 - $20 = $60. Same result. So, either way, Susan pays $60. So that's correct.

Similarly for Pam. Original total for 5 tickets is 5 * 20 = $100. 30% discount on $100 is 0.30 * 100 = $30 discount. Therefore, total paid is $100 - $30 = $70. Alternatively, per ticket discount: 30% off each ticket, so each ticket is 70% of $20, which is $14, so 5 tickets is 5 * 14 = $70. Same result. So, Pam pays $70. Therefore, the difference is $70 - $60 = $10. So, Pam pays $10 more than Susan.

Wait, but let me think again. The problem says "how many more dollars does Pam pay than Susan?" So, if Pam pays $70 and Susan pays $60, then the answer is $10. But let me check once more if I didn't mix up the numbers.

Wait, Susan's discount is 25%, so 25% off 4 tickets. Let me compute 25% of 4 tickets. Wait, no, the discount is per ticket. So, 25% off each ticket. So, each ticket is 20 * 0.75 = 15. 4 tickets is 15 * 4 = 60. Correct. Pam's discount is 30% off each ticket. So, 20 * 0.70 = 14 per ticket. 5 tickets is 14 * 5 = 70. Correct. So, difference is 70 - 60 = 10. So, answer is 10. So, \boxed{10}?

Wait, but let me check if there's any trick here. Sometimes problems can have tricks, like maybe the discount is applied differently. For example, maybe the coupon is for a certain amount, but the problem says "a coupon that gives her a 25% discount", so that's a percentage discount. So, that should be straightforward. So, I think my calculation is correct.

Alternatively, maybe I need to check if the question is asking for how many more dollars Pam pays than Susan, so if Pam pays more, then the answer is positive. Since 70 - 60 is 10, so yes, 10 dollars. So, I think that's correct.

But let me think again. Maybe I made a mistake in calculating the percentages? Let me check:

For Susan: 25% discount on each ticket. So, 25% of 20 is 5, so 20 - 5 = 15. 4 tickets: 15 * 4 = 60. Correct.

For Pam: 30% discount on each ticket. 30% of 20 is 6, so 20 - 6 = 14. 5 tickets: 14 * 5 = 70. Correct. 70 - 60 = 10. So, answer is 10. So, I think that's correct.

Alternatively, maybe the problem is in the interpretation of the discount. For example, if the coupon is a 25% discount on the total, but that's the same as 25% off each ticket. So, same result. So, no difference. So, I think my answer is correct.

Therefore, the answer is 10 dollars. So, Pam pays $10 more than Susan.

**Final Answer**
\boxed{10}

To determine how many more dollars Pam pays than Susan, we need to calculate the total cost each person pays after applying their respective discounts and then find the difference between these amounts.

---

### Step 1: Calculate Susan's Total Cost

- **Full price per ticket**: \$20
- **Discount**: 25% off
- **Discounted price per ticket**:
  $$
  20 \times (1 - 0.25) = 20 \times 0.75 = 15
  $$
- **Number of tickets**: 4
- **Total cost for Susan**:
  $$
  4 \times 15 = 60
  $$

---

### Step 2: Calculate Pam's Total Cost

- **Full price per ticket**: \$20
- **Discount**: 30% off
- **Discounted price per ticket**:
  $$
  20 \times (1 - 0.30) = 20 \times 0.70 = 14
  $$
- **Number of tickets**: 5
- **Total cost for Pam**:
  $$
  5 \times 14 = 70
  $$

---

### Step 3: Find the Difference

- **Pam's total**: \$70
- **Susan's total**: \$60
- **Difference**:
  $$
  70 - 60 = 10
  $$

---

### Final Answer

$$
\boxed{10}
$$

Solution:
We need to calculate the total purchase price that Susan paid and that Pam paid.

Susan bought $4$ tickets with $25\%$ discount: $$4 \times \$20 = \$80.$$With a $25$ percent discount, she paid $\$80 * .75 = \$60.$

Pam bought $5$ tickets with a $30\%$ discount: $$5 \times \$20 = \$100$$With a $30$ percent discount, she paid $\$100 * .70 = \$70.$

Thus Pam paid $\$70 - \$60 = \$\boxed{10}$ more than Susan.


Explanation:
Okay, so I need to solve the inequality 1 divided by (x minus 5) is greater than zero. Let me think about how to approach this.

First, I remember that when dealing with rational inequalities, the key is to find the critical points where the expression might change sign. In this case, the expression is 1/(x - 5). The denominator can't be zero, so x can't be 5. That's going to be an important point to note because the function isn't defined there.

Now, the inequality is 1/(x - 5) > 0. Since the numerator is 1, which is always positive, the sign of the entire expression depends on the denominator. If the denominator is positive, then the whole expression is positive. If the denominator is negative, the whole expression is negative. So, I need to figure out when (x - 5) is positive.

Let me write that down. The expression 1/(x - 5) is positive when (x - 5) > 0. Solving that inequality: x - 5 > 0 implies x > 5. So, does that mean the solution is all real numbers greater than 5? But wait, I need to make sure there are no other intervals where the expression could be positive.

Let me think. The denominator is x - 5. If x is less than 5, then x - 5 is negative, so 1 divided by a negative number is negative. Therefore, the expression is negative when x < 5. At x = 5, the expression is undefined. So, the only interval where the expression is positive is when x > 5.

But let me check if there are any other critical points or if I need to consider anything else. The numerator is 1, which doesn't have any roots, so the only critical point is at x = 5. Therefore, the number line is divided into two intervals: (-∞, 5) and (5, ∞).

Testing the intervals:

1. For x < 5, let's pick a test point, say x = 0. Plugging into the expression: 1/(0 - 5) = 1/(-5) = -1/5, which is negative. So, this interval doesn't satisfy the inequality.

2. For x > 5, let's pick x = 6. Plugging in: 1/(6 - 5) = 1/1 = 1, which is positive. So, this interval satisfies the inequality.

At x = 5, the expression is undefined, so we can't include that point.

Therefore, the solution set is all real numbers greater than 5. In interval notation, that would be (5, ∞).

Wait, but let me make sure I didn't miss anything. Sometimes with inequalities, especially rational ones, there can be more to it. For example, if there were a quadratic in the denominator or something, but here it's linear.

Another way to think about it: the inequality 1/(x - 5) > 0 is equivalent to (x - 5) > 0 because 1 is positive. Since multiplying both sides of an inequality by a positive number doesn't change the direction of the inequality. So, if I multiply both sides by (x - 5), but wait, I have to be careful here because if (x - 5) is negative, multiplying both sides would reverse the inequality. However, since I don't know the sign of (x - 5), maybe this approach isn't straightforward.

Alternatively, maybe I can consider the inequality 1/(x - 5) > 0. Since 1 is positive, the fraction is positive when the denominator is positive. So, again, x - 5 > 0 => x > 5.

Alternatively, if I wanted to solve it algebraically, I could note that the inequality is equivalent to 1/(x - 5) > 0. The expression is positive when both numerator and denominator are positive or both are negative. However, since the numerator is 1, which is always positive, the denominator must also be positive. Therefore, x - 5 > 0 => x > 5.

So, all these methods lead me to the same conclusion.

But let me check if there's a possibility of the expression being positive in another interval. For example, if the denominator was negative, then the whole expression would be negative. So, no.

Another thing to consider: sometimes when solving inequalities, especially with variables in denominators, you have to be careful about the domain. Here, the domain is all real numbers except x = 5. So, the solution set is part of that domain.

Therefore, combining all of this, the solution is x > 5, which in interval notation is (5, ∞).

Wait, but let me think again. Suppose someone might think that since 1/(x - 5) is positive when x approaches 5 from the right, but also when x approaches infinity. But in between, it's negative. So, the only interval where it's positive is (5, ∞).

