ACT Math Ratios and Proportions: 15 Practice Problems with Step-by-Step Explanations
Ratios and proportions questions show up throughout the ACT Math section, often disguised inside word problems about maps, mixtures, ages, construction, and sports teams rather than appearing as a labeled "ratio" question. At its core, this chapter tests whether you can split a total into parts based on a given ratio, scale a rate up or down, and recognize the difference between direct variation (two quantities increase together) and inverse variation (one quantity increases as the other decreases).
The most reliable technique for part-to-part ratio problems is to add up the ratio's parts to find how many total "shares" make up the whole, divide the actual total by that number of shares to find the value of one share, and then multiply by however many shares each part represents. This same technique extends to combining two separate ratios that share a common term — for example, a:b and b:c — by scaling one or both ratios so the shared term matches before combining them into a single a:b:c ratio.
Common traps include forgetting to add the ratio's parts before dividing (and instead dividing by the number of parts times something unrelated), mixing up direct and inverse variation, and, in geometry-flavored ratio problems, forgetting that scaling a 3-dimensional shape's ratio scales its volume by the product of all three scale factors, not just one. Work through the 15 problems below, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Part-to-part ratio
- Problem 2 — Scaling a rate (map scale)
- Problem 3 — Splitting a total by ratio
- Problem 4 — Splitting a total, 3-part ratio
- Problem 5 — Rate with unit conversion
- Problem 6 — Scaling a rate (time)
- Problem 7 — Direct variation
- Problem 8 — Splitting a total, part vs. multiple
- Problem 9 — Direct and inverse variation formula
- Problem 10 — Ratio applied to area
- Problem 11 — Direct variation with a square root
- Problem 12 — Ratio applied to angle sum
- Problem 13 — Combining two ratios
- Problem 14 — Ratio applied to volume
- Problem 15 — Weighted average from a ratio
Practice Problems
If a dodgeball team has 6 girls and 11 total team members, what is the ratio of boys to girls?
- A) 5:6
- B) 6:5
- C) 6:11
- D) 12:10
The number of boys is the total members minus the girls: 11 − 6 = 5. The ratio of boys to girls is therefore 5:6.
Answer B reverses the order (girls to boys instead of boys to girls) — always check which quantity the question asks for first.
On a map, 1/2 inch represents 15 miles. How many miles apart are two gas stations that are 3 1/2 inches apart on the map?
- A) 15
- B) 25
- C) 55
- D) 105
If 1/2 inch represents 15 miles, then 1 full inch represents 30 miles. The two gas stations are 3.5 inches apart on the map, so the actual distance is 3.5 × 30 = 105 miles.
The ratio of Joey's age to his brother's age is 4:3. The sum of their ages is 49. How old is Joey?
- A) 21
- B) 24
- C) 28
- D) 30
The ratio 4:3 splits their combined age into 4 + 3 = 7 equal parts. Each part is worth 49 / 7 = 7 years.
Joey's age is 4 parts: 4 × 7 = 28.
The combined length of 3 pieces of fabric is 105 inches. The lengths of the pieces are in a 4:5:6 ratio. What is the length, in inches, of the longest piece of fabric?
- A) 24
- B) 28
- C) 35
- D) 42
The ratio 4:5:6 splits the total length into 4 + 5 + 6 = 15 equal parts. Each part is worth 105 / 15 = 7 inches.
The longest piece corresponds to 6 parts: 6 × 7 = 42 inches.
Atrazine, a weed killer commonly used by farmers, is such a concentrated substance that only 1 liter can be used to spray up to 4 hectares of farm fields. If one hectare is equal to approximately 2 1/2 acres, how many acres of farm fields could 25 liters of atrazine be used to spray?
- A) 100
- B) 150
- C) 250
- D) 625
Since 1 liter sprays 4 hectares, 25 liters can spray 25 × 4 = 100 hectares.
Convert hectares to acres using the given rate of 2.5 acres per hectare: 100 × 2.5 = 250 acres.
Samantha rows 30 meters in 6.8 seconds. If she rows at the same rate for 2 minutes, which of the following is closest to the number of meters she will row?
- A) 200
- B) 410
- C) 530
- D) 720
First convert 2 minutes to seconds so the units match: 2 × 60 = 120 seconds. Samantha's rate is 30 meters per 6.8 seconds, so set up a proportion: 30/6.8 = x/120.
Solving: x = 120 × (30/6.8) = 3,600/6.8 ≈ 529.4, which rounds to the closest answer choice, 530.
