Ratios & Proportions - ACT Math

1. If $\frac{8}{16} = \frac{x}{32}$, what is the value of $x$?
   Answer: 16 In the proportion $8/16 = x/32$, cross-multiply to get $8 \times 32 = 16x$, so $256 = 16x$. Dividing both sides by 16: $x = 16$. Students might make errors in cross-multiplication or simplify the left fraction incorrectly.
2. A recipe uses a ratio of flour to sugar of $5:2$. If you use 30 cups of flour, how many cups of sugar are needed to keep the ratio the same?
   Answer: 12 This recipe ratio problem shows flour to sugar as 5:2, meaning for every 5 cups of flour, we need 2 cups of sugar. Set up the proportion: 5/2 = 30/x, where x is the sugar needed. Cross-multiply: 5x = 2 × 30 = 60, so x = 60 ÷ 5 = 12 cups of sugar. A common error would be setting up the ratio backward as 2:5 instead of 5:2.
3. A photograph is enlarged with a scale factor of 2. If the original width is 6 inches, what is the width of the enlarged photograph?
   Answer: 12 inches A scale factor of 2 means the enlarged photograph is 2 times the original size. If the original width is 6 inches, the enlarged width is 6 × 2 = 12 inches. Students might mistakenly add the scale factor instead of multiplying.
4. A model car is built at a scale of $1:12$ (model:real). If the real car is $180$ inches long, how long is the model in inches?
   Answer: 15 The scale 1:12 means the model is 1/12 the size of the real car. If the real car is 180 inches long, divide by the scale factor: 180 ÷ 12 = 15 inches for the model length. We divide because we're going from real to model size. A common mistake would be to multiply 180 × 12 = 2160, which would make the model larger than the real car.
5. What is the value of $x$ in the proportion $\frac{11}{x}=\frac{22}{18}$? (If $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$.)
   Answer: 9 This proportion requires solving for x in the denominator. Set up the proportion: $11/x = 22/18$. Cross-multiply: $11 \times 18 = 22x$, so $198 = 22x$, and $x = 198 \div 22 = 9$. Cross-multiplication creates an equation that can be solved using basic algebra. A common error would be incorrectly setting up the cross-multiplication products.
6. Which ratio is equivalent to $4:7$?
   Answer: $\frac{8}{14}$ To find an equivalent ratio to $4:7$, we need a fraction that simplifies to $4/7$ or can be written as $4:7$. Check each option: $8/14 = 4/7$ when simplified (dividing both by 2). The other options don't simplify to $4/7$: $7/4$ is the inverse, $12/28 = 3/7$, and $16/21$ doesn't reduce to $4/7$. Remember that equivalent ratios maintain the same proportional relationship.
7. A model car is built at a scale of $1:18$ (model:real). If the real car is $162$ inches long, how long is the model car in inches?
   Answer: 9 The scale is 1:18 (model:real), meaning the model is 1/18 the size of the real car. To find the model length, divide the real car length by 18: 162 inches ÷ 18 = 9 inches. A common mistake is multiplying by 18 instead of dividing, which would give 2,916 inches.
8. A painting is reduced with a scale factor of 1/3. If the original height is 9 feet, what is the height of the reduced painting?
   Answer: 3 feet A scale factor of 1/3 means the reduced painting is 1/3 the original size. If the original height is 9 feet, the reduced height is 9 × (1/3) = 3 feet. Students might confuse fractional scale factors with division or multiplication.
9. What is the scale factor from a rectangle with dimensions 4 cm by 8 cm to one with dimensions 16 cm by 32 cm?
   Answer: 4 The scale factor is the ratio of corresponding dimensions from the smaller to larger rectangle. Comparing the lengths: 16/4 = 4, and comparing the widths: 32/8 = 4. Both ratios give us a scale factor of 4, confirming the rectangles are similar. This means each dimension of the larger rectangle is 4 times the corresponding dimension of the smaller one. A common mistake would be taking the reciprocal, giving 1/4 instead of 4.
10. In a class, the ratio of students wearing sneakers to students wearing boots is $9:6$. If there are $45$ students wearing sneakers, how many students are wearing boots?
    Answer: 30 The ratio of students wearing sneakers to boots is 9:6, which simplifies to 3:2. With 45 students wearing sneakers, set up the proportion: 9/6 = 45/b, where b is students wearing boots. Cross-multiply: 9b = 270, so b = 30 students wearing boots. An error would be using the simplified ratio incorrectly or reversing the ratio.