Alternatively, if I graph the function f(x) = 1/(x - 5), it has a vertical asymptote at x = 5. For x > 5, the function is positive and approaches zero as x approaches infinity. For x < 5, the function is negative and approaches zero from below as x approaches 5 from the left and approaches negative infinity as x approaches negative infinity.

So, the graph confirms that the function is positive only when x > 5.

Therefore, I think I can confidently say that the solution is (5, ∞).

But just to make sure, let me test another point in the interval. Let's take x = 10. Then 1/(10 - 5) = 1/5, which is positive. If I take x = 5.5, 1/(5.5 - 5) = 1/0.5 = 2, which is positive. If I take x = 5.1, 1/(0.1) = 10, still positive. If I take x = 5, it's undefined. If I take x = 4, 1/(4 - 5) = -1, which is negative. So, yes, only x > 5 works.

Therefore, the answer is (5, ∞).

**Final Answer**
The solution to the inequality is \boxed{(5, \infty)}.

To solve the inequality
$$
\frac{1}{x - 5} > 0,
$$
we begin by analyzing the structure of the expression.

---

### Step 1: Identify the Domain

The expression $\frac{1}{x - 5}$ is undefined when the denominator is zero. That occurs when:

$$
x - 5 = 0 \Rightarrow x = 5.
$$

So, the domain of the expression is all real numbers **except** $x = 5$.

---

### Step 2: Determine When the Expression is Positive

We are looking for values of $x$ such that:

$$
\frac{1}{x - 5} > 0.
$$

Since the numerator is 1 (a positive constant), the sign of the entire expression depends **only** on the sign of the denominator $x - 5$.

- If $x - 5 > 0$, then $\frac{1}{x - 5} > 0$.
- If $x - 5 < 0$, then $\frac{1}{x - 5} < 0$.
- If $x - 5 = 0$, the expression is undefined.

So, the inequality is satisfied when:

$$
x - 5 > 0 \Rightarrow x > 5.
$$

---

### Step 3: Write the Solution in Interval Notation

The solution set is all real numbers **greater than 5**, excluding 5 itself. In interval notation, this is:

$$
(5, \infty).
$$

---

### Final Answer

$$
\boxed{(5, \infty)}
$$

[Assistant]
Explanation:
Okay, so I need to solve the inequality 1 divided by (x minus 5) is greater than zero. Let me think about how to approach this.

First, I remember that when dealing with rational inequalities, the key is to find the critical points where the expression might change sign. In this case, the expression is 1/(x - 5). The denominator can't be zero, so x can't be 5. That's going to be an important point to note because the function isn't defined there.

Now, the inequality is 1/(x - 5) > 0. Since the numerator is 1, which is always positive, the sign of the entire expression depends on the denominator. If the denominator is positive, then the whole expression is positive. If the denominator is negative, the whole expression is negative. So, I need to figure out when (x - 5) is positive.

Let me write that down. The expression 1/(x - 5) is positive when (x - 5) > 0. Solving that inequality: x - 5 > 0 implies x > 5. So, does that mean the solution is all real numbers greater than 5? But wait, I need to make sure there are no other intervals where the expression could be positive.

Let me think. The denominator is x - 5. If x is less than 5, then x - 5 is negative, so 1 divided by a negative number is negative. Therefore, the expression is negative when x < 5. At x = 5, the expression is undefined. So, the only interval where the expression is positive is when x > 5.

But let me check if there are any other critical points or if I need to consider anything else. The numerator is 1, which doesn't have any roots, so the only critical point is at x = 5. Therefore, the number line is divided into two intervals: (-∞, 5) and (5, ∞).

Testing the intervals:

1. For x < 5, let's pick a test point, say x = 0. Plugging into the expression: 1/(0 - 5) = 1/(-5) = -1/5, which is negative. So, this interval doesn't satisfy the inequality.

2. For x > 5, let's pick x = 6. Plugging in: 1/(6 - 5) = 1/1 = 1, which is positive. So, this interval satisfies the inequality.

At x = 5, the expression is undefined, so we can't include that point.

Therefore, the solution set is all real numbers greater than 5. In interval notation, that would be (5, ∞).

Wait, but let me make sure I didn't miss anything. Sometimes with inequalities, especially rational ones, there can be more to it. For example, if there were a quadratic in the denominator or something, but here it's linear.

Another way to think about it: the inequality 1/(x - 5) > 0 is equivalent to (x - 5) > 0 because 1 is positive. Since multiplying both sides of an inequality by a positive number doesn't change the direction of the inequality. So, if I multiply both sides by (x - 5), but wait, I have to be careful here because if (x - 5) is negative, multiplying both sides would reverse the inequality. However, since I don't know the sign of (x - 5), maybe this approach isn't straightforward.

Alternatively, maybe I can consider the inequality 1/(x - 5) > 0. Since 1 is positive, the fraction is positive when the denominator is positive. So, again, x - 5 > 0 => x > 5.

Alternatively, if I wanted to solve it algebraically, I could note that the inequality is equivalent to 1/(x - 5) > 0. The expression is positive when both numerator and denominator are positive or both are negative. However, since the numerator is 1, which is always positive, the denominator must also be positive. Therefore, x - 5 > 0 => x > 5.

So, all these methods lead me to the same conclusion.

But let me check if there's a possibility of the expression being positive in another interval. For example, if the denominator was negative, then the whole expression would be negative. So, no.

Another thing to consider: sometimes when solving inequalities, especially with variables in denominators, you have to be careful about the domain. Here, the domain is all real numbers except x = 5. So, the solution set is part of that domain.

Therefore, combining all of this, the solution is x > 5, which in interval notation is (5, ∞).

Wait, but let me think again. Suppose someone might think that since 1/(x - 5) is positive when x approaches 5 from the right, but also when x approaches infinity. But in between, it's negative. So, the only interval where it's positive is (5, ∞).

Alternatively, if I graph the function f(x) = 1/(x - 5), it has a vertical asymptote at x = 5. For x > 5, the function is positive and approaches zero as x approaches infinity. For x < 5, the function is negative and approaches zero from below as x approaches 5 from the left and approaches negative infinity as x approaches negative infinity.

So, the graph confirms that the function is positive only when x > 5.

Therefore, I think I can confidently say that the solution is (5, ∞).

But just to make sure, let me test another point in the interval. Let's take x = 10. Then 1/(10 - 5) = 1/5, which is positive. If I take x = 5.5, 1/(5.5 - 5) = 1/0.5 = 2, which is positive. If I take x = 5.1, 1/(0.1) = 10, still positive. If I take x = 5, it's undefined. If I take x = 4, 1/(4 - 5) = -1, which is negative. So, yes, only x > 5 works.

Therefore, the answer is (5, ∞).

**Final Answer**
The solution to the inequality is \boxed{(5, \infty)}.

To solve the inequality
$$
\frac{1}{x - 5} > 0,
$$
we begin by analyzing the structure of the expression.

---

### Step 1: Identify the Domain

The expression $\frac{1}{x - 5}$ is undefined when the denominator is zero. That occurs when:

$$
x - 5 = 0 \Rightarrow x = 5.
$$

So, the domain of the expression is all real numbers **except** $x = 5$.

---

### Step 2: Determine When the Expression is Positive

We are looking for values of $x$ such that:

$$
\frac{1}{x - 5} > 0.
$$

Since the numerator is 1 (a positive constant), the sign of the entire expression depends **only** on the sign of the denominator $x - 5$.

- If $x - 5 > 0$, then $\frac{1}{x - 5} > 0$.
- If $x - 5 < 0$, then $\frac{1}{x - 5} < 0$.
- If $x - 5 = 0$, the expression is undefined.

