Explanation:
Okay, so I need to simplify this expression: (1/5) multiplied by (8/7) divided by (12/20). Let me think about how to approach this.

First, I remember that dividing by a fraction is the same as multiplying by its reciprocal. So maybe I can rewrite the division part as a multiplication. Let me write that down.

The original problem is (1/5) * (8/7) ÷ (12/20). So, dividing by (12/20) is the same as multiplying by (20/12). Let me check that. Yes, because dividing by a fraction flips the numerator and denominator. So, ÷ (a/b) becomes * (b/a).

So, rewriting the expression, it becomes (1/5) * (8/7) * (20/12). Now, I need to multiply these three fractions together. To multiply fractions, I multiply the numerators together and the denominators together.

Let me write that out. Numerators: 1 * 8 * 20. Denominators: 5 * 7 * 12. Let me compute those.

First, the numerator: 1 * 8 is 8, then 8 * 20 is 160. So the numerator is 160.

Denominator: 5 * 7 is 35, then 35 * 12. Let me calculate that. 35 * 12. Hmm, 35*10 is 350, 35*2 is 70, so 350 + 70 is 420. So denominator is 420.

So now the fraction is 160/420. Now I need to simplify this fraction. Let me see if I can divide numerator and denominator by a common factor.

First, let me check if both numbers are divisible by 10. 160 ÷ 10 is 16, and 420 ÷ 10 is 42. So that simplifies to 16/42.

Wait, can I simplify that further? Let me check. 16 and 42. Let's see, 16 factors are 2^4, and 42 factors are 2*3*7. So they have a common factor of 2.

Divide numerator and denominator by 2: 16 ÷ 2 is 8, 42 ÷ 2 is 21. So simplified to 8/21.

Is 8 and 21 have any common factors? 8 is 2^3, 21 is 3*7. No common factors, so 8/21 is the simplified form.

Wait, but let me check my steps again to make sure I didn't make a mistake.

Original problem: (1/5)*(8/7) divided by (12/20).

Convert division to multiplication by reciprocal: (1/5)*(8/7)*(20/12).

Multiply numerators: 1*8*20. Let me check that again. 1*8 is 8, 8*20 is 160. Correct.

Denominators: 5*7*12. 5*7 is 35, 35*12. Let me verify 35*12. 35*10=350, 35*2=70, 350+70=420. Correct.

So 160/420. Divided by 10 is 16/42. Divided by 2 is 8/21. That seems right.

Alternatively, maybe I can simplify before multiplying? Let me try that approach to cross-verify.

Original expression: (1/5)*(8/7)*(20/12).

Let me see if I can cancel factors before multiplying.

Looking at numerators and denominators.

First, 1/5 * 8/7 * 20/12.

Let me write all numerators: 1, 8, 20. Denominators: 5, 7, 12.

Let me see if any numerator and denominator can be simplified.

For example, 20 in the numerator and 5 in the denominator. 20 divided by 5 is 4, and 5 divided by 5 is 1. So that simplifies.

So, let me do that.

So, 20 divided by 5 is 4, so replace 20 with 4 and 5 with 1.

So now, the expression becomes (1/1) * (8/7) * (4/12). Wait, no. Wait, original denominators are 5, 7, 12. So if I cancel 20 (numerator) with 5 (denominator), then 20 becomes 4 and 5 becomes 1.

So the expression becomes (1/1) * (8/7) * (4/12). Wait, but the denominator was 5*7*12. So after canceling 5 and 20, it's 1*7*12? Wait, maybe I need to think more carefully.

Alternatively, maybe factor all numbers into primes.

Let me try that.

Original numerators: 1, 8, 20.

Denominators: 5, 7, 12.

Breaking down into prime factors:

Numerators:

1 is 1.

8 is 2^3.

20 is 2^2 * 5.

Denominators:

5 is 5.

7 is 7.

12 is 2^2 * 3.

So combining all numerators: 1 * 2^3 * 2^2 * 5 = 2^(3+2) * 5 = 2^5 * 5.

Denominators: 5 * 7 * 2^2 * 3 = 2^2 * 3 * 5 * 7.

So the entire fraction is (2^5 * 5) / (2^2 * 3 * 5 * 7).

Now, simplifying by canceling common factors.

Cancel 2^2 from numerator and denominator: numerator becomes 2^(5-2) = 2^3, denominator becomes 1.

Cancel 5 from numerator and denominator: numerator becomes 1, denominator becomes 1.

So remaining numerator: 2^3 = 8.

Denominator: 3 * 7 = 21.

So the simplified fraction is 8/21.

Same result as before. So that's reassuring.

Alternatively, maybe I can simplify step by step.

Original expression: (1/5) * (8/7) * (20/12).

First, multiply (1/5) and (8/7).

(1/5)*(8/7) = (1*8)/(5*7) = 8/35.

Then, multiply that by (20/12).

So 8/35 * 20/12.

Again, let me simplify before multiplying.

Looking at 8 and 12: 8 and 12 have a common factor of 4.

Divide 8 by 4 is 2, divide 12 by 4 is 3.

Also, 20 and 35 have a common factor of 5.

Divide 20 by 5 is 4, divide 35 by 5 is 7.

So rewriting the fractions:

(2/7) * (4/3).

Wait, let me check that.

Original fractions after simplifying:

Numerator: 8/35 * 20/12.

After simplifying 8 and 12 by 4: 2/35 * 20/3? Wait, no. Wait, let me think again.

Wait, when you have 8/35 multiplied by 20/12, you can cross-cancel.

So 8 and 12: GCD is 4. So divide 8 by 4 is 2, divide 12 by 4 is 3.

Then 20 and 35: GCD is 5. Divide 20 by 5 is 4, divide 35 by 5 is 7.

So now, the fractions become (2/7) * (4/3).

Wait, because 8/35 becomes 2/7 (since 8 divided by 4 is 2, and 35 divided by 5 is 7? Wait, no. Wait, the original fractions are 8/35 and 20/12.

So cross-canceling between numerator of first fraction (8) and denominator of second fraction (12): 8 and 12 have GCD 4. So divide 8 by 4 is 2, divide 12 by 4 is 3.

Similarly, cross-cancel between denominator of first fraction (35) and numerator of second fraction (20): 35 and 20 have GCD 5. Divide 35 by 5 is 7, divide 20 by 5 is 4.

So now, the fractions become (2/7) * (4/3).

So multiplying those: 2*4 = 8, 7*3 = 21. So 8/21.

Same result again.

So regardless of the method, I get 8/21.

Therefore, the simplified form is 8/21.

But let me check once more with decimal conversion to make sure.

Original expression: (1/5)*(8/7)/(12/20).

Compute each part step by step.

First, compute 1/5: 0.2.

Then 8/7: approximately 1.142857.

Then 12/20: 0.6.

So the original expression is 0.2 * 1.142857 / 0.6.

First compute 0.2 * 1.142857.

0.2 * 1.142857 = 0.2285714.

Then divide by 0.6: 0.2285714 / 0.6.

Let me compute that.

Dividing by 0.6 is the same as multiplying by 10/6 = 5/3 ≈ 1.666666...

So 0.2285714 * 1.666666...

Let me compute that.

First, 0.2285714 * 1.666666...

Well, 0.2285714 * 5/3.

Let me compute 0.2285714 * 5 = 1.142857. Then divide by 3: 1.142857 / 3 ≈ 0.380952.

Now, let me compute 8/21 as a decimal.

8 divided by 21.

21 goes into 8 zero. 21 into 80 is 3 times (3*21=63), remainder 17.

Bring down 0: 170. 21*8=168, remainder 2. Bring down 0: 20. 21 into 20 is 0. Bring down next 0: 200. 21*9=189, remainder 11. Bring down 0: 110. 21*5=105, remainder 5. Bring down 0: 50. 21*2=42, remainder 8.

So decimal is 0.38095238095..., repeating.

Which matches the earlier decimal calculation of approximately 0.380952.

Therefore, 8/21 is indeed the correct simplified fraction.

So all methods lead to the same answer, so I think that's correct.

Therefore, the simplified form of the original expression is 8/21.

**Final Answer**
The simplified expression is \boxed{\dfrac{8}{21}}.

To simplify the expression:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20}
$$

we begin by applying the rule that **dividing by a fraction is the same as multiplying by its reciprocal**. This allows us to rewrite the expression as:

$$
\frac{1}{5} \cdot \frac{8}{7} \cdot \frac{20}{12}
$$

---

### Step 1: Multiply the Numerators and Denominators

We now multiply the three fractions:

- **Numerators**: $1 \cdot 8 \cdot 20 = 160$
- **Denominators**: $5 \cdot 7 \cdot 12 = 420$

So the expression becomes:

$$
\frac{160}{420}
$$

---

### Step 2: Simplify the Fraction

We simplify $\frac{160}{420}$ by finding the greatest common divisor (GCD) of 160 and 420. The GCD is 20.

- $160 \div 20 = 8$
- $420 \div 20 = 21$

So the simplified fraction is:

$$
\frac{8}{21}
$$

---

### Step 3: Final Check

To ensure accuracy, we can also simplify before multiplying by canceling common factors:

- $20$ in the numerator and $5$ in the denominator can be simplified to $4$ and $1$, respectively.
- $8$ and $12$ can be simplified to $2$ and $3$, respectively.
- $35$ and $20$ can be simplified to $7$ and $4$, respectively.

This leads to the same simplified result:

$$
\frac{8}{21}
$$

---

### Final Answer

$$
\boxed{\dfrac{8}{21}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: (1/5) multiplied by (8/7) divided by (12/20). Let me think about how to approach this.

First, I remember that dividing by a fraction is the same as multiplying by its reciprocal. So maybe I can rewrite the division part as a multiplication. Let me write that down.

The original problem is (1/5) * (8/7) ÷ (12/20). So, dividing by (12/20) is the same as multiplying by (20/12). Let me check that. Yes, because dividing by a fraction flips the numerator and denominator. So, ÷ (a/b) becomes * (b/a).

So, rewriting the expression, it becomes (1/5) * (8/7) * (20/12). Now, I need to multiply these three fractions together. To multiply fractions, I multiply the numerators together and the denominators together.

Let me write that out. Numerators: 1 * 8 * 20. Denominators: 5 * 7 * 12. Let me compute those.

First, the numerator: 1 * 8 is 8, then 8 * 20 is 160. So the numerator is 160.

Denominator: 5 * 7 is 35, then 35 * 12. Let me calculate that. 35 * 12. Hmm, 35*10 is 350, 35*2 is 70, so 350 + 70 is 420. So denominator is 420.

So now the fraction is 160/420. Now I need to simplify this fraction. Let me see if I can divide numerator and denominator by a common factor.

First, let me check if both numbers are divisible by 10. 160 ÷ 10 is 16, and 420 ÷ 10 is 42. So that simplifies to 16/42.

Wait, can I simplify that further? Let me check. 16 and 42. Let's see, 16 factors are 2^4, and 42 factors are 2*3*7. So they have a common factor of 2.

Divide numerator and denominator by 2: 16 ÷ 2 is 8, 42 ÷ 2 is 21. So simplified to 8/21.

Is 8 and 21 have any common factors? 8 is 2^3, 21 is 3*7. No common factors, so 8/21 is the simplified form.

Wait, but let me check my steps again to make sure I didn't make a mistake.

Original problem: (1/5)*(8/7) divided by (12/20).

Convert division to multiplication by reciprocal: (1/5)*(8/7)*(20/12).

Multiply numerators: 1*8*20. Let me check that again. 1*8 is 8, 8*20 is 160. Correct.

Denominators: 5*7*12. 5*7 is 35, 35*12. Let me verify 35*12. 35*10=350, 35*2=70, 350+70=420. Correct.

So 160/420. Divided by 10 is 16/42. Divided by 2 is 8/21. That seems right.

Alternatively, maybe I can simplify before multiplying? Let me try that approach to cross-verify.

Original expression: (1/5)*(8/7)*(20/12).

Let me see if I can cancel factors before multiplying.

Looking at numerators and denominators.

First, 1/5 * 8/7 * 20/12.

Let me write all numerators: 1, 8, 20. Denominators: 5, 7, 12.

Let me see if any numerator and denominator can be simplified.

For example, 20 in the numerator and 5 in the denominator. 20 divided by 5 is 4, and 5 divided by 5 is 1. So that simplifies.

So, let me do that.

So, 20 divided by 5 is 4, so replace 20 with 4 and 5 with 1.

