tell whether the two rates form a proportion. 15. 7 inches in 9 hours; 42 inches in 54 hours 16. 12 players from 21 teams; 15 players from 24 teams 17. 440 calories in 4 servings; 300 calories in 3 servings 18. 120 units made in 5 days; 88 units made in 4 days 19. 66 wins in 82 games; 99 wins in 123 games 20. 68 hits in 172 at bats; 43 hits in 123 at bats

Question

Accepted Answer

### Problem 15: 7 inches in 9 hours; 42 inches in 54 hours
# Explanation:
## Step1: Write the rates as fractions
The first rate is $\frac{7}{9}$, the second is $\frac{42}{54}$.
## Step2: Simplify the second fraction
Simplify $\frac{42}{54}$ by dividing numerator and denominator by 6: $\frac{42\div6}{54\div6}=\frac{7}{9}$.
## Step3: Compare the fractions
Since $\frac{7}{9}=\frac{7}{9}$, the rates form a proportion.
# Answer: Yes

### Problem 16: 12 players from 21 teams; 15 players from 24 teams
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{12}{21}$, second rate: $\frac{15}{24}$.
## Step2: Simplify both fractions
Simplify $\frac{12}{21}$ by dividing by 3: $\frac{4}{7}$. Simplify $\frac{15}{24}$ by dividing by 3: $\frac{5}{8}$.
## Step3: Compare the fractions
Since $\frac{4}{7}
eq\frac{5}{8}$, the rates do not form a proportion.
# Answer: No

### Problem 17: 440 calories in 4 servings; 300 calories in 3 servings
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{440}{4}$, second rate: $\frac{300}{3}$.
## Step2: Simplify both fractions
$\frac{440}{4} = 110$, $\frac{300}{3}=100$.
## Step3: Compare the results
Since $110
eq100$, the rates do not form a proportion.
# Answer: No

### Problem 18: 120 units made in 5 days; 88 units made in 4 days
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{120}{5}$, second rate: $\frac{88}{4}$.
## Step2: Simplify both fractions
$\frac{120}{5}=24$, $\frac{88}{4} = 22$.
## Step3: Compare the results
Since $24
eq22$, the rates do not form a proportion.
# Answer: No

### Problem 19: 66 wins in 82 games; 99 wins in 123 games
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{66}{82}$, second rate: $\frac{99}{123}$.
## Step2: Simplify the second fraction (or cross - multiply)
Cross - multiply: $66	imes123 = 8118$, $99	imes82=8118$.
## Step3: Compare the products
Since $66	imes123 = 99	imes82$, the rates form a proportion.
# Answer: Yes

### Problem 20: 68 hits in 172 at bats; 43 hits in 123 at bats
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{68}{172}$, second rate: $\frac{43}{123}$.
## Step2: Simplify the first fraction
Simplify $\frac{68}{172}$ by dividing by 4: $\frac{17}{43}$. Simplify $\frac{43}{123}$ by dividing by 43: $\frac{1}{3}$. Wait, no, simplify $\frac{43}{123}$ by dividing numerator and denominator by 43: $\frac{43\div43}{123\div43}=\frac{1}{3}$? No, 123÷43 = 3, so $\frac{43}{123}=\frac{1}{3}$? Wait, no, 68 and 172: 172÷4 = 43, 68÷4 = 17. So $\frac{68}{172}=\frac{17}{43}$. $\frac{43}{123}=\frac{1}{3}$? No, 43×3 = 129≠123. Wait, cross - multiply: $68	imes123=8364$, $43	imes172 = 7396$. Since $8364
eq7396$, or simplify $\frac{68}{172}=\frac{17}{43}$, $\frac{43}{123}=\frac{43}{123}$ (can't simplify to $\frac{17}{43}$). So $\frac{17}{43}
eq\frac{43}{123}$.
Wait, correct simplification: $\frac{68}{172}$: GCD of 68 and 172 is 4, so $\frac{17}{43}$. $\frac{43}{123}$: GCD of 43 and 123 is 43, so $\frac{1}{3}$. Wait, no, 43×3 = 129, not 123. Wait, 123÷43 = 2.86... No, 43×2 = 86, 123 - 86 = 37. So my mistake, GCD of 43 and 123: 43 is prime, 123÷43 = 2 with remainder 37, so GCD is 1. So $\frac{43}{123}$ can't be simplified. $\frac{17}{43}\approx0.395$, $\frac{43}{123}\approx0.349$. So they are not equal.
## Step3: Compare the fractions
Since $\frac{17}{43}
eq\frac{43}{123}$, the rates do not form a proportion.
# Answer: No

Wait, for problem 19, let's re - check: 66 wins in 82 games: $\frac{66}{82}$, 99 wins in 123 games: $\frac{99}{123}$. Simplify $\frac{66}{82}$ by dividing by 2: $\frac{33}{41}$. Simplify $\frac{99}{123}$ by dividing by 3: $\frac{33}{41}$. Oh! I made a mistake earlier. 99÷3 = 33, 123÷3 = 41. So $\frac{99}{123}=\frac{33}{41}$, and $\frac{66}{82}=\frac{33}{41}$. So they are equal. So my earlier cross - multiply was wrong. Let's correct problem 19:

### Problem 19: 66 wins in 82 games; 99 wins in 123 games
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{66}{82}$, second rate: $\frac{99}{123}$.
## Step2: Simplify both fractions
Simplify $\frac{66}{82}$ by dividing numerator and denominator by 2: $\frac{33}{41}$. Simplify $\frac{99}{123}$ by dividing numerator and denominator by 3: $\frac{33}{41}$.
## Step3: Compare the fractions
Since $\frac{33}{41}=\frac{33}{41}$, the rates form a proportion.
# Answer: Yes

### Problem 20: 68 hits in 172 at bats; 43 hits in 123 at bats
# Explanation:
## Step1: Write the rates as fractions
First rate: $\frac{68}{172}$, second rate: $\frac{43}{123}$.
## Step2: Simplify the first fraction
Divide numerator and denominator of $\frac{68}{172}$ by 4: $\frac{17}{43}$.
## Step3: Simplify the second fraction
Divide numerator and denominator of $\frac{43}{123}$ by 43: $\frac{1}{3}$? No, 123÷43 = 3, so $\frac{43}{123}=\frac{1}{3}$? Wait, 43×3 = 129, not 123. Wait, 123 = 43×2+37, so GCD is 1. Wait, no, 68 and 172: GCD 4, so $\frac{17}{43}$. 43 and 123: 123 = 43×2 + 37, 43 = 37×1+6, 37 = 6×6+1, so GCD is 1. So $\frac{17}{43}\approx0.395$, $\frac{43}{123}\approx0.349$. Not equal. Or cross - multiply: 68×123 = 8364, 43×172 = 7396. 8364≠7396. So the rates do not form a proportion.
# Answer: No

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Question

Explanation:

Problem 15: 7 inches in 9 hours; 42 inches in 54 hours

Step1: Write the rates as fractions

Step2: Simplify the second fraction

Step3: Compare the fractions

Step1: Write the rates as fractions

Step2: Simplify both fractions

Step3: Compare the fractions

Step1: Write the rates as fractions

Step2: Simplify both fractions

Step3: Compare the results

Answer:

Problem 16: 12 players from 21 teams; 15 players from 24 teams