So, the inequality is satisfied when:

$$
x - 5 > 0 \Rightarrow x > 5.
$$

---

### Step 3: Write the Solution in Interval Notation

The solution set is all real numbers **greater than 5**, excluding 5 itself. In interval notation, this is:

$$
(5, \infty).
$$

---

### Final Answer

$$
\boxed{(5, \infty)}
$$

Solution: If $x < 5,$ then $\frac{1}{x - 5} < 0,$ and if $x > 5,$ then $\frac{1}{x - 5} > 0,$ so the solution is $x \in \boxed{(5,\infty)}.$


Explanation:
Okay, so I need to find the value of 513 squared minus 487 squared. Hmm, let me think. I remember there's a formula for the difference of squares, right? It's something like a squared minus b squared equals (a minus b)(a plus b). Let me check if I remember that correctly. Yeah, I think that's right. So if I can apply that here, it might make the calculation easier instead of having to compute each square separately, which could be time-consuming and prone to errors.

Alright, so applying the formula, 513 squared minus 487 squared would be equal to (513 - 487)(513 + 487). Let me write that down:

513² - 487² = (513 - 487)(513 + 487)

Now, let me compute each part step by step. First, let's find 513 minus 487. Let me do that subtraction. 513 minus 487... Let me subtract the numbers. 513 minus 400 is 113, then minus 87 more. Wait, maybe it's easier to do it directly. 513 minus 487. Let me subtract the units digit: 3 - 7. Hmm, can't do that, so I need to borrow. Let me write it out:

513
-487
------
Starting from the rightmost digit: 3 - 7. Since 3 is less than 7, borrow 1 from the tens place. The tens digit is 1, so it becomes 0, and the units digit becomes 13. Then 13 - 7 = 6. Next, the tens place: 0 (after borrowing) minus 8. Again, can't do that, so need to borrow from the hundreds place. The hundreds digit is 5, so it becomes 4, and the tens digit becomes 10. Then 10 - 8 = 2. Finally, the hundreds place: 4 - 4 = 0. Wait, so the result is 26? Let me check with another method. 487 + 26 = 513? Let me add 487 + 20 = 507, then +6 = 513. Yes, that's correct. So 513 - 487 = 26.

Okay, that was straightforward. Now the other part is 513 + 487. Let me compute that. Adding 513 and 487. Let me add the units digits: 3 + 7 = 10. Carry over 1. Then the tens digits: 1 + 8 = 9, plus the carried 1 makes 10. Carry over 1 again. Then the hundreds digits: 5 + 4 = 9, plus the carried 1 makes 10. So the total is 1000? Wait, let me check again. 513 + 487. Let me think of 500 + 400 = 900, 13 + 87 = 100. So total is 900 + 100 = 1000. Yes, that's correct. So 513 + 487 = 1000. That's a nice round number, which makes sense because 513 and 487 are both 1000 - 487 and 1000 - 513? Wait, no, maybe not. Anyway, the sum is 1000. So that's helpful.

So putting it all together, the original expression becomes 26 multiplied by 1000. Which is 26,000. Wait, so 26 * 1000 is 26,000? Let me confirm. Yes, multiplying by 1000 just adds three zeros, so 26 * 1000 = 26,000. Therefore, 513 squared minus 487 squared equals 26,000.

But just to be thorough, maybe I should check this another way to make sure I didn't make a mistake. Let me try calculating each square separately and then subtracting. Although that might take longer, but it's a good way to verify.

First, let's compute 513 squared. Hmm, 513 squared. Let me recall that (a + b)^2 = a² + 2ab + b². Maybe I can break down 513 into 500 + 13. So:

(500 + 13)² = 500² + 2*500*13 + 13²

Calculating each term:

500² = 250,000

2*500*13 = 1000*13 = 13,000

13² = 169

Adding them up: 250,000 + 13,000 = 263,000; 263,000 + 169 = 263,169. So 513 squared is 263,169.

Now let's compute 487 squared. Similarly, maybe break it down. Let me think. 487 is 500 - 13. So:

(500 - 13)² = 500² - 2*500*13 + 13²

Which is the same as:

250,000 - 13,000 + 169

Calculating that: 250,000 - 13,000 = 237,000; 237,000 + 169 = 237,169. Therefore, 487 squared is 237,169.

Now subtract the two results: 263,169 - 237,169. Let me do that subtraction.

263,169 minus 237,169. Let me subtract the numbers:

Starting from the right, 9 - 9 = 0. Then 6 - 6 = 0. Then 1 - 1 = 0. Then 3 - 7. Wait, hold on. Wait, let me write them out:

263,169
-237,169
------------

Subtracting digit by digit:

Units place: 9 - 9 = 0
Tens place: 6 - 6 = 0
Hundreds place: 1 - 1 = 0
Thousands place: 3 - 7. Hmm, 3 is less than 7, so need to borrow. Wait, let me check the numbers again. Wait, maybe I should write them properly:

The first number is 263,169. Let me break it down as 2 6 3 , 1 6 9. The second number is 237,169, which is 2 3 7 , 1 6 9. So aligning them:

  2 6 3 , 1 6 9
- 2 3 7 , 1 6 9
------------------

Starting from the rightmost digit:

Units: 9 - 9 = 0
Tens: 6 - 6 = 0
Hundreds: 1 - 1 = 0
Thousands: 3 - 7. Hmm, 3 is less than 7. So need to borrow from the ten-thousands place. The ten-thousands place is 6 (in 263,169). So borrow 1 from there, making it 5, and the thousands place becomes 13. Then 13 - 7 = 6.
Now, the ten-thousands place: 5 (after borrowing) minus 3 (from the second number's ten-thousands place). Wait, the second number is 237,169. Let me check the digits again. The second number is 2 (hundred thousands), 3 (ten thousands), 7 (thousands), 1 (hundreds), 6 (tens), 9 (units). The first number is 2 (hundred thousands), 6 (ten thousands), 3 (thousands), 1 (hundreds), 6 (tens), 9 (units). So when we subtract:

Hundred thousands: 2 - 2 = 0
Ten thousands: 6 - 3 = 3, but wait, we had borrowed 1 earlier? Wait, no. Wait, when we did the thousands place, we borrowed 1 from the ten-thousands place. So originally, the ten-thousands digit was 6, then we borrowed 1, making it 5. So now, in the ten-thousands place, we have 5 - 3 = 2? Wait, no. Wait, let me retrace.

After borrowing for the thousands place, the ten-thousands digit of the first number is 5. The second number's ten-thousands digit is 3. So 5 - 3 = 2. Then the hundred thousands place is 2 - 2 = 0. So putting it all together, the result is 26,000? Wait, let me write the result:

Starting from the hundred thousands: 0
Ten thousands: 2
Thousands: 6
Hundreds: 0
Tens: 0
Units: 0
So that's 26,000. Yes! So 263,169 - 237,169 = 26,000. Which matches the previous result. So that confirms that the answer is indeed 26,000.

Therefore, using both methods—applying the difference of squares formula and calculating each square separately—both lead to the same result. That gives me confidence that the answer is correct.

Alternatively, another way to think about it is recognizing that 513 and 487 are both numbers that are 1000 apart? Wait, no. Wait, 513 + 487 = 1000, as we saw earlier. So they are symmetric around 500. Because 500 + 513 = 1013? Wait, no. Wait, 513 is 13 more than 500, and 487 is 13 less than 500. Wait, 500 - 13 = 487. Yes! So 513 = 500 + 13, and 487 = 500 - 13. Therefore, their squares difference can be calculated as (a + b)^2 - (a - b)^2, where a = 500 and b = 13.

Let me compute that. (a + b)^2 - (a - b)^2. Expanding both:

(a² + 2ab + b²) - (a² - 2ab + b²) = a² + 2ab + b² - a² + 2ab - b² = 4ab.

Oh! So the difference of squares when the numbers are a + b and a - b is 4ab. Therefore, in this case, 4 * 500 * 13. Let me compute that.