So now, the expression becomes (1/1) * (8/7) * (4/12). Wait, no. Wait, original denominators are 5, 7, 12. So if I cancel 20 (numerator) with 5 (denominator), then 20 becomes 4 and 5 becomes 1.

So the expression becomes (1/1) * (8/7) * (4/12). Wait, but the denominator was 5*7*12. So after canceling 5 and 20, it's 1*7*12? Wait, maybe I need to think more carefully.

Alternatively, maybe factor all numbers into primes.

Let me try that.

Original numerators: 1, 8, 20.

Denominators: 5, 7, 12.

Breaking down into prime factors:

Numerators:

1 is 1.

8 is 2^3.

20 is 2^2 * 5.

Denominators:

5 is 5.

7 is 7.

12 is 2^2 * 3.

So combining all numerators: 1 * 2^3 * 2^2 * 5 = 2^(3+2) * 5 = 2^5 * 5.

Denominators: 5 * 7 * 2^2 * 3 = 2^2 * 3 * 5 * 7.

So the entire fraction is (2^5 * 5) / (2^2 * 3 * 5 * 7).

Now, simplifying by canceling common factors.

Cancel 2^2 from numerator and denominator: numerator becomes 2^(5-2) = 2^3, denominator becomes 1.

Cancel 5 from numerator and denominator: numerator becomes 1, denominator becomes 1.

So remaining numerator: 2^3 = 8.

Denominator: 3 * 7 = 21.

So the simplified fraction is 8/21.

Same result as before. So that's reassuring.

Alternatively, maybe I can simplify step by step.

Original expression: (1/5) * (8/7) * (20/12).

First, multiply (1/5) and (8/7).

(1/5)*(8/7) = (1*8)/(5*7) = 8/35.

Then, multiply that by (20/12).

So 8/35 * 20/12.

Again, let me simplify before multiplying.

Looking at 8 and 12: 8 and 12 have a common factor of 4.

Divide 8 by 4 is 2, divide 12 by 4 is 3.

Also, 20 and 35 have a common factor of 5.

Divide 20 by 5 is 4, divide 35 by 5 is 7.

So rewriting the fractions:

(2/7) * (4/3).

Wait, let me check that.

Original fractions after simplifying:

Numerator: 8/35 * 20/12.

After simplifying 8 and 12 by 4: 2/35 * 20/3? Wait, no. Wait, let me think again.

Wait, when you have 8/35 multiplied by 20/12, you can cross-cancel.

So 8 and 12: GCD is 4. So divide 8 by 4 is 2, divide 12 by 4 is 3.

Then 20 and 35: GCD is 5. Divide 20 by 5 is 4, divide 35 by 5 is 7.

So now, the fractions become (2/7) * (4/3).

Wait, because 8/35 becomes 2/7 (since 8 divided by 4 is 2, and 35 divided by 5 is 7? Wait, no. Wait, the original fractions are 8/35 and 20/12.

So cross-canceling between numerator of first fraction (8) and denominator of second fraction (12): 8 and 12 have GCD 4. So divide 8 by 4 is 2, divide 12 by 4 is 3.

Similarly, cross-cancel between denominator of first fraction (35) and numerator of second fraction (20): 35 and 20 have GCD 5. Divide 35 by 5 is 7, divide 20 by 5 is 4.

So now, the fractions become (2/7) * (4/3).

So multiplying those: 2*4 = 8, 7*3 = 21. So 8/21.

Same result again.

So regardless of the method, I get 8/21.

Therefore, the simplified form is 8/21.

But let me check once more with decimal conversion to make sure.

Original expression: (1/5)*(8/7)/(12/20).

Compute each part step by step.

First, compute 1/5: 0.2.

Then 8/7: approximately 1.142857.

Then 12/20: 0.6.

So the original expression is 0.2 * 1.142857 / 0.6.

First compute 0.2 * 1.142857.

0.2 * 1.142857 = 0.2285714.

Then divide by 0.6: 0.2285714 / 0.6.

Let me compute that.

Dividing by 0.6 is the same as multiplying by 10/6 = 5/3 ≈ 1.666666...

So 0.2285714 * 1.666666...

Let me compute that.

First, 0.2285714 * 1.666666...

Well, 0.2285714 * 5/3.

Let me compute 0.2285714 * 5 = 1.142857. Then divide by 3: 1.142857 / 3 ≈ 0.380952.

Now, let me compute 8/21 as a decimal.

8 divided by 21.

21 goes into 8 zero. 21 into 80 is 3 times (3*21=63), remainder 17.

Bring down 0: 170. 21*8=168, remainder 2. Bring down 0: 20. 21 into 20 is 0. Bring down next 0: 200. 21*9=189, remainder 11. Bring down 0: 110. 21*5=105, remainder 5. Bring down 0: 50. 21*2=42, remainder 8.

So decimal is 0.38095238095..., repeating.

Which matches the earlier decimal calculation of approximately 0.380952.

Therefore, 8/21 is indeed the correct simplified fraction.

So all methods lead to the same answer, so I think that's correct.

Therefore, the simplified form of the original expression is 8/21.

**Final Answer**
The simplified expression is \boxed{\dfrac{8}{21}}.

To simplify the expression:

$$
\frac{1}{5} \cdot \frac{8}{7} \div \frac{12}{20}
$$

we begin by applying the rule that **dividing by a fraction is the same as multiplying by its reciprocal**. This allows us to rewrite the expression as:

$$
\frac{1}{5} \cdot \frac{8}{7} \cdot \frac{20}{12}
$$

---

### Step 1: Multiply the Numerators and Denominators

We now multiply the three fractions:

- **Numerators**: $1 \cdot 8 \cdot 20 = 160$
- **Denominators**: $5 \cdot 7 \cdot 12 = 420$

So the expression becomes:

$$
\frac{160}{420}
$$

---

### Step 2: Simplify the Fraction

We simplify $\frac{160}{420}$ by finding the greatest common divisor (GCD) of 160 and 420. The GCD is 20.

- $160 \div 20 = 8$
- $420 \div 20 = 21$

So the simplified fraction is:

$$
\frac{8}{21}
$$

---

### Step 3: Final Check

To ensure accuracy, we can also simplify before multiplying by canceling common factors:

- $20$ in the numerator and $5$ in the denominator can be simplified to $4$ and $1$, respectively.
- $8$ and $12$ can be simplified to $2$ and $3$, respectively.
- $35$ and $20$ can be simplified to $7$ and $4$, respectively.

This leads to the same simplified result:

$$
\frac{8}{21}
$$

---

### Final Answer

$$
\boxed{\dfrac{8}{21}}
$$

Solution: First, we can use the division rule so that we have an expression with only multiplication of fractions.  We get $$\frac{1}{5}\cdot \frac{8}{7}\div \frac{12}{20}=\frac{1}{5}\cdot \frac{8}{7}\cdot \frac{20}{12}.$$Now, notice that $5$ and $20$ have a common factor of $5$.  We can also see that $8$ and $12$ have a common factor of $4$.  Therefore, we can simplify to get $$\frac{1}{5}\cdot \frac{8}{7}\cdot \frac{20}{12}=\frac{1}{\cancel{5}}\cdot \frac{\cancelto{2}{8}}{7}\cdot \frac{\cancelto{4}{20}}{\cancelto{3}{12}}=\frac{1\cdot 2 \cdot 4}{7\cdot 3}=\boxed{\frac{8}{21}}.$$


Explanation:
Okay, so I need to figure out how many blue marbles were in the bag originally. Let me start by understanding the problem.

We have a bag with yellow and blue marbles. The current ratio of blue to yellow marbles is 4:3. Then, if we add 5 blue marbles and remove 3 yellow marbles, the new ratio becomes 7:3. I need to find the original number of blue marbles.

Hmm, ratios. Let me recall that ratios can be represented as fractions. So, if the ratio of blue to yellow is 4:3, that means for every 4 blue marbles, there are 3 yellow marbles. So maybe I can let the number of blue marbles be 4x and the number of yellow marbles be 3x, where x is some common multiplier. That seems like a standard approach for ratio problems.

Let me write that down:

Let the original number of blue marbles be 4x.

Let the original number of yellow marbles be 3x.

Okay, so then if we add 5 blue marbles, the new number of blue marbles is 4x + 5.

And if we remove 3 yellow marbles, the new number of yellow marbles is 3x - 3.

And according to the problem, after this change, the ratio of blue to yellow marbles becomes 7:3.

So, the new ratio is (4x + 5) / (3x - 3) = 7/3.

So, I can set up the equation:

(4x + 5) / (3x - 3) = 7/3.

Now, I need to solve for x. Let me cross-multiply to eliminate the fractions.

So, 3*(4x + 5) = 7*(3x - 3).

Let me compute both sides:

Left side: 3*4x + 3*5 = 12x + 15.

Right side: 7*3x - 7*3 = 21x - 21.

So, the equation becomes:

12x + 15 = 21x - 21.

Now, let me solve for x. Let me subtract 12x from both sides:

15 = 9x - 21.

Then, add 21 to both sides:

15 + 21 = 9x.

Which is 36 = 9x.

Divide both sides by 9:

x = 4.

Okay, so x is 4. Therefore, the original number of blue marbles is 4x, which is 4*4 = 16.

Wait, let me check if that makes sense. Let me verify the numbers.

Original blue marbles: 16, original yellow marbles: 3x = 12. So ratio is 16:12, which simplifies to 4:3. That checks out.

Then, adding 5 blue marbles: 16 + 5 = 21. Removing 3 yellow marbles: 12 - 3 = 9. So new ratio is 21:9. Let me divide both by 3: 7:3. Perfect, that's the ratio given in the problem. So that seems correct.

Therefore, the original number of blue marbles is 16.

Wait, but let me just make sure I didn't make any calculation errors. Let me go through the steps again.

Original ratio: blue to yellow is 4:3. So blue = 4x, yellow = 3x. After adding 5 blue and removing 3 yellow, blue becomes 4x + 5, yellow becomes 3x - 3. The new ratio is 7:3. So equation:

(4x + 5)/(3x - 3) = 7/3.

Cross-multiplying: 3*(4x + 5) = 7*(3x - 3).

Left side: 12x + 15. Right side: 21x - 21. Then moving terms:

12x + 15 = 21x - 21.

Subtract 12x: 15 = 9x - 21.

Add 21: 36 = 9x. So x = 4. Yep, same result. So original blue marbles: 4x = 16. Seems correct.

But let me check if there's another way to approach this problem, just to confirm.

Alternatively, maybe using variables without the x. Let me try that.

Let the original number of blue marbles be B, and yellow marbles be Y.

Given that B/Y = 4/3. So 3B = 4Y. That's one equation.

After adding 5 blue and removing 3 yellow, the new numbers are B + 5 and Y - 3. The ratio is 7/3, so (B + 5)/(Y - 3) = 7/3. So cross-multiplying: 3(B + 5) = 7(Y - 3).

So, expanding that: 3B + 15 = 7Y - 21.

So, 3B + 15 = 7Y - 21.

Now, from the first equation, 3B = 4Y. So we can substitute 3B with 4Y in the second equation.

So replacing 3B with 4Y:

4Y + 15 = 7Y - 21.

Subtract 4Y from both sides:

15 = 3Y - 21.

Add 21 to both sides:

36 = 3Y => Y = 12.

Then, since 3B = 4Y, so 3B = 4*12 = 48 => B = 16. Same answer. So that's consistent.

Therefore, the original number of blue marbles is 16. So I think that's correct.

But let me check once more with actual numbers. Original blue: 16, original yellow: 12. Ratio 16:12 = 4:3. Correct. Add 5 blue: 21, remove 3 yellow: 9. Ratio 21:9 = 7:3. Correct. So all checks out.

Therefore, the answer is 16. So I think that's solid.

**Final Answer**
The number of blue marbles before adding more was \boxed{16}.

To determine the original number of blue marbles in the bag, we begin by analyzing the given ratio and the changes described.

---

### Step 1: Represent the Original Quantities

We are told that the **initial ratio of blue to yellow marbles is 4:3**.
Let the number of blue marbles be $ 4x $, and the number of yellow marbles be $ 3x $, where $ x $ is a common multiplier.