If x varies directly with y and x = 15 when y = 3, what is the value of x when y = 15?
- A) 3
- B) 5
- C) 15
- D) 75
Direct variation means x = ky for some constant k. Using the given values: 15 = k(3), so k = 5.
When y = 15: x = 5 × 15 = 75.
Four friends buy a winning lottery ticket worth $25,000. They agree to give four times as much to charity as they will spend for fun. How much do they give to charity?
- A) $5,000
- B) $12,500
- C) $18,000
- D) $20,000
If "fun" spending is 1 part, "charity" is 4 parts (four times as much), for a total of 1 + 4 = 5 parts. Each part is worth $25,000 / 5 = $5,000.
Charity gets 4 parts: 4 × $5,000 = $20,000.
A recent study examined how the amount of sunlight (S), pesticides (P), and organic matter (M) affected the height of corn plants (H). It was discovered that sunlight and organic matter varied directly with plant height while pesticides varied inversely. Which of the following formulas is correct?
- A) H = kSMP
- B) H = kSM/P
- C) H = kP/SM
- D) H = k/SMP
Variables that vary directly with H (sunlight and organic matter) belong in the numerator, multiplied together. A variable that varies inversely with H (pesticides) belongs in the denominator.
That gives H = kSM/P, where k is the constant of variation.
In a certain rectangle, the ratio of the lengths of the 2 sides is 7:4. If the area of the rectangle is 252 square inches, what is the length of the shorter side?
- A) 12
- B) 14
- C) 18
- D) 21
Let the sides be 7x and 4x, matching the given ratio. The area is length × width: (7x)(4x) = 28x² = 252, so x² = 9 and x = 3.
The shorter side is 4x = 4(3) = 12 inches.
If m is directly proportional with the square root of n and m = 40 when n = 25, what is the value of m when n = 13?
- A) 20.8
- B) 2√13
- C) 8√13
- D) 8√21
Direct variation with a square root means m = k√n. Using the given values: 40 = k√25 = k(5), so k = 8.
When n = 13: m = 8√13. Since 13 isn't a perfect square, leave the answer in radical form rather than trying to force a decimal.
The degree measures of the interior angles in a hexagon are in the ratio of 1:2:2:3:5:7. What is the measure of the largest interior angle of the hexagon?
- A) 48
- B) 96
- C) 144
- D) 252
The interior angles of any hexagon sum to (6 − 2) × 180 = 720 degrees. The given ratio has 1 + 2 + 2 + 3 + 5 + 7 = 20 total parts, so each part is worth 720 / 20 = 36 degrees.
The largest angle corresponds to 7 parts: 7 × 36 = 252 degrees.
The ratio of a to b is 4 to 1. The ratio of b to c is 3 to 10. What is the value of (2a − c) / (3b)?
- A) −2
- B) −2/3
- C) 7/9
- D) 14/9
To combine a:b = 4:1 with b:c = 3:10, make the b terms match. Scale a:b = 4:1 by 3 to get a:b = 12:3, which now shares b = 3 with the second ratio (b:c = 3:10). Combined: a:b:c = 12:3:10.
Substitute a = 12, b = 3, c = 10 into the expression: (2(12) − 10) / (3(3)) = (24 − 10) / 9 = 14/9.
Javon is constructing a box. The ratio of the length to the width to the height of the box must be 5 to 4 to 2. If the volume of the box is 2,560 cubic inches, what is the height, in inches, of the box?
- A) 4
- B) 8
- C) 20
- D) 32
Let the dimensions be 5x, 4x, and 2x, matching the ratio. Volume is length × width × height: (5x)(4x)(2x) = 40x³ = 2,560, so x³ = 64 and x = 4.
The height is 2x = 2(4) = 8 inches.
The high school football team is raising money to upgrade the turf on the field. The team has only sophomore, junior, and senior members. The ratio of sophomores to juniors to seniors on the football team is 2:4:5. Sophomores on average raised $40 per person, juniors on average raised $55 per person, and seniors on average raised $72 per person. What is the average amount raised per team member?
- A) 55
- B) 57
- C) 59
- D) 60
Let the group sizes be 2k, 4k, and 5k, matching the ratio, for a total of 11k team members. The total amount raised is (2k)(40) + (4k)(55) + (5k)(72) = 80k + 220k + 360k = 660k.
The average per member is total raised divided by total members: 660k / 11k = 60. The k cancels out, so the actual team size never needs to be known.
Want more? Practice 2,500+ free ACT Math questions with instant explanations.
Try the Free ACT Math QBank →