First, 4 * 500 = 2000. Then 2000 * 13. Let me calculate 2000 * 10 = 20,000 and 2000 * 3 = 6,000. Adding them together gives 20,000 + 6,000 = 26,000. So that's another way to see it. Therefore, regardless of the method, the answer is 26,000.

This third method is even more straightforward. Recognizing that the two numbers are equidistant from 500, so using the identity for the difference of squares when expressed as (a + b)^2 - (a - b)^2 = 4ab. That's a nice formula to remember.

So all three methods—direct calculation, difference of squares formula, and the symmetric difference around a midpoint—lead to the same answer. Therefore, I can be confident that 513² - 487² equals 26,000.

Just to recap, the key steps were:

1. Recognize the difference of squares formula.
2. Apply the formula by calculating (513 - 487) and (513 + 487).
3. Simplify those differences to 26 and 1000, multiply them to get 26,000.
4. Verify by expanding both squares and subtracting, which also gave 26,000.
5. Use an alternative approach by expressing the numbers as a + b and a - b, leading to 4ab = 26,000.

All methods concur. Therefore, the answer is definitely 26,000.

I think that's thorough enough. I don't see any mistakes in my reasoning, and multiple methods confirm the result. So I can safely conclude the value is 26,000.

**Final Answer**
The value of $513^2 - 487^2$ is \boxed{26000}.

To evaluate the expression $ 513^2 - 487^2 $, we can take advantage of the **difference of squares** identity:

$$
a^2 - b^2 = (a - b)(a + b)
$$

---

### Step 1: Apply the Identity

Let $ a = 513 $ and $ b = 487 $. Then:

$$
513^2 - 487^2 = (513 - 487)(513 + 487)
$$

---

### Step 2: Compute the Differences

- $ 513 - 487 = 26 $
- $ 513 + 487 = 1000 $

So the expression becomes:

$$
(513 - 487)(513 + 487) = 26 \times 1000 = 26000
$$

---

### Step 3: Verification via Alternative Method

We can also verify this by expanding both squares:

- $ 513^2 = (500 + 13)^2 = 500^2 + 2 \cdot 500 \cdot 13 + 13^2 = 250000 + 13000 + 169 = 263169 $
- $ 487^2 = (500 - 13)^2 = 500^2 - 2 \cdot 500 \cdot 13 + 13^2 = 250000 - 13000 + 169 = 237169 $

Now subtract:

$$
263169 - 237169 = 26000
$$

---

### Step 4: Alternative Insight

Recognizing that $ 513 = 500 + 13 $ and $ 487 = 500 - 13 $, we can use the identity:

$$
(a + b)^2 - (a - b)^2 = 4ab
$$

Here, $ a = 500 $, $ b = 13 $, so:

$$
4 \cdot 500 \cdot 13 = 2000 \cdot 13 = 26000
$$

---

### Final Answer

$$
\boxed{26000}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of 513 squared minus 487 squared. Hmm, let me think. I remember there's a formula for the difference of squares, right? It's something like a squared minus b squared equals (a minus b)(a plus b). Let me check if I remember that correctly. Yeah, I think that's right. So if I can apply that here, it might make the calculation easier instead of having to compute each square separately, which could be time-consuming and prone to errors.

Alright, so applying the formula, 513 squared minus 487 squared would be equal to (513 - 487)(513 + 487). Let me write that down:

513² - 487² = (513 - 487)(513 + 487)

Now, let me compute each part step by step. First, let's find 513 minus 487. Let me do that subtraction. 513 minus 487... Let me subtract the numbers. 513 minus 400 is 113, then minus 87 more. Wait, maybe it's easier to do it directly. 513 minus 487. Let me subtract the units digit: 3 - 7. Hmm, can't do that, so I need to borrow. Let me write it out:

513
-487
------
Starting from the rightmost digit: 3 - 7. Since 3 is less than 7, borrow 1 from the tens place. The tens digit is 1, so it becomes 0, and the units digit becomes 13. Then 13 - 7 = 6. Next, the tens place: 0 (after borrowing) minus 8. Again, can't do that, so need to borrow from the hundreds place. The hundreds digit is 5, so it becomes 4, and the tens digit becomes 10. Then 10 - 8 = 2. Finally, the hundreds place: 4 - 4 = 0. Wait, so the result is 26? Let me check with another method. 487 + 26 = 513? Let me add 487 + 20 = 507, then +6 = 513. Yes, that's correct. So 513 - 487 = 26.

Okay, that was straightforward. Now the other part is 513 + 487. Let me compute that. Adding 513 and 487. Let me add the units digits: 3 + 7 = 10. Carry over 1. Then the tens digits: 1 + 8 = 9, plus the carried 1 makes 10. Carry over 1 again. Then the hundreds digits: 5 + 4 = 9, plus the carried 1 makes 10. So the total is 1000? Wait, let me check again. 513 + 487. Let me think of 500 + 400 = 900, 13 + 87 = 100. So total is 900 + 100 = 1000. Yes, that's correct. So 513 + 487 = 1000. That's a nice round number, which makes sense because 513 and 487 are both 1000 - 487 and 1000 - 513? Wait, no, maybe not. Anyway, the sum is 1000. So that's helpful.

So putting it all together, the original expression becomes 26 multiplied by 1000. Which is 26,000. Wait, so 26 * 1000 is 26,000? Let me confirm. Yes, multiplying by 1000 just adds three zeros, so 26 * 1000 = 26,000. Therefore, 513 squared minus 487 squared equals 26,000.

But just to be thorough, maybe I should check this another way to make sure I didn't make a mistake. Let me try calculating each square separately and then subtracting. Although that might take longer, but it's a good way to verify.

First, let's compute 513 squared. Hmm, 513 squared. Let me recall that (a + b)^2 = a² + 2ab + b². Maybe I can break down 513 into 500 + 13. So:

(500 + 13)² = 500² + 2*500*13 + 13²

Calculating each term:

500² = 250,000

2*500*13 = 1000*13 = 13,000

13² = 169

Adding them up: 250,000 + 13,000 = 263,000; 263,000 + 169 = 263,169. So 513 squared is 263,169.

Now let's compute 487 squared. Similarly, maybe break it down. Let me think. 487 is 500 - 13. So:

(500 - 13)² = 500² - 2*500*13 + 13²

Which is the same as:

250,000 - 13,000 + 169

Calculating that: 250,000 - 13,000 = 237,000; 237,000 + 169 = 237,169. Therefore, 487 squared is 237,169.

Now subtract the two results: 263,169 - 237,169. Let me do that subtraction.

263,169 minus 237,169. Let me subtract the numbers:

Starting from the right, 9 - 9 = 0. Then 6 - 6 = 0. Then 1 - 1 = 0. Then 3 - 7. Wait, hold on. Wait, let me write them out:

263,169
-237,169
------------

Subtracting digit by digit:

Units place: 9 - 9 = 0
Tens place: 6 - 6 = 0
Hundreds place: 1 - 1 = 0
Thousands place: 3 - 7. Hmm, 3 is less than 7, so need to borrow. Wait, let me check the numbers again. Wait, maybe I should write them properly:

The first number is 263,169. Let me break it down as 2 6 3 , 1 6 9. The second number is 237,169, which is 2 3 7 , 1 6 9. So aligning them:

  2 6 3 , 1 6 9
- 2 3 7 , 1 6 9
------------------

Starting from the rightmost digit:

Units: 9 - 9 = 0
Tens: 6 - 6 = 0
Hundreds: 1 - 1 = 0
Thousands: 3 - 7. Hmm, 3 is less than 7. So need to borrow from the ten-thousands place. The ten-thousands place is 6 (in 263,169). So borrow 1 from there, making it 5, and the thousands place becomes 13. Then 13 - 7 = 6.
Now, the ten-thousands place: 5 (after borrowing) minus 3 (from the second number's ten-thousands place). Wait, the second number is 237,169. Let me check the digits again. The second number is 2 (hundred thousands), 3 (ten thousands), 7 (thousands), 1 (hundreds), 6 (tens), 9 (units). The first number is 2 (hundred thousands), 6 (ten thousands), 3 (thousands), 1 (hundreds), 6 (tens), 9 (units). So when we subtract:

Hundred thousands: 2 - 2 = 0
Ten thousands: 6 - 3 = 3, but wait, we had borrowed 1 earlier? Wait, no. Wait, when we did the thousands place, we borrowed 1 from the ten-thousands place. So originally, the ten-thousands digit was 6, then we borrowed 1, making it 5. So now, in the ten-thousands place, we have 5 - 3 = 2? Wait, no. Wait, let me retrace.

