8. In a poll that interviewed 1,000 people about their favorite color, 640 people said they liked blue. a. Simplify this ratio to lowest terms. b. Assuming the same ratio holds true in another group of 125 people, how many of those people can we expect to like blue?

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**Problem 8**

In a poll that interviewed 1,000 people about their favorite color, 640 people said they liked blue.

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**a. Simplify this ratio to lowest terms.**

To simplify the ratio of people who like blue to the total number of people surveyed:

1. **Original Ratio:** 640 (blue) : 1,000 (total)

2. **Find the Greatest Common Divisor (GCD) of 640 and 1,000:**
   - Prime factors of 640: $$ 640 = 2^7 \times 5 $$
   - Prime factors of 1,000: $$ 1,000 = 2^3 \times 5^3 $$
   - **GCD:** $$ 2^3 \times 5 = 8 \times 5 = 40 $$

3. **Divide both terms by the GCD:**
   $$
   \frac{640}{40} : \frac{1,000}{40} = 16 : 25
   $$

**Simplified Ratio:** **16 : 25**

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**b. Assuming the same ratio holds true in another group of 125 people, how many of those people can we expect to like blue?**

Given the simplified ratio **16 : 25**, we can set up a proportion to find the expected number of people who like blue in a group of 125.

1. **Set Up the Proportion:**
   $$
   \frac{16}{25} = \frac{x}{125}
   $$
   
   Where $$ x $$ is the number of people expected to like blue.

2. **Solve for $$ x $$:**
   $$
   x = \frac{16}{25} \times 125
   $$
   
3. **Calculate:**
   $$
   x = 16 \times 5 = 80
   $$

**Expected Number of People Who Like Blue:** **80**

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**Summary:**
- **a.** The simplified ratio is **16 : 25**.
- **b.** In a group of 125 people, we can expect **80** to like blue.
**a. Simplified Ratio:** 16 : 25 **b. Expected Number of People Who Like Blue in 125 People:** 80

In a poll that interviewed 1,000 people about their favorite color, 640 people said they liked blue.
a. Simplify this ratio to lowest terms.
b. Assuming the same ratio holds true in another group of 125 people,
how many of those people can we expect to like blue?

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