---

### Step 2: Apply the Changes

- **Add 5 blue marbles**: New number of blue marbles = $ 4x + 5 $
- **Remove 3 yellow marbles**: New number of yellow marbles = $ 3x - 3 $

After these changes, the **new ratio of blue to yellow marbles is 7:3**.
So, we can write the equation:

$$
\frac{4x + 5}{3x - 3} = \frac{7}{3}
$$

---

### Step 3: Solve the Equation

Cross-multiply to eliminate the fractions:

$$
3(4x + 5) = 7(3x - 3)
$$

Simplify both sides:

$$
12x + 15 = 21x - 21
$$

Subtract $ 12x $ from both sides:

$$
15 = 9x - 21
$$

Add 21 to both sides:

$$
36 = 9x
$$

Divide both sides by 9:

$$
x = 4
$$

---

### Step 4: Find the Original Number of Blue Marbles

Recall that the original number of blue marbles is $ 4x $. Substituting $ x = 4 $:

$$
4x = 4 \times 4 = 16
$$

---

### Step 5: Verify the Solution

- Original blue marbles: 16, yellow marbles: 12 → Ratio = $ \frac{16}{12} = \frac{4}{3} $ ✅
- After adding 5 blue and removing 3 yellow:
  - Blue: $ 16 + 5 = 21 $
  - Yellow: $ 12 - 3 = 9 $
  - New ratio: $ \frac{21}{9} = \frac{7}{3} $ ✅

All conditions are satisfied.

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many blue marbles were in the bag originally. Let me start by understanding the problem.

We have a bag with yellow and blue marbles. The current ratio of blue to yellow marbles is 4:3. Then, if we add 5 blue marbles and remove 3 yellow marbles, the new ratio becomes 7:3. I need to find the original number of blue marbles.

Hmm, ratios. Let me recall that ratios can be represented as fractions. So, if the ratio of blue to yellow is 4:3, that means for every 4 blue marbles, there are 3 yellow marbles. So maybe I can let the number of blue marbles be 4x and the number of yellow marbles be 3x, where x is some common multiplier. That seems like a standard approach for ratio problems.

Let me write that down:

Let the original number of blue marbles be 4x.

Let the original number of yellow marbles be 3x.

Okay, so then if we add 5 blue marbles, the new number of blue marbles is 4x + 5.

And if we remove 3 yellow marbles, the new number of yellow marbles is 3x - 3.

And according to the problem, after this change, the ratio of blue to yellow marbles becomes 7:3.

So, the new ratio is (4x + 5) / (3x - 3) = 7/3.

So, I can set up the equation:

(4x + 5) / (3x - 3) = 7/3.

Now, I need to solve for x. Let me cross-multiply to eliminate the fractions.

So, 3*(4x + 5) = 7*(3x - 3).

Let me compute both sides:

Left side: 3*4x + 3*5 = 12x + 15.

Right side: 7*3x - 7*3 = 21x - 21.

So, the equation becomes:

12x + 15 = 21x - 21.

Now, let me solve for x. Let me subtract 12x from both sides:

15 = 9x - 21.

Then, add 21 to both sides:

15 + 21 = 9x.

Which is 36 = 9x.

Divide both sides by 9:

x = 4.

Okay, so x is 4. Therefore, the original number of blue marbles is 4x, which is 4*4 = 16.

Wait, let me check if that makes sense. Let me verify the numbers.

Original blue marbles: 16, original yellow marbles: 3x = 12. So ratio is 16:12, which simplifies to 4:3. That checks out.

Then, adding 5 blue marbles: 16 + 5 = 21. Removing 3 yellow marbles: 12 - 3 = 9. So new ratio is 21:9. Let me divide both by 3: 7:3. Perfect, that's the ratio given in the problem. So that seems correct.

Therefore, the original number of blue marbles is 16.

Wait, but let me just make sure I didn't make any calculation errors. Let me go through the steps again.

Original ratio: blue to yellow is 4:3. So blue = 4x, yellow = 3x. After adding 5 blue and removing 3 yellow, blue becomes 4x + 5, yellow becomes 3x - 3. The new ratio is 7:3. So equation:

(4x + 5)/(3x - 3) = 7/3.

Cross-multiplying: 3*(4x + 5) = 7*(3x - 3).

Left side: 12x + 15. Right side: 21x - 21. Then moving terms:

12x + 15 = 21x - 21.

Subtract 12x: 15 = 9x - 21.

Add 21: 36 = 9x. So x = 4. Yep, same result. So original blue marbles: 4x = 16. Seems correct.

But let me check if there's another way to approach this problem, just to confirm.

Alternatively, maybe using variables without the x. Let me try that.

Let the original number of blue marbles be B, and yellow marbles be Y.

Given that B/Y = 4/3. So 3B = 4Y. That's one equation.

After adding 5 blue and removing 3 yellow, the new numbers are B + 5 and Y - 3. The ratio is 7/3, so (B + 5)/(Y - 3) = 7/3. So cross-multiplying: 3(B + 5) = 7(Y - 3).

So, expanding that: 3B + 15 = 7Y - 21.

So, 3B + 15 = 7Y - 21.

Now, from the first equation, 3B = 4Y. So we can substitute 3B with 4Y in the second equation.

So replacing 3B with 4Y:

4Y + 15 = 7Y - 21.

Subtract 4Y from both sides:

15 = 3Y - 21.

Add 21 to both sides:

36 = 3Y => Y = 12.

Then, since 3B = 4Y, so 3B = 4*12 = 48 => B = 16. Same answer. So that's consistent.

Therefore, the original number of blue marbles is 16. So I think that's correct.

But let me check once more with actual numbers. Original blue: 16, original yellow: 12. Ratio 16:12 = 4:3. Correct. Add 5 blue: 21, remove 3 yellow: 9. Ratio 21:9 = 7:3. Correct. So all checks out.

Therefore, the answer is 16. So I think that's solid.

**Final Answer**
The number of blue marbles before adding more was \boxed{16}.

To determine the original number of blue marbles in the bag, we begin by analyzing the given ratio and the changes described.

---

### Step 1: Represent the Original Quantities

We are told that the **initial ratio of blue to yellow marbles is 4:3**.
Let the number of blue marbles be $ 4x $, and the number of yellow marbles be $ 3x $, where $ x $ is a common multiplier.

---

### Step 2: Apply the Changes

- **Add 5 blue marbles**: New number of blue marbles = $ 4x + 5 $
- **Remove 3 yellow marbles**: New number of yellow marbles = $ 3x - 3 $

After these changes, the **new ratio of blue to yellow marbles is 7:3**.
So, we can write the equation:

$$
\frac{4x + 5}{3x - 3} = \frac{7}{3}
$$

---

### Step 3: Solve the Equation

Cross-multiply to eliminate the fractions:

$$
3(4x + 5) = 7(3x - 3)
$$

Simplify both sides:

$$
12x + 15 = 21x - 21
$$

Subtract $ 12x $ from both sides:

$$
15 = 9x - 21
$$

Add 21 to both sides:

$$
36 = 9x
$$

Divide both sides by 9:

$$
x = 4
$$

---

### Step 4: Find the Original Number of Blue Marbles

Recall that the original number of blue marbles is $ 4x $. Substituting $ x = 4 $:

$$
4x = 4 \times 4 = 16
$$

---

### Step 5: Verify the Solution

- Original blue marbles: 16, yellow marbles: 12 → Ratio = $ \frac{16}{12} = \frac{4}{3} $ ✅
- After adding 5 blue and removing 3 yellow:
  - Blue: $ 16 + 5 = 21 $
  - Yellow: $ 12 - 3 = 9 $
  - New ratio: $ \frac{21}{9} = \frac{7}{3} $ ✅

All conditions are satisfied.

---

### Final Answer

$$
\boxed{16}
$$

Solution: Let $x$ be the number of blue marbles and $y$ the number of yellow marbles before I added more. We are given that the ratio of blue to yellow is 4:3, so $\dfrac{x}{y}=\dfrac{4}{3}$. Additionally, after we add blue marbles and remove yellow marbles the total number of blue marbles and yellow marbles will be x+5 and y-3 respectively. We're given that at this point the ratio will be $7:3$, so $\dfrac{x+5}{y-3}=\dfrac{7}{3}$. Cross multiplying the first equation gives $3x=4y$ and cross multiplying the second gives $3(x+5)=7(y-3)$. Solving two linear equations on two variables is routine; we get the solution $y=12$, $x=16$. Since $x$ represents the number of blue marbles before more were added, the answer to the problem is just $\boxed{16}$.


Explanation:
Okay, so I need to find the angle ABC, which is the angle at point B formed by points A, B, and C. Let me recall how to approach this. I think the angle between two lines can be found using vectors. So, maybe I need to find vectors BA and BC, then use the dot product formula to find the angle between them. Let me try to remember the exact steps.

First, let me note down the coordinates of the points again to make sure I have them right. Point A is (1, -11, 2), point B is (3, -4, 1), and point C is (-2, 1, -1). So angle ABC is at point B, so the two sides of the angle are BA and BC. Therefore, I need vectors BA and BC.

To find vector BA, I subtract coordinates of A from B? Wait, no. Wait, vector BA goes from B to A, so it's A - B. Similarly, vector BC goes from B to C, so it's C - B. Let me confirm that. Yes, if you have a vector from point X to point Y, it's Y - X. So vector BA is A - B, and vector BC is C - B. Let me compute those.

First, vector BA: A is (1, -11, 2), B is (3, -4, 1). So subtracting B from A: (1 - 3, -11 - (-4), 2 - 1) = (-2, -7, 1). Let me check that again. For the x-coordinate: 1 - 3 is -2. Y-coordinate: -11 - (-4) is -11 + 4 = -7. Z-coordinate: 2 - 1 = 1. Yes, that seems right. So vector BA is (-2, -7, 1).

Now vector BC: C is (-2, 1, -1), B is (3, -4, 1). So subtracting B from C: (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2). Let me check again. X-coordinate: -2 - 3 = -5. Y-coordinate: 1 - (-4) = 1 + 4 = 5. Z-coordinate: -1 - 1 = -2. Yep, that looks correct. So vector BC is (-5, 5, -2).

Now, to find the angle between vectors BA and BC. The formula for the angle θ between two vectors u and v is:

cosθ = (u · v) / (|u| |v|)

So I need to compute the dot product of BA and BC, then divide by the product of their magnitudes.

First, let me compute the dot product. The dot product of BA and BC is:

(-2)(-5) + (-7)(5) + (1)(-2)

Calculating each term:

First term: (-2)(-5) = 10

Second term: (-7)(5) = -35

Third term: (1)(-2) = -2

Adding them up: 10 - 35 - 2 = 10 - 37 = -27. Wait, 10 - 35 is -25, then -25 - 2 is -27. So the dot product is -27.

Now, compute the magnitudes of BA and BC.

First, |BA|: sqrt[(-2)^2 + (-7)^2 + (1)^2] = sqrt[4 + 49 + 1] = sqrt[54]. Let me compute that. 54 is 9*6, so sqrt(54) = 3*sqrt(6). Let me check:

(-2)^2 = 4, (-7)^2 = 49, 1^2 = 1. 4 + 49 = 53, 53 + 1 = 54. Yes, so sqrt(54) is indeed 3√6.

Now |BC|: sqrt[(-5)^2 + (5)^2 + (-2)^2] = sqrt[25 + 25 + 4] = sqrt[54]. Wait, same as |BA|? Let me check:

(-5)^2 = 25, 5^2 = 25, (-2)^2 = 4. 25 + 25 = 50, 50 + 4 = 54. Yes, so sqrt(54) again. So both vectors have magnitude sqrt(54). Interesting.

So then cosθ = (-27) / (sqrt(54) * sqrt(54)) = (-27)/54 = -0.5.

So cosθ = -0.5. Now, what angle has cosine equal to -0.5? Let me recall the unit circle. Cosine is -0.5 at 120 degrees and 240 degrees. But since angles between vectors are between 0 and 180 degrees, it must be 120 degrees. Therefore, θ = 120 degrees.

Wait, let me confirm. The arccos of -0.5 is 120 degrees. Yes, because cos(120°) = cos(180° - 60°) = -cos(60°) = -0.5. So that's correct.

Therefore, the angle ABC is 120 degrees. Let me just check my steps again to make sure I didn't make any calculation errors.

First, vectors BA and BC:

BA: A - B = (1-3, -11 - (-4), 2 - 1) = (-2, -7, 1). Correct.

BC: C - B = (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2). Correct.

Dot product: (-2)(-5) + (-7)(5) + (1)(-2) = 10 - 35 - 2 = -27. Correct.

Magnitudes:

|BA|: sqrt(4 + 49 + 1) = sqrt(54). Correct.

|BC|: sqrt(25 + 25 + 4) = sqrt(54). Correct.