After borrowing for the thousands place, the ten-thousands digit of the first number is 5. The second number's ten-thousands digit is 3. So 5 - 3 = 2. Then the hundred thousands place is 2 - 2 = 0. So putting it all together, the result is 26,000? Wait, let me write the result:

Starting from the hundred thousands: 0
Ten thousands: 2
Thousands: 6
Hundreds: 0
Tens: 0
Units: 0
So that's 26,000. Yes! So 263,169 - 237,169 = 26,000. Which matches the previous result. So that confirms that the answer is indeed 26,000.

Therefore, using both methods—applying the difference of squares formula and calculating each square separately—both lead to the same result. That gives me confidence that the answer is correct.

Alternatively, another way to think about it is recognizing that 513 and 487 are both numbers that are 1000 apart? Wait, no. Wait, 513 + 487 = 1000, as we saw earlier. So they are symmetric around 500. Because 500 + 513 = 1013? Wait, no. Wait, 513 is 13 more than 500, and 487 is 13 less than 500. Wait, 500 - 13 = 487. Yes! So 513 = 500 + 13, and 487 = 500 - 13. Therefore, their squares difference can be calculated as (a + b)^2 - (a - b)^2, where a = 500 and b = 13.

Let me compute that. (a + b)^2 - (a - b)^2. Expanding both:

(a² + 2ab + b²) - (a² - 2ab + b²) = a² + 2ab + b² - a² + 2ab - b² = 4ab.

Oh! So the difference of squares when the numbers are a + b and a - b is 4ab. Therefore, in this case, 4 * 500 * 13. Let me compute that.

First, 4 * 500 = 2000. Then 2000 * 13. Let me calculate 2000 * 10 = 20,000 and 2000 * 3 = 6,000. Adding them together gives 20,000 + 6,000 = 26,000. So that's another way to see it. Therefore, regardless of the method, the answer is 26,000.

This third method is even more straightforward. Recognizing that the two numbers are equidistant from 500, so using the identity for the difference of squares when expressed as (a + b)^2 - (a - b)^2 = 4ab. That's a nice formula to remember.

So all three methods—direct calculation, difference of squares formula, and the symmetric difference around a midpoint—lead to the same answer. Therefore, I can be confident that 513² - 487² equals 26,000.

Just to recap, the key steps were:

1. Recognize the difference of squares formula.
2. Apply the formula by calculating (513 - 487) and (513 + 487).
3. Simplify those differences to 26 and 1000, multiply them to get 26,000.
4. Verify by expanding both squares and subtracting, which also gave 26,000.
5. Use an alternative approach by expressing the numbers as a + b and a - b, leading to 4ab = 26,000.

All methods concur. Therefore, the answer is definitely 26,000.

I think that's thorough enough. I don't see any mistakes in my reasoning, and multiple methods confirm the result. So I can safely conclude the value is 26,000.

**Final Answer**
The value of $513^2 - 487^2$ is \boxed{26000}.

To evaluate the expression $ 513^2 - 487^2 $, we can take advantage of the **difference of squares** identity:

$$
a^2 - b^2 = (a - b)(a + b)
$$

---

### Step 1: Apply the Identity

Let $ a = 513 $ and $ b = 487 $. Then:

$$
513^2 - 487^2 = (513 - 487)(513 + 487)
$$

---

### Step 2: Compute the Differences

- $ 513 - 487 = 26 $
- $ 513 + 487 = 1000 $

So the expression becomes:

$$
(513 - 487)(513 + 487) = 26 \times 1000 = 26000
$$

---

### Step 3: Verification via Alternative Method

We can also verify this by expanding both squares:

- $ 513^2 = (500 + 13)^2 = 500^2 + 2 \cdot 500 \cdot 13 + 13^2 = 250000 + 13000 + 169 = 263169 $
- $ 487^2 = (500 - 13)^2 = 500^2 - 2 \cdot 500 \cdot 13 + 13^2 = 250000 - 13000 + 169 = 237169 $

Now subtract:

$$
263169 - 237169 = 26000
$$

---

### Step 4: Alternative Insight

Recognizing that $ 513 = 500 + 13 $ and $ 487 = 500 - 13 $, we can use the identity:

$$
(a + b)^2 - (a - b)^2 = 4ab
$$

Here, $ a = 500 $, $ b = 13 $, so:

$$
4 \cdot 500 \cdot 13 = 2000 \cdot 13 = 26000
$$

---

### Final Answer

$$
\boxed{26000}
$$

Solution: We note that this is a difference of squares, so $513^2 - 487^2 = (513+487)(513-487) = (1000)(26) = \boxed{26000}$.


Explanation:
Okay, so there's this problem about two candidates running for class president: Dan and Donald. Then there are two other candidates for vice president: Freddie and Bernie. Each candidate has a 50% chance of winning. The question is asking for the probability that both Dan and Freddie win. Hmm, let me think about how to approach this.

First, I need to make sure I understand the problem correctly. There are two separate elections here: one for president and one for vice president. Each has two candidates. For president, it's Dan vs. Donald. For vice president, it's Freddie vs. Bernie. Each candidate has a 50% chance of winning their respective election. So, the question is asking for the probability that Dan wins the presidency and Freddie wins the vice presidency. Is that right? The wording says "both Dan and Freddie win," so I think that's what it means. So, we need the probability that Dan is president and Freddie is vice president.

Since these are separate elections, I think the outcomes are independent. That is, the result of the presidential election doesn't affect the vice presidential election, and vice versa. So, if that's the case, then the probability of both events happening would be the product of their individual probabilities.

So, the probability that Dan wins the presidency is 50%, which is 1/2. Similarly, the probability that Freddie wins the vice presidency is also 50%, which is 1/2. Therefore, the probability that both happen is 1/2 * 1/2 = 1/4. So, 1/4? That seems straightforward, but let me check if there's something I'm missing here.

Wait, the problem says "each candidate has a 50% chance of winning." Does that mean that each candidate has a 50% chance regardless of the other candidates? For example, in the presidential election, Dan and Donald are the only candidates, so each has a 50% chance. Similarly, for vice president, Freddie and Bernie are the only candidates, so each has a 50% chance. Therefore, the two events are independent, so multiplying them is correct.

Alternatively, maybe there's some trick here. Let me think again. Suppose that the elections are not independent? But the problem doesn't mention any dependence between them. It just says each candidate has a 50% chance. So, I think independence is a safe assumption here.

Alternatively, maybe the problem is considering that there are four candidates in total? Wait, no. The problem says two candidates run for president (Dan and Donald) and two other candidates run for vice president (Freddie and Bernie). So, they are separate positions. So, the president and vice president are different positions, so the elections are separate. Therefore, the outcomes are independent.

Therefore, the probability that Dan wins president is 1/2, the probability that Freddie wins vice president is 1/2, so the probability that both happen is 1/2 * 1/2 = 1/4. Therefore, the answer is 1/4. But let me think again if there's another way to model this.