11. In a school, the ratio of teachers to students is 1:20. If there are 5 teachers, how many students are there?
    Answer: 100 The ratio of teachers to students is 1:20, meaning for every 1 teacher there are 20 students. With 5 teachers, set up the proportion: 1/20 = 5/x. Cross-multiply: x = 100 students. Students might confuse the ratio direction or make multiplication errors.
12. The ratio of apples to oranges in a basket is $3:2$. If there are 15 apples, how many oranges are there?
    Answer: 10 The ratio of apples to oranges is $3:2$, which means for every 3 apples there are 2 oranges. Set up the proportion: $3/2 = 15/x$, where x is the number of oranges. Cross-multiply: $3x = 30$, so x = 10. A common error would be setting up the ratio incorrectly as $2/3 = 15/x$, which would give x = 22.5.
13. If the ratio of red to blue marbles is 4:7 and there are 28 blue marbles, how many red marbles are there?
    Answer: 16 The ratio of red to blue marbles is 4:7, and there are 28 blue marbles. Set up the proportion: 4/7 = x/28, where x is the number of red marbles. Cross-multiply: 7x = 112, so x = 16. Students might reverse the ratio or make calculation errors.
14. If the ratio of cats to dogs in a shelter is 4:3, how many cats are there when there are 21 dogs?
    Answer: 28 With a ratio of cats to dogs of 4:3, we set up the proportion 4/3 = x/21 where x is the number of cats. Cross-multiplying gives us 3x = 4 × 21, so 3x = 84. Dividing both sides by 3, we get x = 28 cats. We can verify: 28:21 simplifies to 4:3 when divided by 7. A common mistake would be reversing the ratio or incorrectly setting up the proportion.
15. The ratio of red balls to green balls in a box is 2:5. If there are 20 red balls, how many green balls are there?
    Answer: 50 The ratio of red to green balls is 2:5, and there are 20 red balls. Set up the proportion: 2/5 = 20/x, where x is the number of green balls. Cross-multiply: 2x = 100, so x = 50. Students might reverse the ratio or make calculation errors.
16. A toy is made with a scale of 1:50. If the actual building is 100 meters tall, how tall is the toy model?
    Answer: 2 meters The scale $1:50$ means the toy is $1/50$ the size of the actual building. If the building is 100 meters tall, the toy height is $100 \div 50 = 2$ meters. Students might confuse scale direction or make division errors.
17. If a recipe calls for a ratio of butter to sugar of 1:4, how much sugar is needed for 8 cups of butter?
    Answer: 32 cups With a butter to sugar ratio of 1:4 and 8 cups of butter, we set up the proportion 1/4 = 8/x where x is cups of sugar needed. Cross-multiplying gives us 1 × x = 4 × 8, so x = 32 cups of sugar. We can verify: 8:32 simplifies to 1:4 when divided by 8. A common error would be reversing the ratio or confusing which quantity corresponds to which part of the ratio.
18. A model car is built to a scale of $1:18$, meaning 1 inch on the model represents 18 inches on the real car. If the real car is 162 inches long, how long is the model car in inches?
    Answer: 9 This scale problem involves a ratio relationship where 1 inch on the model represents 18 inches on the real car. Set up the proportion: 1/18 = x/162, where x is the model car length. Cross-multiply: 18x = 162, so x = 162 ÷ 18 = 9 inches. A common error would be reversing the ratio and multiplying 162 × 18 instead of dividing.
19. In a class, the ratio of students wearing sneakers to students not wearing sneakers is $9:7$. If $63$ students are wearing sneakers, how many students are not wearing sneakers?
    Answer: 49 The ratio of students wearing sneakers to not wearing sneakers is 9:7, with 63 students wearing sneakers. Set up the proportion: 9/7 = 63/x, where x is students not wearing sneakers. Cross-multiply: 9x = 441, so x = 49 students not wearing sneakers. A common error is using 63 as the total instead of just those wearing sneakers.
20. A sports drink is mixed in a ratio of water to concentrate of $7:1$. If you have 6 cups of concentrate, how many cups of water do you need?
    Answer: 42 The sports drink ratio of water to concentrate is 7:1, meaning for every 7 cups of water, there is 1 cup of concentrate. With 6 cups of concentrate, set up the proportion: 7/1 = x/6. Cross-multiply: 1x = 7 × 6 = 42, so x = 42 cups of water. Ratio problems require identifying which quantity corresponds to which part of the ratio.

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