So cosθ = -27 / 54 = -0.5. Correct. Then arccos(-0.5) is 120 degrees. So yes, that seems right.

Wait, but just to make sure, maybe I should check if I had the vectors in the right direction? Because angle ABC is at point B, so the vectors should be BA and BC emanating from B. Wait, but when we compute the angle between BA and BC, is that the same as angle ABC?

Wait, actually, vectors BA and BC start at B, so the angle between them is indeed angle ABC. So yes, that's correct. So the calculation is correct.

Alternatively, sometimes people might get confused whether to use AB and BC or BA and BC, but since angle is at B, the vectors should be from B to A and from B to C, which are BA and BC. Wait, but actually, vectors BA and BC are from B to A and from B to C? Wait, no. Wait, vector BA is from B to A? Wait, no. Wait, vector BA is from B to A? Wait, no. Wait, vector BA is from point B to point A? Wait, no. Wait, the vector BA is from B to A? Wait, no. Wait, the vector BA is from point B to point A? Wait, no. Wait, the vector BA is from point A to point B? Wait, confusion here.

Wait, hold on. Let me clarify. The notation can be confusing. The vector BA is from point B to point A? Or from point A to point B?

Wait, in standard notation, vector BA is from point B to point A. Wait, no. Wait, the vector BA is the vector that starts at point B and ends at point A. So to get that vector, you subtract coordinates of B from A? Wait, no. Wait, if you have a vector from point B to point A, then it's A - B. Because if you think of position vectors, the position vector of A is OA, and position vector of B is OB. Then vector BA is OA - OB. So yes, vector BA is A - B. Similarly, vector BC is C - B. So that is correct. So the vectors BA and BC are from point B to A and from point B to C. Therefore, the angle between them is indeed angle ABC. So that is correct.

Alternatively, if I had used vectors AB and BC, that would be different. Vector AB is from A to B, which is B - A. But since angle is at B, we need vectors going out from B, so BA and BC. So yes, the calculation is correct.

Therefore, I think my answer is correct. The angle is 120 degrees. Let me just check once again the dot product and magnitudes.

Dot product: (-2)(-5) = 10, (-7)(5) = -35, (1)(-2) = -2. Total: 10 - 35 - 2 = -27. Correct.

|BA|: sqrt( (-2)^2 + (-7)^2 + 1^2 ) = sqrt(4 + 49 + 1) = sqrt(54). Correct.

|BC|: sqrt( (-5)^2 + 5^2 + (-2)^2 ) = sqrt(25 + 25 + 4) = sqrt(54). Correct.

So cosθ = -27 / (sqrt(54)*sqrt(54)) = -27 / 54 = -0.5. Correct. Arccos(-0.5) is 120 degrees. So yes, that's the answer.

Therefore, I think the angle ABC is 120 degrees. I don't see any mistakes in my reasoning. Let me just check if there's another way to approach this problem, maybe using coordinates or something else, but I think the vector approach is solid here.

Alternatively, maybe I can compute the lengths of AB, BC, and AC, then use the Law of Cosines? Let me try that as a check.

First, compute AB, BC, and AC.

Wait, AB is the distance between A and B. Let me compute AB, BC, and AC.

Wait, but angle ABC is at B, so sides are AB, BC, and AC. Wait, Law of Cosines says that for triangle ABC, AC² = AB² + BC² - 2 AB * BC * cos(angle ABC). So if I compute AB, BC, and AC, then I can solve for angle ABC.

Let me try that.

First, compute AB: distance between A and B.

Coordinates of A: (1, -11, 2), coordinates of B: (3, -4, 1). So AB distance squared is (3 - 1)^2 + (-4 - (-11))^2 + (1 - 2)^2.

Calculating:

(2)^2 + (7)^2 + (-1)^2 = 4 + 49 + 1 = 54. So AB = sqrt(54). Which matches the magnitude of vector BA, which was sqrt(54). Makes sense because AB is the same as BA in length.

Similarly, BC distance squared is between B and C: coordinates of B (3, -4, 1) and C (-2, 1, -1). So squared distance is (-2 - 3)^2 + (1 - (-4))^2 + (-1 - 1)^2 = (-5)^2 + (5)^2 + (-2)^2 = 25 + 25 + 4 = 54. So BC is sqrt(54). Same as before.

Now AC distance squared: between A (1, -11, 2) and C (-2, 1, -1). So squared distance is (-2 - 1)^2 + (1 - (-11))^2 + (-1 - 2)^2 = (-3)^2 + (12)^2 + (-3)^2 = 9 + 144 + 9 = 162. Therefore, AC = sqrt(162) = sqrt(81*2) = 9*sqrt(2).

Now, using Law of Cosines:

AC² = AB² + BC² - 2 AB * BC * cos(angle ABC)

Plugging in the numbers:

162 = 54 + 54 - 2 * sqrt(54) * sqrt(54) * cos(angle ABC)

Simplify:

162 = 108 - 2 * 54 * cos(angle ABC)

Wait, because sqrt(54)*sqrt(54) is 54. So:

162 = 108 - 108 cos(angle ABC)

Subtract 108 from both sides:

162 - 108 = -108 cos(angle ABC)

54 = -108 cos(angle ABC)

Divide both sides by -108:

54 / (-108) = cos(angle ABC)

Which simplifies to:

-0.5 = cos(angle ABC)

Which is the same result as before! Therefore, angle ABC is arccos(-0.5) = 120 degrees. So this confirms the previous result. Therefore, I can be confident that the angle is indeed 120 degrees.

Therefore, the answer is 120 degrees.

**Final Answer**
The measure of angle $ ABC $ is \boxed{120} degrees.

To find the angle $ \angle ABC $, we are given three points in 3D space:

- $ A = (1, -11, 2) $
- $ B = (3, -4, 1) $
- $ C = (-2, 1, -1) $

---

### Step 1: Determine the vectors from point $ B $

Since the angle is at point $ B $, we need the vectors that originate from $ B $ and point to $ A $ and $ C $, i.e., vectors $ \vec{BA} $ and $ \vec{BC} $.

- $ \vec{BA} = A - B = (1 - 3, -11 - (-4), 2 - 1) = (-2, -7, 1) $
- $ \vec{BC} = C - B = (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2) $

---

### Step 2: Compute the dot product of $ \vec{BA} $ and $ \vec{BC} $

$$
\vec{BA} \cdot \vec{BC} = (-2)(-5) + (-7)(5) + (1)(-2) = 10 - 35 - 2 = -27
$$

---

### Step 3: Compute the magnitudes of the vectors

$$
|\vec{BA}| = \sqrt{(-2)^2 + (-7)^2 + 1^2} = \sqrt{4 + 49 + 1} = \sqrt{54}
$$
$$
|\vec{BC}| = \sqrt{(-5)^2 + 5^2 + (-2)^2} = \sqrt{25 + 25 + 4} = \sqrt{54}
$$

---

### Step 4: Use the dot product formula to find the cosine of the angle

$$
\cos \theta = \frac{\vec{BA} \cdot \vec{BC}}{|\vec{BA}| \cdot |\vec{BC}|} = \frac{-27}{\sqrt{54} \cdot \sqrt{54}} = \frac{-27}{54} = -0.5
$$

---

### Step 5: Find the angle from the cosine value

$$
\theta = \cos^{-1}(-0.5) = 120^\circ
$$

This is the angle between the vectors $ \vec{BA} $ and $ \vec{BC} $, which corresponds to $ \angle ABC $.

---

### Final Answer

$$
\boxed{120}
$$

[Assistant]
Explanation:
Okay, so I need to find the angle ABC, which is the angle at point B formed by points A, B, and C. Let me recall how to approach this. I think the angle between two lines can be found using vectors. So, maybe I need to find vectors BA and BC, then use the dot product formula to find the angle between them. Let me try to remember the exact steps.

First, let me note down the coordinates of the points again to make sure I have them right. Point A is (1, -11, 2), point B is (3, -4, 1), and point C is (-2, 1, -1). So angle ABC is at point B, so the two sides of the angle are BA and BC. Therefore, I need vectors BA and BC.

To find vector BA, I subtract coordinates of A from B? Wait, no. Wait, vector BA goes from B to A, so it's A - B. Similarly, vector BC goes from B to C, so it's C - B. Let me confirm that. Yes, if you have a vector from point X to point Y, it's Y - X. So vector BA is A - B, and vector BC is C - B. Let me compute those.

First, vector BA: A is (1, -11, 2), B is (3, -4, 1). So subtracting B from A: (1 - 3, -11 - (-4), 2 - 1) = (-2, -7, 1). Let me check that again. For the x-coordinate: 1 - 3 is -2. Y-coordinate: -11 - (-4) is -11 + 4 = -7. Z-coordinate: 2 - 1 = 1. Yes, that seems right. So vector BA is (-2, -7, 1).

Now vector BC: C is (-2, 1, -1), B is (3, -4, 1). So subtracting B from C: (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2). Let me check again. X-coordinate: -2 - 3 = -5. Y-coordinate: 1 - (-4) = 1 + 4 = 5. Z-coordinate: -1 - 1 = -2. Yep, that looks correct. So vector BC is (-5, 5, -2).

Now, to find the angle between vectors BA and BC. The formula for the angle θ between two vectors u and v is:

cosθ = (u · v) / (|u| |v|)

So I need to compute the dot product of BA and BC, then divide by the product of their magnitudes.

First, let me compute the dot product. The dot product of BA and BC is:

(-2)(-5) + (-7)(5) + (1)(-2)

Calculating each term:

First term: (-2)(-5) = 10

Second term: (-7)(5) = -35

Third term: (1)(-2) = -2

Adding them up: 10 - 35 - 2 = 10 - 37 = -27. Wait, 10 - 35 is -25, then -25 - 2 is -27. So the dot product is -27.

Now, compute the magnitudes of BA and BC.

First, |BA|: sqrt[(-2)^2 + (-7)^2 + (1)^2] = sqrt[4 + 49 + 1] = sqrt[54]. Let me compute that. 54 is 9*6, so sqrt(54) = 3*sqrt(6). Let me check:

(-2)^2 = 4, (-7)^2 = 49, 1^2 = 1. 4 + 49 = 53, 53 + 1 = 54. Yes, so sqrt(54) is indeed 3√6.

Now |BC|: sqrt[(-5)^2 + (5)^2 + (-2)^2] = sqrt[25 + 25 + 4] = sqrt[54]. Wait, same as |BA|? Let me check:

(-5)^2 = 25, 5^2 = 25, (-2)^2 = 4. 25 + 25 = 50, 50 + 4 = 54. Yes, so sqrt(54) again. So both vectors have magnitude sqrt(54). Interesting.

So then cosθ = (-27) / (sqrt(54) * sqrt(54)) = (-27)/54 = -0.5.

So cosθ = -0.5. Now, what angle has cosine equal to -0.5? Let me recall the unit circle. Cosine is -0.5 at 120 degrees and 240 degrees. But since angles between vectors are between 0 and 180 degrees, it must be 120 degrees. Therefore, θ = 120 degrees.

Wait, let me confirm. The arccos of -0.5 is 120 degrees. Yes, because cos(120°) = cos(180° - 60°) = -cos(60°) = -0.5. So that's correct.

Therefore, the angle ABC is 120 degrees. Let me just check my steps again to make sure I didn't make any calculation errors.

First, vectors BA and BC:

BA: A - B = (1-3, -11 - (-4), 2 - 1) = (-2, -7, 1). Correct.

BC: C - B = (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2). Correct.

Dot product: (-2)(-5) + (-7)(5) + (1)(-2) = 10 - 35 - 2 = -27. Correct.

Magnitudes:

|BA|: sqrt(4 + 49 + 1) = sqrt(54). Correct.

|BC|: sqrt(25 + 25 + 4) = sqrt(54). Correct.

So cosθ = -27 / 54 = -0.5. Correct. Then arccos(-0.5) is 120 degrees. So yes, that seems right.

Wait, but just to make sure, maybe I should check if I had the vectors in the right direction? Because angle ABC is at point B, so the vectors should be BA and BC emanating from B. Wait, but when we compute the angle between BA and BC, is that the same as angle ABC?

Wait, actually, vectors BA and BC start at B, so the angle between them is indeed angle ABC. So yes, that's correct. So the calculation is correct.