Alternatively, maybe we can think of all possible outcomes. For the president, there are two possibilities: Dan or Donald. For vice president, there are two possibilities: Freddie or Bernie. Therefore, the total number of possible outcomes is 2 * 2 = 4. Each outcome is equally likely, since each candidate has a 50% chance. Therefore, the possible outcomes are:

1. Dan (President) and Freddie (Vice President)
2. Dan (President) and Bernie (Vice President)
3. Donald (President) and Freddie (Vice President)
4. Donald (President) and Bernie (Vice President)

Each of these four outcomes has a probability of 1/4. Therefore, the probability that both Dan and Freddie win is 1/4. So, that's the same answer as before. Therefore, 1/4 is the probability.

Wait, but the problem says "each candidate has a 50% chance of winning." So, does that mean that each candidate has a 50% chance regardless of the other candidates? For example, in the presidential election, Dan has 50% chance, Donald has 50% chance. Similarly, Freddie has 50% chance, Bernie has 50% chance. Therefore, the combined probability is 1/2 * 1/2 = 1/4. So, that seems to check out.

Alternatively, if the problem had said that each candidate has a 50% chance of winning against the others, but that's already the case here because there are only two candidates in each election. So, each has a 50% chance. Therefore, the answer is 1/4.

But let me think again if there's a different interpretation. For example, maybe the problem is considering that there are four candidates in total, and they are electing both president and vice president? But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, they are separate positions. So, president and vice president are separate elections. Therefore, the answer is 1/4.

Alternatively, maybe the problem is considering that the same person can't be both president and vice president? But the problem doesn't mention that. It just says two candidates for president and two for vice president. So, unless there's a rule that a person can't hold both positions, but since they are different positions, maybe they can. However, the problem doesn't specify any such restriction, so we can assume that the elections are independent. Therefore, the answer is 1/4.

Alternatively, maybe the problem is in the way the question is phrased. It says "each candidate has a 50% chance of winning." So, maybe that's not referring to each candidate in each election, but each candidate overall? Wait, but there are four candidates: Dan, Donald, Freddie, Bernie. But they are running for different positions. So, maybe the problem is that each candidate has a 50% chance of winning their respective position? Which is the same as before.

Alternatively, maybe the problem is that each candidate has a 50% chance of winning the entire election? But that doesn't make much sense. Because there are two positions. But the problem says "each candidate has a 50% chance of winning." So, maybe the problem is that for each position, each candidate has a 50% chance? Which is the same as before.

Alternatively, maybe the problem is that there is a single election where each candidate is running for both positions? But that seems unlikely. The problem says "run for class president" and "run for vice president," so they are separate.

Therefore, I think my original reasoning is correct. The answer is 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

But let me check once more. Let me think of the sample space. For president, two possibilities: Dan or Donald. For vice president, two possibilities: Freddie or Bernie. So, the total number of possible outcomes is 2 * 2 = 4. Each outcome is equally likely. Therefore, the probability of each outcome is 1/4. Therefore, the probability that Dan is president and Freddie is vice president is 1/4. Therefore, the answer is 1/4.

Alternatively, if the problem had said that the candidates are running for both positions, but that's not the case here. So, I think that's solid.

Wait, but maybe the problem is in the wording: "each candidate has a 50% chance of winning." So, maybe the problem is that each candidate has a 50% chance of winning their respective election. So, for example, Dan has 50% chance to win president, Donald 50% chance, Freddie 50% chance to win vice president, Bernie 50% chance. Therefore, the probability that Dan and Freddie both win is 0.5 * 0.5 = 0.25, which is 1/4. So, same answer.

Alternatively, if the problem had said that each candidate has a 50% chance of winning the entire election, but that would be different. But that's not the case here. The problem says "each candidate has a 50% chance of winning." Since there are two candidates for president, and two for vice president, it's natural to assume that each has a 50% chance in their respective elections.

Therefore, I think the answer is 1/4. Therefore, the probability is 1/4.

But just to make sure, let me think of another approach. Suppose we model the probability as follows. The probability that Dan wins is 1/2. Given that Dan wins, what is the probability that Freddie wins? Since the elections are independent, the probability that Freddie wins is still 1/2. Therefore, the conditional probability is 1/2, so the joint probability is 1/2 * 1/2 = 1/4. So, same answer.

Alternatively, if there was some dependence, like if the president and vice president had to be different people, but again, the problem doesn't mention that. If that were the case, then we would have to adjust the probabilities. For example, if Dan is president, then Freddie can still be vice president, so that's allowed. So, even if there was a rule that you can't be both president and vice president, since Dan and Freddie are different people, that wouldn't affect the probability. Therefore, even with that rule, the probability would still be 1/2 * 1/2 = 1/4.

Therefore, I think the answer is definitely 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

**Final Answer**
The probability that both Dan and Freddie win is \boxed{\dfrac{1}{4}}.

To determine the probability that both **Dan** and **Freddie** win their respective positions, we need to analyze the structure of the election and the independence of the events.

---

### Step 1: Understand the Election Structure

- **President**: Two candidates — **Dan** and **Donald**.
- **Vice President**: Two candidates — **Freddie** and **Bernie**.

Each candidate has a **50% chance of winning** their respective election. This implies:

- **Dan** has a 50% chance of winning the presidency.
- **Freddie** has a 50% chance of winning the vice presidency.

---

### Step 2: Are the Events Independent?

The key assumption here is that the outcome of the **presidential election** does **not affect** the outcome of the **vice presidential election**. Since the two positions are separate and the candidates are distinct, we can reasonably assume that the results are **independent**.

This independence allows us to compute the joint probability of both events occurring by multiplying their individual probabilities.

---

### Step 3: Compute the Joint Probability

- Probability that **Dan wins president**: $ \frac{1}{2} $
- Probability that **Freddie wins vice president**: $ \frac{1}{2} $

Since the events are independent:

$$
P(\text{Dan wins and Freddie wins}) = P(\text{Dan wins}) \times P(\text{Freddie wins}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
$$

---

### Step 4: Final Answer

Thus, the probability that both **Dan** and **Freddie** win their respective positions is:

$$
\boxed{\dfrac{1}{4}}
$$

[Assistant]
Explanation:
Okay, so there's this problem about two candidates running for class president: Dan and Donald. Then there are two other candidates for vice president: Freddie and Bernie. Each candidate has a 50% chance of winning. The question is asking for the probability that both Dan and Freddie win. Hmm, let me think about how to approach this.

First, I need to make sure I understand the problem correctly. There are two separate elections here: one for president and one for vice president. Each has two candidates. For president, it's Dan vs. Donald. For vice president, it's Freddie vs. Bernie. Each candidate has a 50% chance of winning their respective election. So, the question is asking for the probability that Dan wins the presidency and Freddie wins the vice presidency. Is that right? The wording says "both Dan and Freddie win," so I think that's what it means. So, we need the probability that Dan is president and Freddie is vice president.

Since these are separate elections, I think the outcomes are independent. That is, the result of the presidential election doesn't affect the vice presidential election, and vice versa. So, if that's the case, then the probability of both events happening would be the product of their individual probabilities.

So, the probability that Dan wins the presidency is 50%, which is 1/2. Similarly, the probability that Freddie wins the vice presidency is also 50%, which is 1/2. Therefore, the probability that both happen is 1/2 * 1/2 = 1/4. So, 1/4? That seems straightforward, but let me check if there's something I'm missing here.

Wait, the problem says "each candidate has a 50% chance of winning." Does that mean that each candidate has a 50% chance regardless of the other candidates? For example, in the presidential election, Dan and Donald are the only candidates, so each has a 50% chance. Similarly, for vice president, Freddie and Bernie are the only candidates, so each has a 50% chance. Therefore, the two events are independent, so multiplying them is correct.

Alternatively, maybe there's some trick here. Let me think again. Suppose that the elections are not independent? But the problem doesn't mention any dependence between them. It just says each candidate has a 50% chance. So, I think independence is a safe assumption here.

Alternatively, maybe the problem is considering that there are four candidates in total? Wait, no. The problem says two candidates run for president (Dan and Donald) and two other candidates run for vice president (Freddie and Bernie). So, they are separate positions. So, the president and vice president are different positions, so the elections are separate. Therefore, the outcomes are independent.

Therefore, the probability that Dan wins president is 1/2, the probability that Freddie wins vice president is 1/2, so the probability that both happen is 1/2 * 1/2 = 1/4. Therefore, the answer is 1/4. But let me think again if there's another way to model this.

Alternatively, maybe we can think of all possible outcomes. For the president, there are two possibilities: Dan or Donald. For vice president, there are two possibilities: Freddie or Bernie. Therefore, the total number of possible outcomes is 2 * 2 = 4. Each outcome is equally likely, since each candidate has a 50% chance. Therefore, the possible outcomes are:

1. Dan (President) and Freddie (Vice President)
2. Dan (President) and Bernie (Vice President)
3. Donald (President) and Freddie (Vice President)
4. Donald (President) and Bernie (Vice President)

Each of these four outcomes has a probability of 1/4. Therefore, the probability that both Dan and Freddie win is 1/4. So, that's the same answer as before. Therefore, 1/4 is the probability.

Wait, but the problem says "each candidate has a 50% chance of winning." So, does that mean that each candidate has a 50% chance regardless of the other candidates? For example, in the presidential election, Dan has 50% chance, Donald has 50% chance. Similarly, Freddie has 50% chance, Bernie has 50% chance. Therefore, the combined probability is 1/2 * 1/2 = 1/4. So, that seems to check out.

Alternatively, if the problem had said that each candidate has a 50% chance of winning against the others, but that's already the case here because there are only two candidates in each election. So, each has a 50% chance. Therefore, the answer is 1/4.

But let me think again if there's a different interpretation. For example, maybe the problem is considering that there are four candidates in total, and they are electing both president and vice president? But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, they are separate positions. So, president and vice president are separate elections. Therefore, the answer is 1/4.

Alternatively, maybe the problem is considering that the same person can't be both president and vice president? But the problem doesn't mention that. It just says two candidates for president and two for vice president. So, unless there's a rule that a person can't hold both positions, but since they are different positions, maybe they can. However, the problem doesn't specify any such restriction, so we can assume that the elections are independent. Therefore, the answer is 1/4.

Alternatively, maybe the problem is in the way the question is phrased. It says "each candidate has a 50% chance of winning." So, maybe that's not referring to each candidate in each election, but each candidate overall? Wait, but there are four candidates: Dan, Donald, Freddie, Bernie. But they are running for different positions. So, maybe the problem is that each candidate has a 50% chance of winning their respective position? Which is the same as before.

Alternatively, maybe the problem is that each candidate has a 50% chance of winning the entire election? But that doesn't make much sense. Because there are two positions. But the problem says "each candidate has a 50% chance of winning." So, maybe the problem is that for each position, each candidate has a 50% chance? Which is the same as before.

Alternatively, maybe the problem is that there is a single election where each candidate is running for both positions? But that seems unlikely. The problem says "run for class president" and "run for vice president," so they are separate.

Therefore, I think my original reasoning is correct. The answer is 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

But let me check once more. Let me think of the sample space. For president, two possibilities: Dan or Donald. For vice president, two possibilities: Freddie or Bernie. So, the total number of possible outcomes is 2 * 2 = 4. Each outcome is equally likely. Therefore, the probability of each outcome is 1/4. Therefore, the probability that Dan is president and Freddie is vice president is 1/4. Therefore, the answer is 1/4.

Alternatively, if the problem had said that the candidates are running for both positions, but that's not the case here. So, I think that's solid.

Wait, but maybe the problem is in the wording: "each candidate has a 50% chance of winning." So, maybe the problem is that each candidate has a 50% chance of winning their respective election. So, for example, Dan has 50% chance to win president, Donald 50% chance, Freddie 50% chance to win vice president, Bernie 50% chance. Therefore, the probability that Dan and Freddie both win is 0.5 * 0.5 = 0.25, which is 1/4. So, same answer.

Alternatively, if the problem had said that each candidate has a 50% chance of winning the entire election, but that would be different. But that's not the case here. The problem says "each candidate has a 50% chance of winning." Since there are two candidates for president, and two for vice president, it's natural to assume that each has a 50% chance in their respective elections.

Therefore, I think the answer is 1/4. Therefore, the probability is 1/4.

But just to make sure, let me think of another approach. Suppose we model the probability as follows. The probability that Dan wins is 1/2. Given that Dan wins, what is the probability that Freddie wins? Since the elections are independent, the probability that Freddie wins is still 1/2. Therefore, the conditional probability is 1/2, so the joint probability is 1/2 * 1/2 = 1/4. So, same answer.

Alternatively, if there was some dependence, like if the president and vice president had to be different people, but again, the problem doesn't mention that. If that were the case, then we would have to adjust the probabilities. For example, if Dan is president, then Freddie can still be vice president, so that's allowed. So, even if there was a rule that you can't be both president and vice president, since Dan and Freddie are different people, that wouldn't affect the probability. Therefore, even with that rule, the probability would still be 1/2 * 1/2 = 1/4.

Therefore, I think the answer is definitely 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

**Final Answer**
The probability that both Dan and Freddie win is \boxed{\dfrac{1}{4}}.

To determine the probability that both **Dan** and **Freddie** win their respective positions, we need to analyze the structure of the election and the independence of the events.

---

### Step 1: Understand the Election Structure

- **President**: Two candidates — **Dan** and **Donald**.
- **Vice President**: Two candidates — **Freddie** and **Bernie**.

Each candidate has a **50% chance of winning** their respective election. This implies:

- **Dan** has a 50% chance of winning the presidency.
- **Freddie** has a 50% chance of winning the vice presidency.

---

### Step 2: Are the Events Independent?

The key assumption here is that the outcome of the **presidential election** does **not affect** the outcome of the **vice presidential election**. Since the two positions are separate and the candidates are distinct, we can reasonably assume that the results are **independent**.

This independence allows us to compute the joint probability of both events occurring by multiplying their individual probabilities.

---

### Step 3: Compute the Joint Probability

- Probability that **Dan wins president**: $ \frac{1}{2} $
- Probability that **Freddie wins vice president**: $ \frac{1}{2} $

Since the events are independent:

$$
P(\text{Dan wins and Freddie wins}) = P(\text{Dan wins}) \times P(\text{Freddie wins}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
$$

---

### Step 4: Final Answer

Thus, the probability that both **Dan** and **Freddie** win their respective positions is:

$$
\boxed{\dfrac{1}{4}}
$$

Solution: The probability that Dan wins is $\frac12$. The probability that Freddie wins is also $\frac12$. Therefore, the probability that both win is $\frac12 \cdot \frac12 =\boxed{\frac14}$.


Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if that's correct.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that sounds right. I think that's the formula. Let me make sure there isn't a different formula or something else involved. Hmm, maybe there's something with angles or magnitudes? Wait, no, the dot product can also be expressed as |a||b|cosθ, where θ is the angle between them. But since I don't know the angle here, I think I should stick with the component-wise multiplication and addition method.

Alright, so the problem gives me two vectors:

First vector:
$$
\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}
$$
Second vector:
$$
\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

So, let me label them for clarity. Let me call the first vector **u** and the second vector **v**. So,

**u** = [-2, 0, 7]

**v** = [3, 4, -5]

Now, applying the formula for the dot product, which is u₁*v₁ + u₂*v₂ + u₃*v₃.

Let me compute each term step by step.

First term: u₁*v₁ = (-2) * 3. Let me calculate that. -2 multiplied by 3 is... -6. Okay, that's straightforward.