Alternatively, sometimes people might get confused whether to use AB and BC or BA and BC, but since angle is at B, the vectors should be from B to A and from B to C, which are BA and BC. Wait, but actually, vectors BA and BC are from B to A and from B to C? Wait, no. Wait, vector BA is from B to A? Wait, no. Wait, vector BA is from B to A? Wait, no. Wait, vector BA is from point B to point A? Wait, no. Wait, the vector BA is from B to A? Wait, no. Wait, the vector BA is from point B to point A? Wait, no. Wait, the vector BA is from point A to point B? Wait, confusion here.

Wait, hold on. Let me clarify. The notation can be confusing. The vector BA is from point B to point A? Or from point A to point B?

Wait, in standard notation, vector BA is from point B to point A. Wait, no. Wait, the vector BA is the vector that starts at point B and ends at point A. So to get that vector, you subtract coordinates of B from A? Wait, no. Wait, if you have a vector from point B to point A, then it's A - B. Because if you think of position vectors, the position vector of A is OA, and position vector of B is OB. Then vector BA is OA - OB. So yes, vector BA is A - B. Similarly, vector BC is C - B. So that is correct. So the vectors BA and BC are from point B to A and from point B to C. Therefore, the angle between them is indeed angle ABC. So that is correct.

Alternatively, if I had used vectors AB and BC, that would be different. Vector AB is from A to B, which is B - A. But since angle is at B, we need vectors going out from B, so BA and BC. So yes, the calculation is correct.

Therefore, I think my answer is correct. The angle is 120 degrees. Let me just check once again the dot product and magnitudes.

Dot product: (-2)(-5) = 10, (-7)(5) = -35, (1)(-2) = -2. Total: 10 - 35 - 2 = -27. Correct.

|BA|: sqrt( (-2)^2 + (-7)^2 + 1^2 ) = sqrt(4 + 49 + 1) = sqrt(54). Correct.

|BC|: sqrt( (-5)^2 + 5^2 + (-2)^2 ) = sqrt(25 + 25 + 4) = sqrt(54). Correct.

So cosθ = -27 / (sqrt(54)*sqrt(54)) = -27 / 54 = -0.5. Correct. Arccos(-0.5) is 120 degrees. So yes, that's the answer.

Therefore, I think the angle ABC is 120 degrees. I don't see any mistakes in my reasoning. Let me just check if there's another way to approach this problem, maybe using coordinates or something else, but I think the vector approach is solid here.

Alternatively, maybe I can compute the lengths of AB, BC, and AC, then use the Law of Cosines? Let me try that as a check.

First, compute AB, BC, and AC.

Wait, AB is the distance between A and B. Let me compute AB, BC, and AC.

Wait, but angle ABC is at B, so sides are AB, BC, and AC. Wait, Law of Cosines says that for triangle ABC, AC² = AB² + BC² - 2 AB * BC * cos(angle ABC). So if I compute AB, BC, and AC, then I can solve for angle ABC.

Let me try that.

First, compute AB: distance between A and B.

Coordinates of A: (1, -11, 2), coordinates of B: (3, -4, 1). So AB distance squared is (3 - 1)^2 + (-4 - (-11))^2 + (1 - 2)^2.

Calculating:

(2)^2 + (7)^2 + (-1)^2 = 4 + 49 + 1 = 54. So AB = sqrt(54). Which matches the magnitude of vector BA, which was sqrt(54). Makes sense because AB is the same as BA in length.

Similarly, BC distance squared is between B and C: coordinates of B (3, -4, 1) and C (-2, 1, -1). So squared distance is (-2 - 3)^2 + (1 - (-4))^2 + (-1 - 1)^2 = (-5)^2 + (5)^2 + (-2)^2 = 25 + 25 + 4 = 54. So BC is sqrt(54). Same as before.

Now AC distance squared: between A (1, -11, 2) and C (-2, 1, -1). So squared distance is (-2 - 1)^2 + (1 - (-11))^2 + (-1 - 2)^2 = (-3)^2 + (12)^2 + (-3)^2 = 9 + 144 + 9 = 162. Therefore, AC = sqrt(162) = sqrt(81*2) = 9*sqrt(2).

Now, using Law of Cosines:

AC² = AB² + BC² - 2 AB * BC * cos(angle ABC)

Plugging in the numbers:

162 = 54 + 54 - 2 * sqrt(54) * sqrt(54) * cos(angle ABC)

Simplify:

162 = 108 - 2 * 54 * cos(angle ABC)

Wait, because sqrt(54)*sqrt(54) is 54. So:

162 = 108 - 108 cos(angle ABC)

Subtract 108 from both sides:

162 - 108 = -108 cos(angle ABC)

54 = -108 cos(angle ABC)

Divide both sides by -108:

54 / (-108) = cos(angle ABC)

Which simplifies to:

-0.5 = cos(angle ABC)

Which is the same result as before! Therefore, angle ABC is arccos(-0.5) = 120 degrees. So this confirms the previous result. Therefore, I can be confident that the angle is indeed 120 degrees.

Therefore, the answer is 120 degrees.

**Final Answer**
The measure of angle $ ABC $ is \boxed{120} degrees.

To find the angle $ \angle ABC $, we are given three points in 3D space:

- $ A = (1, -11, 2) $
- $ B = (3, -4, 1) $
- $ C = (-2, 1, -1) $

---

### Step 1: Determine the vectors from point $ B $

Since the angle is at point $ B $, we need the vectors that originate from $ B $ and point to $ A $ and $ C $, i.e., vectors $ \vec{BA} $ and $ \vec{BC} $.

- $ \vec{BA} = A - B = (1 - 3, -11 - (-4), 2 - 1) = (-2, -7, 1) $
- $ \vec{BC} = C - B = (-2 - 3, 1 - (-4), -1 - 1) = (-5, 5, -2) $

---

### Step 2: Compute the dot product of $ \vec{BA} $ and $ \vec{BC} $

$$
\vec{BA} \cdot \vec{BC} = (-2)(-5) + (-7)(5) + (1)(-2) = 10 - 35 - 2 = -27
$$

---

### Step 3: Compute the magnitudes of the vectors

$$
|\vec{BA}| = \sqrt{(-2)^2 + (-7)^2 + 1^2} = \sqrt{4 + 49 + 1} = \sqrt{54}
$$
$$
|\vec{BC}| = \sqrt{(-5)^2 + 5^2 + (-2)^2} = \sqrt{25 + 25 + 4} = \sqrt{54}
$$

---

### Step 4: Use the dot product formula to find the cosine of the angle

$$
\cos \theta = \frac{\vec{BA} \cdot \vec{BC}}{|\vec{BA}| \cdot |\vec{BC}|} = \frac{-27}{\sqrt{54} \cdot \sqrt{54}} = \frac{-27}{54} = -0.5
$$

---

### Step 5: Find the angle from the cosine value

$$
\theta = \cos^{-1}(-0.5) = 120^\circ
$$

This is the angle between the vectors $ \vec{BA} $ and $ \vec{BC} $, which corresponds to $ \angle ABC $.

---

### Final Answer

$$
\boxed{120}
$$

Solution:
From the distance formula, we compute that $AB = 3 \sqrt{6},$ $AC = 9 \sqrt{2},$ and $BC = 3 \sqrt{6}.$  Then from the Law of Cosines,
\[\cos \angle ABC = \frac{(3 \sqrt{6})^2 + (3 \sqrt{6})^2 - (9 \sqrt{2})^2}{2 \cdot 3 \sqrt{6} \cdot 3 \sqrt{6}} = -\frac{1}{2}.\]Therefore, $\angle ABC = \boxed{120^\circ}.$


Explanation:
Okay, so I need to find the measure of angle B in triangle ABC, where angle A is 14 degrees, and AB is the diameter of the circle in which the triangle is inscribed. Hmm, let me think about how to approach this.

First, I remember that when a triangle is inscribed in a circle and one of its sides is the diameter, there's a special property related to right angles. Is it the Thales' theorem? Yeah, Thales' theorem states that if you have a triangle inscribed in a circle where one side is the diameter, then the angle opposite that diameter is a right angle. So, that means angle C must be 90 degrees because AB is the diameter. Wait, let me confirm that. If AB is the diameter, then the triangle ABC with AB as a side and C on the circle would have angle C being 90 degrees. Yes, that seems right.

So, triangle ABC is a right-angled triangle at point C. Therefore, angles A, B, and C add up to 180 degrees. Since angle C is 90 degrees, angles A and B must add up to 90 degrees. The problem says angle A is 14 degrees, so angle B should be 90 - 14 = 76 degrees? Wait, that seems straightforward, but let me make sure I didn't skip any steps or make any wrong assumptions.

Let me recap. AB is the diameter of the circumscribed circle. Therefore, by Thales' theorem, angle ACB is 90 degrees. Therefore, triangle ABC is right-angled at C. Therefore, the sum of angles A and B is 90 degrees. Given angle A is 14 degrees, angle B is 90 - 14 = 76 degrees. So, is that the answer? 76 degrees?

Wait, but maybe I need to verify using other methods? Let me think. Maybe using the Law of Sines or something else?

In a circle, the Law of Sines relates the sides of a triangle to the sine of their opposite angles. But since AB is the diameter, let me denote AB as 2R, where R is the radius of the circle. Then, the length of AB is 2R. In triangle ABC, angle C is 90 degrees, so by Pythagoras, AB² = AC² + BC². But maybe that's not necessary here.

Alternatively, since AB is the diameter, the central angle subtended by AB is 180 degrees. The inscribed angle subtended by AB would be half of that, which is 90 degrees. Wait, that's angle C again. So that's consistent with Thales' theorem.

Alternatively, maybe using coordinates? Let me try to imagine placing the triangle in a coordinate system. Let me suppose that the circle has center at the origin (0,0), and AB is the diameter along the x-axis. Let me say point A is at (-R, 0) and point B is at (R, 0). Then, point C is somewhere on the circle, not on the x-axis. The coordinates of C can be represented as (R cos θ, R sin θ) for some angle θ. Then, angle at A is 14 degrees. Let me try to compute angle A.

Wait, angle at A is the angle between sides AB and AC. Let me compute vectors. Vector AB is from A to B: (R - (-R), 0 - 0) = (2R, 0). Vector AC is from A to C: (R cos θ - (-R), R sin θ - 0) = (R cos θ + R, R sin θ) = R(cos θ + 1, sin θ). Then, the angle between AB and AC is angle A, which is 14 degrees. So, using the dot product formula:

cos(angle A) = (AB · AC) / (|AB| |AC|)

Compute AB · AC: (2R)(R(cos θ + 1)) + (0)(R sin θ) = 2R² (cos θ + 1)

|AB| is sqrt((2R)^2 + 0^2) = 2R

|AC| is sqrt( [R(cos θ + 1)]² + [R sin θ]^2 ) = R sqrt( (cos θ + 1)^2 + sin² θ )

Let me compute that:

(cos θ + 1)^2 + sin² θ = cos² θ + 2 cos θ + 1 + sin² θ = (cos² θ + sin² θ) + 2 cos θ + 1 = 1 + 2 cos θ + 1 = 2 + 2 cos θ = 2(1 + cos θ)

Therefore, |AC| = R sqrt(2(1 + cos θ)) = R * sqrt(2) * sqrt(1 + cos θ)

Therefore, putting back into the cosine formula:

cos(angle A) = [2R² (cos θ + 1)] / [2R * R * sqrt(2) * sqrt(1 + cos θ)) ] = [2R² (cos θ + 1)] / [2R² sqrt(2) sqrt(1 + cos θ)) ] = [ (cos θ + 1) ] / [ sqrt(2) sqrt(1 + cos θ) ) ] = sqrt(1 + cos θ) / sqrt(2)

Therefore, cos(angle A) = sqrt( (1 + cos θ)/2 )

But angle A is 14 degrees, so:

cos(14°) = sqrt( (1 + cos θ)/2 )

Let me square both sides:

cos²(14°) = (1 + cos θ)/2

Multiply both sides by 2:

2 cos²(14°) = 1 + cos θ

Therefore, cos θ = 2 cos²(14°) - 1

But 2 cos² x - 1 is equal to cos(2x). Therefore, cos θ = cos(28°)

Therefore, θ = 28° or θ = -28°, but since θ is an angle parameterizing point C on the circle, we can take θ = 28°, assuming it's in the upper half of the circle (since angle C is 90°, and if θ is 28°, then point C is in the upper half).

Therefore, θ = 28°, so coordinates of C are (R cos 28°, R sin 28°). Now, we need to find angle B.

Angle B is the angle at point B, between sides BA and BC. Let me compute that. Let me use vectors again. Vector BA is from B to A: (-R - R, 0 - 0) = (-2R, 0). Vector BC is from B to C: (R cos θ - R, R sin θ - 0) = (R (cos θ - 1), R sin θ). So, similar to before.