Second term: u₂*v₂ = 0 * 4. Hmm, 0 times anything is 0, right? So that term is 0.

Third term: u₃*v₃ = 7 * (-5). Let me compute that. 7 times -5 is... -35.

Now, adding all these terms together: -6 + 0 + (-35). Let me do that step by step.

Starting with -6 + 0, that's still -6. Then adding -35 to that. So, -6 + (-35) is the same as -(6 + 35) which is -41.

Wait, let me check that again. If I have -6 and then subtract 35, that's like -6 - 35. Which is indeed -41. Yeah, that seems right.

So, putting it all together, the dot product is -41.

Let me just verify my calculations once more to make sure I didn't make any mistakes.

First component: -2 * 3 = -6. Correct.

Second component: 0 * 4 = 0. Correct.

Third component: 7 * (-5) = -35. Correct.

Adding them: -6 + 0 = -6; -6 + (-35) = -41. Yep, that seems consistent.

I don't think I made any errors here. Let me think if there's another way to check. Maybe by using the alternative formula with magnitudes and the cosine of the angle? But since I don't know the angle, that might not be helpful. Alternatively, maybe I can compute the magnitudes of both vectors and see if the dot product makes sense? Let me try that.

First, compute the magnitude of **u**.

|**u**| = sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

Similarly, |**v**| = sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So, if the dot product is |**u**||**v**|cosθ, then -41 = sqrt(53)*sqrt(50)*cosθ.

But unless I know θ, I can't check this. However, maybe I can compute the value of sqrt(53)*sqrt(50) and see if -41 is a reasonable value.

sqrt(53) is approximately 7.28, sqrt(50) is approximately 7.07. Multiplying them gives roughly 7.28 * 7.07 ≈ 51.4. So, the maximum possible value of the dot product would be around 51.4 (if θ=0°) and the minimum would be around -51.4 (if θ=180°). Since -41 is between -51.4 and 51.4, it's a reasonable value. So that seems to check out.

Alternatively, maybe I can check with another method? For example, if I had a different angle or something. But I think that's overcomplicating.

Alternatively, maybe I can check if I multiplied the components correctly. Let me write them again:

First vector components: -2, 0, 7.

Second vector components: 3, 4, -5.

Multiplying each pair:

-2 * 3 = -6. Correct.

0 * 4 = 0. Correct.

7 * (-5) = -35. Correct.

Adding them: -6 + 0 + (-35) = -41. Correct.

So, I think that's solid.

Wait, just to make sure, maybe I can think of another example. Suppose I have two vectors, say [1, 2, 3] and [4, 5, 6]. Their dot product would be 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32. If I do the same steps here, it's straightforward. So, same method applies here.

Therefore, I think my calculation is correct.

Another thing to check: signs. The first term is negative, the second is zero, the third is negative. So adding two negative numbers, which gives a more negative number. So -6 -35 is -41. Correct.

I think I've verified it enough. I don't see any mistakes here.

So, the final answer should be -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the formula:

$$
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3
$$

---

### Step 1: Identify the vectors

We are given the vectors:

$$
\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

---

### Step 2: Multiply corresponding components

We compute each term of the dot product:

- First component: $ (-2) \cdot 3 = -6 $
- Second component: $ 0 \cdot 4 = 0 $
- Third component: $ 7 \cdot (-5) = -35 $

---

### Step 3: Add the results

$$
\mathbf{u} \cdot \mathbf{v} = -6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

[Assistant]
Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if that's correct.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that sounds right. I think that's the formula. Let me make sure there isn't a different formula or something else involved. Hmm, maybe there's something with angles or magnitudes? Wait, no, the dot product can also be expressed as |a||b|cosθ, where θ is the angle between them. But since I don't know the angle here, I think I should stick with the component-wise multiplication and addition method.

Alright, so the problem gives me two vectors:

First vector:
$$
\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}
$$
Second vector:
$$
\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

So, let me label them for clarity. Let me call the first vector **u** and the second vector **v**. So,

**u** = [-2, 0, 7]

**v** = [3, 4, -5]

Now, applying the formula for the dot product, which is u₁*v₁ + u₂*v₂ + u₃*v₃.

Let me compute each term step by step.

First term: u₁*v₁ = (-2) * 3. Let me calculate that. -2 multiplied by 3 is... -6. Okay, that's straightforward.

Second term: u₂*v₂ = 0 * 4. Hmm, 0 times anything is 0, right? So that term is 0.

Third term: u₃*v₃ = 7 * (-5). Let me compute that. 7 times -5 is... -35.

Now, adding all these terms together: -6 + 0 + (-35). Let me do that step by step.

Starting with -6 + 0, that's still -6. Then adding -35 to that. So, -6 + (-35) is the same as -(6 + 35) which is -41.

Wait, let me check that again. If I have -6 and then subtract 35, that's like -6 - 35. Which is indeed -41. Yeah, that seems right.

So, putting it all together, the dot product is -41.

Let me just verify my calculations once more to make sure I didn't make any mistakes.

First component: -2 * 3 = -6. Correct.

Second component: 0 * 4 = 0. Correct.

Third component: 7 * (-5) = -35. Correct.

Adding them: -6 + 0 = -6; -6 + (-35) = -41. Yep, that seems consistent.

I don't think I made any errors here. Let me think if there's another way to check. Maybe by using the alternative formula with magnitudes and the cosine of the angle? But since I don't know the angle, that might not be helpful. Alternatively, maybe I can compute the magnitudes of both vectors and see if the dot product makes sense? Let me try that.

First, compute the magnitude of **u**.

|**u**| = sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

Similarly, |**v**| = sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So, if the dot product is |**u**||**v**|cosθ, then -41 = sqrt(53)*sqrt(50)*cosθ.

But unless I know θ, I can't check this. However, maybe I can compute the value of sqrt(53)*sqrt(50) and see if -41 is a reasonable value.

sqrt(53) is approximately 7.28, sqrt(50) is approximately 7.07. Multiplying them gives roughly 7.28 * 7.07 ≈ 51.4. So, the maximum possible value of the dot product would be around 51.4 (if θ=0°) and the minimum would be around -51.4 (if θ=180°). Since -41 is between -51.4 and 51.4, it's a reasonable value. So that seems to check out.

Alternatively, maybe I can check with another method? For example, if I had a different angle or something. But I think that's overcomplicating.

Alternatively, maybe I can check if I multiplied the components correctly. Let me write them again:

First vector components: -2, 0, 7.

Second vector components: 3, 4, -5.

Multiplying each pair:

-2 * 3 = -6. Correct.

0 * 4 = 0. Correct.

7 * (-5) = -35. Correct.

Adding them: -6 + 0 + (-35) = -41. Correct.

So, I think that's solid.

Wait, just to make sure, maybe I can think of another example. Suppose I have two vectors, say [1, 2, 3] and [4, 5, 6]. Their dot product would be 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32. If I do the same steps here, it's straightforward. So, same method applies here.

Therefore, I think my calculation is correct.

Another thing to check: signs. The first term is negative, the second is zero, the third is negative. So adding two negative numbers, which gives a more negative number. So -6 -35 is -41. Correct.

I think I've verified it enough. I don't see any mistakes here.

So, the final answer should be -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the formula:

$$
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3
$$

---

### Step 1: Identify the vectors

We are given the vectors:

$$
\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

---

### Step 2: Multiply corresponding components

We compute each term of the dot product:

- First component: $ (-2) \cdot 3 = -6 $
- Second component: $ 0 \cdot 4 = 0 $
- Third component: $ 7 \cdot (-5) = -35 $

---

### Step 3: Add the results

$$
\mathbf{u} \cdot \mathbf{v} = -6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

Solution:
The dot product of $\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$ is
\[(-2)(3) + (0)(4) + (7)(-5) = \boxed{-41}.\]