So, angle at B is between BA and BC. Let me compute the angle between vectors BA and BC.

Again, using the dot product formula:

cos(angle B) = (BA · BC) / (|BA| |BC|)

Compute BA · BC: (-2R)(R (cos θ - 1)) + (0)(R sin θ) = -2R² (cos θ - 1) = 2R² (1 - cos θ)

|BA| is sqrt( (-2R)^2 + 0^2 ) = 2R

|BC| is sqrt( [R (cos θ - 1)]² + [R sin θ]^2 ) = R sqrt( (cos θ - 1)^2 + sin² θ )

Compute inside the sqrt:

(cos θ - 1)^2 + sin² θ = cos² θ - 2 cos θ + 1 + sin² θ = (cos² θ + sin² θ) - 2 cos θ + 1 = 1 - 2 cos θ + 1 = 2(1 - cos θ)

Therefore, |BC| = R sqrt(2(1 - cos θ)) = R * sqrt(2) * sqrt(1 - cos θ)

Therefore, putting back into the cosine formula:

cos(angle B) = [2R² (1 - cos θ)] / [2R * R * sqrt(2) * sqrt(1 - cos θ)) ] = [2R² (1 - cos θ)] / [2R² sqrt(2) sqrt(1 - cos θ)) ] = [ (1 - cos θ) ] / [ sqrt(2) sqrt(1 - cos θ) ) ] = sqrt(1 - cos θ)/sqrt(2)

Therefore, cos(angle B) = sqrt( (1 - cos θ)/2 )

But earlier, we found that θ = 28°, so let's compute this:

cos(angle B) = sqrt( (1 - cos 28°)/2 )

But (1 - cos θ)/2 is equal to sin²(θ/2). Therefore, sqrt( sin²(θ/2) ) = |sin(θ/2)|. Since θ is 28°, which is between 0 and 180°, sin(θ/2) is positive, so:

cos(angle B) = sin(θ/2) = sin(14°)

Therefore, angle B is arccos( sin(14°) )

Wait, but arccos(sin(14°))? Let me think. Alternatively, since sin(14°) = cos(90° - 14°) = cos(76°). Therefore, cos(angle B) = cos(76°), so angle B is 76°, since angles in a triangle are between 0 and 180°, so arccos(cos(76°)) is 76°. Therefore, angle B is 76 degrees. Which matches the previous result.

So, that confirms that angle B is 76 degrees. Therefore, the answer is 76 degrees. But let me check once more with another approach to be thorough.

Alternatively, since triangle ABC is right-angled at C, angles A and B are complementary. So angle A + angle B = 90°, so angle B = 90° - angle A = 90° - 14° = 76°, which is the same answer. Therefore, regardless of the method, it seems that angle B is 76 degrees.

But just to make sure there isn't any trick or something I missed. The problem says the triangle is inscribed in a circle where AB is a diameter. So, AB is the diameter, so the circle's diameter is AB, so the center is the midpoint of AB. Then, by Thales' theorem, angle at C is right. Therefore, triangle ABC is right-angled at C. Therefore, angles at A and B add up to 90°, so angle B is 90° - 14° = 76°. So, that's straightforward.

Is there any possibility that angle A is not the angle at A? Wait, the problem says "a triangle triangle ABC with angle A = 14°", so angle at vertex A is 14°, so that's correct. So, no confusion there.

Alternatively, could there be another configuration where AB is the diameter but angle A is not 14°? No, because the problem states angle A is 14°, so we just need to find angle B given that.

Alternatively, maybe using the Law of Sines on the triangle. In triangle ABC, with angle C = 90°, angle A = 14°, angle B = 76°, then sides are in proportion:

a / sin A = b / sin B = c / sin C

Where side a is BC, opposite angle A; side b is AC, opposite angle B; side c is AB, opposite angle C (which is 90°). Since AB is the diameter, let's denote AB = 2R. Then, c = 2R. Then, sin C = sin 90° = 1, so c / sin C = 2R / 1 = 2R. Therefore, a = 2R sin A = 2R sin 14°, and b = 2R sin B = 2R sin 76°. Let me check if these satisfy Pythagoras:

a² + b² = (2R sin 14°)^2 + (2R sin 76°)^2 = 4R² [ sin² 14° + sin² 76° ]

But sin 76° = cos 14°, since sin(90° - x) = cos x. Therefore, sin 76° = cos 14°, so:

sin² 14° + cos² 14° = 1. Therefore, a² + b² = 4R² * 1 = 4R² = c². Which matches Pythagoras. Therefore, this is consistent. Therefore, angle B is indeed 76 degrees.

Therefore, after multiple methods and checks, I can be confident that angle B is 76 degrees.

**Final Answer**
The measure of angle $ \angle B $ is \boxed{76} degrees.

To determine the measure of angle $ \angle B $ in triangle $ \triangle ABC $, we are given the following:

- $ \angle A = 14^\circ $
- $ AB $ is the diameter of the circle in which the triangle is inscribed

---

### Step 1: Apply Thales' Theorem

Thales' Theorem states that if a triangle is inscribed in a circle and one of its sides is the diameter of the circle, then the angle opposite that diameter is a right angle.

In this case, since $ AB $ is the diameter, the angle opposite to it, which is $ \angle C $, must be a right angle:

$$
\angle C = 90^\circ
$$

---

### Step 2: Use the Triangle Angle Sum

In any triangle, the sum of the interior angles is $ 180^\circ $. Therefore, in triangle $ \triangle ABC $:

$$
\angle A + \angle B + \angle C = 180^\circ
$$

Substituting the known values:

$$
14^\circ + \angle B + 90^\circ = 180^\circ
$$

Solving for $ \angle B $:

$$
\angle B = 180^\circ - 14^\circ - 90^\circ = 76^\circ
$$

---

### Step 3: Verification via Complementary Angles

Since $ \angle C = 90^\circ $, angles $ \angle A $ and $ \angle B $ must be complementary (i.e., their sum is $ 90^\circ $):

$$
\angle A + \angle B = 90^\circ
$$

$$
\angle B = 90^\circ - 14^\circ = 76^\circ
$$

This confirms the result.

---

### Final Answer

$$
\boxed{76}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle B in triangle ABC, where angle A is 14 degrees, and AB is the diameter of the circle in which the triangle is inscribed. Hmm, let me think about how to approach this.

First, I remember that when a triangle is inscribed in a circle and one of its sides is the diameter, there's a special property related to right angles. Is it the Thales' theorem? Yeah, Thales' theorem states that if you have a triangle inscribed in a circle where one side is the diameter, then the angle opposite that diameter is a right angle. So, that means angle C must be 90 degrees because AB is the diameter. Wait, let me confirm that. If AB is the diameter, then the triangle ABC with AB as a side and C on the circle would have angle C being 90 degrees. Yes, that seems right.

So, triangle ABC is a right-angled triangle at point C. Therefore, angles A, B, and C add up to 180 degrees. Since angle C is 90 degrees, angles A and B must add up to 90 degrees. The problem says angle A is 14 degrees, so angle B should be 90 - 14 = 76 degrees? Wait, that seems straightforward, but let me make sure I didn't skip any steps or make any wrong assumptions.

Let me recap. AB is the diameter of the circumscribed circle. Therefore, by Thales' theorem, angle ACB is 90 degrees. Therefore, triangle ABC is right-angled at C. Therefore, the sum of angles A and B is 90 degrees. Given angle A is 14 degrees, angle B is 90 - 14 = 76 degrees. So, is that the answer? 76 degrees?

Wait, but maybe I need to verify using other methods? Let me think. Maybe using the Law of Sines or something else?

In a circle, the Law of Sines relates the sides of a triangle to the sine of their opposite angles. But since AB is the diameter, let me denote AB as 2R, where R is the radius of the circle. Then, the length of AB is 2R. In triangle ABC, angle C is 90 degrees, so by Pythagoras, AB² = AC² + BC². But maybe that's not necessary here.

Alternatively, since AB is the diameter, the central angle subtended by AB is 180 degrees. The inscribed angle subtended by AB would be half of that, which is 90 degrees. Wait, that's angle C again. So that's consistent with Thales' theorem.

Alternatively, maybe using coordinates? Let me try to imagine placing the triangle in a coordinate system. Let me suppose that the circle has center at the origin (0,0), and AB is the diameter along the x-axis. Let me say point A is at (-R, 0) and point B is at (R, 0). Then, point C is somewhere on the circle, not on the x-axis. The coordinates of C can be represented as (R cos θ, R sin θ) for some angle θ. Then, angle at A is 14 degrees. Let me try to compute angle A.

Wait, angle at A is the angle between sides AB and AC. Let me compute vectors. Vector AB is from A to B: (R - (-R), 0 - 0) = (2R, 0). Vector AC is from A to C: (R cos θ - (-R), R sin θ - 0) = (R cos θ + R, R sin θ) = R(cos θ + 1, sin θ). Then, the angle between AB and AC is angle A, which is 14 degrees. So, using the dot product formula:

cos(angle A) = (AB · AC) / (|AB| |AC|)

Compute AB · AC: (2R)(R(cos θ + 1)) + (0)(R sin θ) = 2R² (cos θ + 1)

|AB| is sqrt((2R)^2 + 0^2) = 2R

|AC| is sqrt( [R(cos θ + 1)]² + [R sin θ]^2 ) = R sqrt( (cos θ + 1)^2 + sin² θ )

Let me compute that:

(cos θ + 1)^2 + sin² θ = cos² θ + 2 cos θ + 1 + sin² θ = (cos² θ + sin² θ) + 2 cos θ + 1 = 1 + 2 cos θ + 1 = 2 + 2 cos θ = 2(1 + cos θ)

Therefore, |AC| = R sqrt(2(1 + cos θ)) = R * sqrt(2) * sqrt(1 + cos θ)

Therefore, putting back into the cosine formula:

cos(angle A) = [2R² (cos θ + 1)] / [2R * R * sqrt(2) * sqrt(1 + cos θ)) ] = [2R² (cos θ + 1)] / [2R² sqrt(2) sqrt(1 + cos θ)) ] = [ (cos θ + 1) ] / [ sqrt(2) sqrt(1 + cos θ) ) ] = sqrt(1 + cos θ) / sqrt(2)

Therefore, cos(angle A) = sqrt( (1 + cos θ)/2 )

But angle A is 14 degrees, so:

cos(14°) = sqrt( (1 + cos θ)/2 )

Let me square both sides:

cos²(14°) = (1 + cos θ)/2

Multiply both sides by 2:

2 cos²(14°) = 1 + cos θ

Therefore, cos θ = 2 cos²(14°) - 1

But 2 cos² x - 1 is equal to cos(2x). Therefore, cos θ = cos(28°)

Therefore, θ = 28° or θ = -28°, but since θ is an angle parameterizing point C on the circle, we can take θ = 28°, assuming it's in the upper half of the circle (since angle C is 90°, and if θ is 28°, then point C is in the upper half).

Therefore, θ = 28°, so coordinates of C are (R cos 28°, R sin 28°). Now, we need to find angle B.

Angle B is the angle at point B, between sides BA and BC. Let me compute that. Let me use vectors again. Vector BA is from B to A: (-R - R, 0 - 0) = (-2R, 0). Vector BC is from B to C: (R cos θ - R, R sin θ - 0) = (R (cos θ - 1), R sin θ). So, similar to before.

So, angle at B is between BA and BC. Let me compute the angle between vectors BA and BC.

Again, using the dot product formula:

cos(angle B) = (BA · BC) / (|BA| |BC|)

Compute BA · BC: (-2R)(R (cos θ - 1)) + (0)(R sin θ) = -2R² (cos θ - 1) = 2R² (1 - cos θ)

|BA| is sqrt( (-2R)^2 + 0^2 ) = 2R

|BC| is sqrt( [R (cos θ - 1)]² + [R sin θ]^2 ) = R sqrt( (cos θ - 1)^2 + sin² θ )

Compute inside the sqrt:

(cos θ - 1)^2 + sin² θ = cos² θ - 2 cos θ + 1 + sin² θ = (cos² θ + sin² θ) - 2 cos θ + 1 = 1 - 2 cos θ + 1 = 2(1 - cos θ)

Therefore, |BC| = R sqrt(2(1 - cos θ)) = R * sqrt(2) * sqrt(1 - cos θ)

Therefore, putting back into the cosine formula:

cos(angle B) = [2R² (1 - cos θ)] / [2R * R * sqrt(2) * sqrt(1 - cos θ)) ] = [2R² (1 - cos θ)] / [2R² sqrt(2) sqrt(1 - cos θ)) ] = [ (1 - cos θ) ] / [ sqrt(2) sqrt(1 - cos θ) ) ] = sqrt(1 - cos θ)/sqrt(2)

Therefore, cos(angle B) = sqrt( (1 - cos θ)/2 )

But earlier, we found that θ = 28°, so let's compute this:

cos(angle B) = sqrt( (1 - cos 28°)/2 )

But (1 - cos θ)/2 is equal to sin²(θ/2). Therefore, sqrt( sin²(θ/2) ) = |sin(θ/2)|. Since θ is 28°, which is between 0 and 180°, sin(θ/2) is positive, so:

cos(angle B) = sin(θ/2) = sin(14°)

Therefore, angle B is arccos( sin(14°) )

Wait, but arccos(sin(14°))? Let me think. Alternatively, since sin(14°) = cos(90° - 14°) = cos(76°). Therefore, cos(angle B) = cos(76°), so angle B is 76°, since angles in a triangle are between 0 and 180°, so arccos(cos(76°)) is 76°. Therefore, angle B is 76 degrees. Which matches the previous result.

So, that confirms that angle B is 76 degrees. Therefore, the answer is 76 degrees. But let me check once more with another approach to be thorough.

Alternatively, since triangle ABC is right-angled at C, angles A and B are complementary. So angle A + angle B = 90°, so angle B = 90° - angle A = 90° - 14° = 76°, which is the same answer. Therefore, regardless of the method, it seems that angle B is 76 degrees.

But just to make sure there isn't any trick or something I missed. The problem says the triangle is inscribed in a circle where AB is a diameter. So, AB is the diameter, so the circle's diameter is AB, so the center is the midpoint of AB. Then, by Thales' theorem, angle at C is right. Therefore, triangle ABC is right-angled at C. Therefore, angles at A and B add up to 90°, so angle B is 90° - 14° = 76°. So, that's straightforward.

Is there any possibility that angle A is not the angle at A? Wait, the problem says "a triangle triangle ABC with angle A = 14°", so angle at vertex A is 14°, so that's correct. So, no confusion there.

Alternatively, could there be another configuration where AB is the diameter but angle A is not 14°? No, because the problem states angle A is 14°, so we just need to find angle B given that.

Alternatively, maybe using the Law of Sines on the triangle. In triangle ABC, with angle C = 90°, angle A = 14°, angle B = 76°, then sides are in proportion:

a / sin A = b / sin B = c / sin C

Where side a is BC, opposite angle A; side b is AC, opposite angle B; side c is AB, opposite angle C (which is 90°). Since AB is the diameter, let's denote AB = 2R. Then, c = 2R. Then, sin C = sin 90° = 1, so c / sin C = 2R / 1 = 2R. Therefore, a = 2R sin A = 2R sin 14°, and b = 2R sin B = 2R sin 76°. Let me check if these satisfy Pythagoras:

a² + b² = (2R sin 14°)^2 + (2R sin 76°)^2 = 4R² [ sin² 14° + sin² 76° ]

But sin 76° = cos 14°, since sin(90° - x) = cos x. Therefore, sin 76° = cos 14°, so:

sin² 14° + cos² 14° = 1. Therefore, a² + b² = 4R² * 1 = 4R² = c². Which matches Pythagoras. Therefore, this is consistent. Therefore, angle B is indeed 76 degrees.

Therefore, after multiple methods and checks, I can be confident that angle B is 76 degrees.

**Final Answer**
The measure of angle $ \angle B $ is \boxed{76} degrees.

To determine the measure of angle $ \angle B $ in triangle $ \triangle ABC $, we are given the following:

- $ \angle A = 14^\circ $
- $ AB $ is the diameter of the circle in which the triangle is inscribed

---

### Step 1: Apply Thales' Theorem

Thales' Theorem states that if a triangle is inscribed in a circle and one of its sides is the diameter of the circle, then the angle opposite that diameter is a right angle.

In this case, since $ AB $ is the diameter, the angle opposite to it, which is $ \angle C $, must be a right angle:

$$
\angle C = 90^\circ
$$

---

### Step 2: Use the Triangle Angle Sum

In any triangle, the sum of the interior angles is $ 180^\circ $. Therefore, in triangle $ \triangle ABC $:

$$
\angle A + \angle B + \angle C = 180^\circ
$$

Substituting the known values:

$$
14^\circ + \angle B + 90^\circ = 180^\circ
$$

Solving for $ \angle B $:

$$
\angle B = 180^\circ - 14^\circ - 90^\circ = 76^\circ
$$

---

### Step 3: Verification via Complementary Angles

Since $ \angle C = 90^\circ $, angles $ \angle A $ and $ \angle B $ must be complementary (i.e., their sum is $ 90^\circ $):

$$
\angle A + \angle B = 90^\circ
$$

$$
\angle B = 90^\circ - 14^\circ = 76^\circ
$$

This confirms the result.

---

### Final Answer

$$
\boxed{76}
$$

Solution:
If $AB$ is a diameter, that means the triangle must have a right angle at $C.$ Therefore, we have that \begin{align*}
\angle B &= 180^\circ - (\angle A + \angle C) \\
&= 180^\circ - (14^\circ + 90^\circ) = \boxed{76^\circ}.
\end{align*}


Explanation:
Okay, so I need to find the probability that a number selected at random from 1 through 100 is a multiple of 3. Let me think about how to approach this.

First, probability is generally the number of favorable outcomes divided by the total number of possible outcomes. In this case, the total number of possible outcomes is 100 because we're selecting a number from 1 to 100 inclusive. That part seems straightforward.

Now, the tricky part is figuring out how many numbers between 1 and 100 are multiples of 3. Let me recall that a multiple of 3 is any number that can be expressed as 3 times some integer. So, numbers like 3, 6, 9, ..., up to the largest multiple of 3 that's less than or equal to 100.

Hmm, how do I find how many such numbers there are? Maybe I can divide 100 by 3 and take the floor of that? Let me check. If I divide 100 by 3, I get approximately 33.333... So, does that mean there are 33 multiples of 3 between 1 and 100? Wait, let me verify that.

Let me list some multiples of 3 and see. The first multiple is 3*1=3, then 3*2=6, 3*3=9, ..., up to 3*33=99. Wait, 3*33 is 99, which is less than 100. Then the next multiple would be 3*34=102, which is greater than 100. So, yes, the multiples of 3 between 1 and 100 are 3*1 up to 3*33. Therefore, there are 33 multiples of 3.

But hold on, let me make sure I didn't miss anything. Let me check with another method. For example, if I take the number 100 and divide by 3, the quotient is 33 with a remainder of 1. So, that would mean that there are 33 full multiples of 3 in that range. So, that seems to confirm it.

Alternatively, maybe I can use the formula for the number of terms in an arithmetic sequence. The multiples of 3 form an arithmetic sequence where the first term a1 is 3, the common difference d is 3, and the last term an is the largest multiple of 3 less than or equal to 100.

So, to find the number of terms, n, in this sequence, we can use the formula:

an = a1 + (n - 1)d

We know that an is 99, a1 is 3, d is 3. Plugging in:

99 = 3 + (n - 1)*3

Subtract 3 from both sides:

96 = (n - 1)*3

Divide both sides by 3:

32 = n - 1

Add 1 to both sides:

n = 33

So, that gives me 33 terms. Therefore, there are 33 multiples of 3 between 1 and 100.

Wait, but let me check with another approach. Let me count how many numbers between 1 and 100 are divisible by 3. For example, numbers divisible by 3 are every third number. So starting from 3, then 6, 9,... up to 99. So, if I divide 100 by 3, I get 33.333... So, the integer part is 33, which would be the number of multiples. So that seems consistent.

But just to be thorough, let me check with actual counting. Let me think: from 1 to 100, how many numbers are divisible by 3? Let me think of how many complete groups of 3 there are in 100 numbers. Each group of 3 numbers contains exactly one multiple of 3. For example, numbers 1-3: 3 is a multiple; 4-6: 6 is a multiple; 7-9: 9 is a multiple, etc.

So, if there are 100 numbers, how many complete groups of 3 are there? Well, 100 divided by 3 is 33 with a remainder of 1. So, there are 33 complete groups, each contributing one multiple of 3, so 33 multiples. Then, the remaining 1 number (since 3*33=99, so the next number is 100) is not a multiple of 3. Therefore, total multiples of 3 are 33.

Therefore, all methods point to there being 33 multiples of 3 between 1 and 100.

Therefore, the probability would be the number of favorable outcomes (which is 33) divided by the total number of possible outcomes (which is 100). So, probability = 33/100.

But wait, let me check once more. Let me actually list some numbers. For example, from 1 to 10, the multiples of 3 are 3, 6, 9. That's 3 numbers. 10 divided by 3 is approximately 3.333, so floor(10/3)=3. Which matches. Similarly, from 1 to 12, there are 4 multiples: 3,6,9,12. 12/3=4. So that works. So, if I take 100, then floor(100/3)=33. So that seems correct.

Alternatively, if I use the formula floor(100/3) = floor(33.333...) = 33. So that's another way to think about it.

Therefore, I think I can be confident that there are 33 multiples of 3 between 1 and 100.

Therefore, the probability is 33/100.

But let me check if there's a possibility that 100 is a multiple of 3? 100 divided by 3 is 33.333... So no, 100 is not a multiple of 3. Therefore, the last multiple is 99, which is 3*33. So that's correct.

Alternatively, if I wanted to check for numbers from 1 to 99, there would be exactly 33 multiples of 3, since 99 is 3*33. But since we are going up to 100, which is not a multiple of 3, the count remains 33.

Therefore, the probability is 33/100.

But let me think again if there's another way someone might approach this problem and perhaps get a different answer. For example, someone might think that since 3*33=99, and 3*34=102, which is beyond 100, so there are 33 numbers. But maybe someone could think that 100 divided by 3 is 33.333, so maybe they think there are 33.333 multiples? But of course, you can't have a fraction of a number, so you have to take the floor.

Alternatively, someone might think that since 3*33=99, and 3*34=102, so between 1 and 100, there are 33 multiples. So that's consistent.

Alternatively, someone might think that maybe 0 is included? But the problem says numbers from 1 through 100 inclusive, so 0 is not included.

Alternatively, someone might think that numbers like 3, 6, 9,... up to 99. Let me count how many numbers that is. Let me think of it as starting at 3 and ending at 99, with a common difference of 3.

The formula for the number of terms is ((last term - first term)/common difference) + 1.

So, ((99 - 3)/3) + 1 = (96/3) + 1 = 32 + 1 = 33.

Yes, that gives 33 terms. So that's another way to calculate it.

Therefore, all methods confirm that there are 33 multiples of 3 between 1 and 100.

Therefore, the probability is 33/100.

But let me check if 33/100 can be simplified. Well, 33 and 100 share no common factors besides 1. Because 33 factors are 3 and 11, and 100 factors are 2 and 5. So, no common factors. Therefore, 33/100 is already in simplest form.

Therefore, the probability is 33/100.

Alternatively, if I wanted to write it as a decimal, it would be 0.33, or 33%. But the question doesn't specify the form, so since it's a probability, fractional form is probably preferred.

Wait, but let me check once again. Maybe I made a mistake in counting? Let me try a smaller range. Let's say numbers from 1 to 6. How many multiples of 3? 3 and 6, so 2. 6 divided by 3 is 2. So that works. From 1 to 7, multiples of 3 are 3, 6. So 2 numbers. 7 divided by 3 is 2.333, floor is 2. Correct. From 1 to 9, multiples are 3,6,9. 9/3=3. Correct. So same pattern. So 100/3 is 33.333, floor is 33. So that seems correct.

Alternatively, if I think of the numbers from 1 to N, the number of multiples of k is floor(N/k). So here N=100, k=3, so floor(100/3)=33.

Therefore, that formula gives 33.

Therefore, I think I can be confident that the number of favorable outcomes is 33.

Therefore, probability is 33/100.

But just to make sure, let me think of another way. Let me think of how many numbers are not multiples of 3, and subtract from total.

Total numbers: 100.

Numbers not divisible by 3: total numbers minus numbers divisible by 3.

But that might not help directly, but let me try.

Alternatively, if I think of numbers from 1 to 100, each number is either divisible by 3 or not.

But since we already calculated numbers divisible by 3 as 33, then numbers not divisible by 3 would be 100 - 33 = 67.

But maybe I can check that with another method. For example, in any set of three consecutive numbers, exactly one is divisible by 3. So, if we have 100 numbers, how many complete groups of three are there?

Well, 100 divided by 3 is 33 groups with a remainder of 1. So, each group has one multiple of 3, so 33 multiples. Then, the remaining 1 number (which is 100) is not a multiple of 3. Therefore, again, 33 multiples.

Therefore, that also gives the same result.

Therefore, I think I can be confident that the answer is 33/100.

But just to make sure, let me check with actual enumeration for a smaller number. Let's take numbers from 1 to 10.

Multiples of 3: 3,6,9. That's 3 numbers. 10 divided by 3 is 3.333, floor is 3. Correct.

Another example: numbers from 1 to 12. Multiples of 3: 3,6,9,12. That's 4 numbers. 12 divided by 3 is 4. Correct.

So, same pattern. Therefore, for 100, floor(100/3)=33.

Therefore, the probability is 33/100.

Therefore, I think that's the answer.

**Final Answer**
The probability is \boxed{\dfrac{33}{100}}.

To determine the probability that a number selected at random from 1 through 100 is a multiple of 3, we follow a structured approach based on the definition of probability:

---

### **Step 1: Total Number of Outcomes**

We are selecting a number at random from the set of integers from 1 to 100, inclusive. Therefore, the total number of possible outcomes is:

$$
\text{Total outcomes} = 100
$$

---

### **Step 2: Count the Favorable Outcomes**

We are interested in the numbers that are **multiples of 3** within this range. A number is a multiple of 3 if it can be written in the form $3k$, where $k$ is an integer.

To find how many such numbers exist between 1 and 100, we can use the formula for the number of terms in an arithmetic sequence:

$$
\text{Number of multiples of 3} = \left\lfloor \frac{100}{3} \right\rfloor
$$

$$
\frac{100}{3} = 33.\overline{3} \Rightarrow \left\lfloor 33.\overline{3} \right\rfloor = 33
$$

So, there are **33** multiples of 3 between 1 and 100.

---

### **Step 3: Compute the Probability**

The probability of selecting a number that is a multiple of 3 is the ratio of favorable outcomes to total outcomes:

$$
\text{Probability} = \frac{\text{Number of multiples of 3}}{\text{Total numbers}} = \frac{33}{100}
$$

This fraction is already in its simplest form, as 33 and 100 have no common factors other than 1.

---

### **Final Answer**

$$
\boxed{\dfrac{33}{100}}
$$

[Assistant]
Explanation:
Okay, so I need to find the probability that a number selected at random from 1 through 100 is a multiple of 3. Let me think about how to approach this.

First, probability is generally the number of favorable outcomes divided by the total number of possible outcomes. In this case, the total number of possible outcomes is 100 because we're selecting a number from 1 to 100 inclusive. That part seems straightforward.

Now, the tricky part is figuring out how many numbers between 1 and 100 are multiples of 3. Let me recall that a multiple of 3 is any number that can be expressed as 3 times some integer. So, numbers like 3, 6, 9, ..., up to the largest multiple of 3 that's less than or equal to 100.

Hmm, how do I find how many such numbers there are? Maybe I can divide 100 by 3 and take the floor of that? Let me check. If I divide 100 by 3, I get approximately 33.333... So, does that mean there are 33 multiples of 3 between 1 and 100? Wait, let me verify that.

Let me list some multiples of 3 and see. The first multiple is 3*1=3, then 3*2=6, 3*3=9, ..., up to 3*33=99. Wait, 3*33 is 99, which is less than 100. Then the next multiple would be 3*34=102, which is greater than 100. So, yes, the multiples of 3 between 1 and 100 are 3*1 up to 3*33. Therefore, there are 33 multiples of 3.

But hold on, let me make sure I didn't miss anything. Let me check with another method. For example, if I take the number 100 and divide by 3, the quotient is 33 with a remainder of 1. So, that would mean that there are 33 full multiples of 3 in that range. So, that seems to confirm it.

Alternatively, maybe I can use the formula for the number of terms in an arithmetic sequence. The multiples of 3 form an arithmetic sequence where the first term a1 is 3, the common difference d is 3, and the last term an is the largest multiple of 3 less than or equal to 100.

So, to find the number of terms, n, in this sequence, we can use the formula:

an = a1 + (n - 1)d

We know that an is 99, a1 is 3, d is 3. Plugging in:

99 = 3 + (n - 1)*3

Subtract 3 from both sides:

96 = (n - 1)*3

Divide both sides by 3:

32 = n - 1

Add 1 to both sides:

n = 33

So, that gives me 33 terms. Therefore, there are 33 multiples of 3 between 1 and 100.

Wait, but let me check with another approach. Let me count how many numbers between 1 and 100 are divisible by 3. For example, numbers divisible by 3 are every third number. So starting from 3, then 6, 9,... up to 99. So, if I divide 100 by 3, I get 33.333... So, the integer part is 33, which would be the number of multiples. So that seems consistent.

But just to be thorough, let me check with actual counting. Let me think: from 1 to 100, how many numbers are divisible by 3? Let me think of how many complete groups of 3 there are in 100 numbers. Each group of 3 numbers contains exactly one multiple of 3. For example, numbers 1-3: 3 is a multiple; 4-6: 6 is a multiple; 7-9: 9 is a multiple, etc.

So, if there are 100 numbers, how many complete groups of 3 are there? Well, 100 divided by 3 is 33 with a remainder of 1. So, there are 33 complete groups, each contributing one multiple of 3, so 33 multiples. Then, the remaining 1 number (since 3*33=99, so the next number is 100) is not a multiple of 3. Therefore, total multiples of 3 are 33.

Therefore, all methods point to there being 33 multiples of 3 between 1 and 100.

Therefore, the probability would be the number of favorable outcomes (which is 33) divided by the total number of possible outcomes (which is 100). So, probability = 33/100.

But wait, let me check once more. Let me actually list some numbers. For example, from 1 to 10, the multiples of 3 are 3, 6, 9. That's 3 numbers. 10 divided by 3 is approximately 3.333, so floor(10/3)=3. Which matches. Similarly, from 1 to 12, there are 4 multiples: 3,6,9,12. 12/3=4. So that works. So, if I take 100, then floor(100/3)=33. So that seems correct.

Alternatively, if I use the formula floor(100/3) = floor(33.333...) = 33. So that's another way to think about it.

Therefore, I think I can be confident that there are 33 multiples of 3 between 1 and 100.

Therefore, the probability is 33/100.

But let me check if there's a possibility that 100 is a multiple of 3? 100 divided by 3 is 33.333... So no, 100 is not a multiple of 3. Therefore, the last multiple is 99, which is 3*33. So that's correct.

Alternatively, if I wanted to check for numbers from 1 to 99, there would be exactly 33 multiples of 3, since 99 is 3*33. But since we are going up to 100, which is not a multiple of 3, the count remains 33.

Therefore, the probability is 33/100.

But let me think again if there's another way someone might approach this problem and perhaps get a different answer. For example, someone might think that since 3*33=99, and 3*34=102, which is beyond 100, so there are 33 numbers. But maybe someone could think that 100 divided by 3 is 33.333, so maybe they think there are 33.333 multiples? But of course, you can't have a fraction of a number, so you have to take the floor.

Alternatively, someone might think that since 3*33=99, and 3*34=102, so between 1 and 100, there are 33 multiples. So that's consistent.

Alternatively, someone might think that maybe 0 is included? But the problem says numbers from 1 through 100 inclusive, so 0 is not included.

Alternatively, someone might think that numbers like 3, 6, 9,... up to 99. Let me count how many numbers that is. Let me think of it as starting at 3 and ending at 99, with a common difference of 3.

The formula for the number of terms is ((last term - first term)/common difference) + 1.

So, ((99 - 3)/3) + 1 = (96/3) + 1 = 32 + 1 = 33.

Yes, that gives 33 terms. So that's another way to calculate it.

Therefore, all methods confirm that there are 33 multiples of 3 between 1 and 100.

Therefore, the probability is 33/100.

But let me check if 33/100 can be simplified. Well, 33 and 100 share no common factors besides 1. Because 33 factors are 3 and 11, and 100 factors are 2 and 5. So, no common factors. Therefore, 33/100 is already in simplest form.

Therefore, the probability is 33/100.

Alternatively, if I wanted to write it as a decimal, it would be 0.33, or 33%. But the question doesn't specify the form, so since it's a probability, fractional form is probably preferred.

Wait, but let me check once again. Maybe I made a mistake in counting? Let me try a smaller range. Let's say numbers from 1 to 6. How many multiples of 3? 3 and 6, so 2. 6 divided by 3 is 2. So that works. From 1 to 7, multiples of 3 are 3, 6. So 2 numbers. 7 divided by 3 is 2.333, floor is 2. Correct. From 1 to 9, multiples are 3,6,9. 9/3=3. Correct. So same pattern. So 100/3 is 33.333, floor is 33. So that seems correct.

Alternatively, if I think of the numbers from 1 to N, the number of multiples of k is floor(N/k). So here N=100, k=3, so floor(100/3)=33.

Therefore, that formula gives 33.

Therefore, I think I can be confident that the number of favorable outcomes is 33.

Therefore, probability is 33/100.

But just to make sure, let me think of another way. Let me think of how many numbers are not multiples of 3, and subtract from total.

Total numbers: 100.

Numbers not divisible by 3: total numbers minus numbers divisible by 3.

But that might not help directly, but let me try.

Alternatively, if I think of numbers from 1 to 100, each number is either divisible by 3 or not.

But since we already calculated numbers divisible by 3 as 33, then numbers not divisible by 3 would be 100 - 33 = 67.

But maybe I can check that with another method. For example, in any set of three consecutive numbers, exactly one is divisible by 3. So, if we have 100 numbers, how many complete groups of three are there?

Well, 100 divided by 3 is 33 groups with a remainder of 1. So, each group has one multiple of 3, so 33 multiples. Then, the remaining 1 number (which is 100) is not a multiple of 3. Therefore, again, 33 multiples.

Therefore, that also gives the same result.

Therefore, I think I can be confident that the answer is 33/100.

But just to make sure, let me check with actual enumeration for a smaller number. Let's take numbers from 1 to 10.

Multiples of 3: 3,6,9. That's 3 numbers. 10 divided by 3 is 3.333, floor is 3. Correct.

Another example: numbers from 1 to 12. Multiples of 3: 3,6,9,12. That's 4 numbers. 12 divided by 3 is 4. Correct.

So, same pattern. Therefore, for 100, floor(100/3)=33.

Therefore, the probability is 33/100.

Therefore, I think that's the answer.

**Final Answer**
The probability is \boxed{\dfrac{33}{100}}.

To determine the probability that a number selected at random from 1 through 100 is a multiple of 3, we follow a structured approach based on the definition of probability:

---

### **Step 1: Total Number of Outcomes**

We are selecting a number at random from the set of integers from 1 to 100, inclusive. Therefore, the total number of possible outcomes is:

$$
\text{Total outcomes} = 100
$$

---

### **Step 2: Count the Favorable Outcomes**

We are interested in the numbers that are **multiples of 3** within this range. A number is a multiple of 3 if it can be written in the form $3k$, where $k$ is an integer.

To find how many such numbers exist between 1 and 100, we can use the formula for the number of terms in an arithmetic sequence:

$$
\text{Number of multiples of 3} = \left\lfloor \frac{100}{3} \right\rfloor
$$

$$
\frac{100}{3} = 33.\overline{3} \Rightarrow \left\lfloor 33.\overline{3} \right\rfloor = 33
$$

So, there are **33** multiples of 3 between 1 and 100.

---

### **Step 3: Compute the Probability**

The probability of selecting a number that is a multiple of 3 is the ratio of favorable outcomes to total outcomes:

$$
\text{Probability} = \frac{\text{Number of multiples of 3}}{\text{Total numbers}} = \frac{33}{100}
$$

This fraction is already in its simplest form, as 33 and 100 have no common factors other than 1.

---

### **Final Answer**

$$
\boxed{\dfrac{33}{100}}
$$

Solution: There are 100 numbers possible between 1 and 100. There are 33 multiples of 3 between 1 and 100: $(3,6,9,\ldots,99)=(1\times 3,2\times 3,3\times 3,\ldots,33\times 3)$.  So the probability that a randomly selected number is a multiple of 3 is $\boxed{\dfrac{33}{100}}$.