ACT Math Word Problems: 23 Practice Problems with Step-by-Step Explanations
Word problems on the ACT don't test one isolated skill — they test whether you can translate a real-world scenario (deliveries, temperatures, savings accounts, seating arrangements) into the right equation or sequence of steps, often combining several topics in one problem. A single question might layer a rate calculation on top of a ratio, or ask you to set up a system of equations from a wordy business scenario. The math itself is usually not the hard part; correctly modeling the situation is.
A few strategies apply across almost all of these: break multi-part scenarios into separate stages and solve each one before combining them (a trip's total time is driving time + unloading time + resting time, calculated separately); watch for language that implies a ratio or a "twice as many" relationship, which usually means setting up a variable and its multiple rather than two independent unknowns; and for "evenly spaced" or "how many gaps" problems, remember that n objects spaced evenly create n−1 gaps, not n gaps — a common off-by-one trap. Optimization problems (like maximizing revenue) usually reduce to a quadratic, where the maximum sits exactly halfway between the two roots, or can be found by testing the vertex.
Common traps include mixing up which quantity a percentage or ratio applies to, forgetting to convert time units consistently (minutes vs. hours), and stopping the calculation one step too early — many of these problems require finding an intermediate value (like a common difference or a per-unit rate) before answering the actual question asked. Work through the 23 problems below, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Proportional payment
- Problem 2 — Solving for a monthly payment
- Problem 3 — Volume and rate
- Problem 4 — Multi-stage time calculation
- Problem 5 — Sum of a growing sequence
- Problem 6 — Area ratio
- Problem 7 — Scaling dimensions twice
- Problem 8 — Exponential growth
- Problem 9 — Comparing two changing quantities
- Problem 10 — Splitting a total by ratio
- Problem 11 — Multi-stage rate change
- Problem 12 — Least common multiple
- Problem 13 — System from a time budget
- Problem 14 — Evenly spaced gaps
- Problem 15 — Unit conversion and multiplication
- Problem 16 — Rate difference over time
- Problem 17 — Multi-stage trip time
- Problem 18 — Comparing two trip times
- Problem 19 — Revenue optimization
- Problem 20 — Percentage of a rate
- Problem 21 — Arithmetic sequence
- Problem 22 — Least common multiple
- Problem 23 — System of equations
Practice Problems
The total price of a pie bought by Alex and her friends was $23.80. The pie was cut into 7 equal slices, and Alex ate 2 slices. Alex paid the portion of the price that was equal to the portion of the pie she ate. What portion of the total price did Alex pay?
- A) $2.64
- B) $3.40
- C) $5.95
- D) $6.80
Alex ate 2 out of 7 slices, so she pays 2/7 of the total price.
(2/7)(23.80) = 6.80.
Amin bought a new motorcycle. He made an initial payment of $500 and then made 48 equal monthly payments. The total that Amin paid for the motorcycle was $7,700. What was the amount of each of his monthly payments?
- A) $10.42
- B) $150.00
- C) $160.42
- D) $172.92
Subtract the initial payment from the total to find the amount covered by the 48 monthly payments: 7,700−500 = 7,200.
Divide by the number of payments: 7,200/48 = 150.
A fish tank has dimensions, in inches, of 50 by 21 by 33. A hose is filling the tank at a rate of 200 cubic inches per minute. Which of the following is closest to the numbers of minutes it will take to fill the tank?
- A) 6
- B) 56
- C) 173
- D) 347
Find the tank's volume: 50 × 21 × 33 = 34,650 cubic inches.
Divide by the fill rate: 34,650/200 = 173.25, which rounds to 173.
To build a model rocket, Sebastian spent 3 hours on each of 6 workdays. On the 7th day, he spent 3/4 of the time he had worked each of the previous workdays to complete the project. How many total hours did it take Sebastian to complete the model rocket?
- A) 20 1/4
- B) 20 3/4
- C) 21
- D) 23 1/4
The first 6 days total 3 × 6 = 18 hours. On the 7th day, he worked 3/4 of a typical day's 3 hours: (3/4)(3) = 2.25 hours.
Total: 18 + 2.25 = 20.25 = 20 1/4 hours.
Silvia received $100 as a present on her 15th birthday and decided to deposit the money in a savings account. To continue to increase her savings, Silvia decides on a savings plan: for each successive birthday, she will deposit $100 more than the amount deposited for the previous birthday. This is the only money deposited into the account. What is the total amount of money that Silvia has in the account on the day after her 19th birthday?
- A) $800
- B) $1,000
- C) $1,100
- D) $1,500
Deposits happen on birthdays 15 through 19: $100, $200, $300, $400, and $500.
Sum: 100+200+300+400+500 = 1,500.
On a canvas, Malik painted a triangle whose width is half the width of the canvas and whose height is the same as the height of the canvas. What is the ratio of the area of the triangle to the area of the entire canvas?
- A) 1/4
- B) 1/3
- C) 1/2
- D) 3/4
Let the canvas have width w and height h, so canvas area = wh. The triangle has base w/2 and height h.
Triangle area = (1/2)(w/2)(h) = wh/4. Ratio of triangle to canvas: (wh/4)/(wh) = 1/4.
Rectangle X has a length of 48 inches and a width of 30 inches. Rectangle Y has a length and width that are both 2/3 the length and width of Rectangle X. Rectangle Z has a length and width that are both 1/2 the length and width of Rectangle Y. What is the perimeter, in inches, of Rectangle Z?
- A) 156
- B) 78
- C) 52
- D) 40
Rectangle Y: length = (2/3)(48)=32, width = (2/3)(30)=20. Rectangle Z: length = (1/2)(32)=16, width = (1/2)(20)=10.
Perimeter of Z: 2(16+10) = 2(26) = 52.
The number of a certain type of bacteria increases exponentially, tripling every 40 minutes. What is the mass, in grams, of the bacteria exactly 2 hours after the mass first reaches 5 grams?
- A) 50
- B) 135
- C) 200
- D) 405
2 hours is 120 minutes, which is 120/40 = 3 tripling periods.
Mass after 3 triplings: 5 × 33 = 5 × 27 = 135.
A cup of hot coffee that is 84°C is placed into a freezer at the same time as can of soda at −3°C is taken out of a cooler. If the coffee temperature drops by 41°C and the soda temperature increases by 41°C, how do the temperatures compare?
- A) The coffee is 10°C cooler than the soda.
- B) The coffee is 1°C cooler than the soda.
- C) The temperatures are the same.
- D) The coffee is 5°C warmer than the soda.
New coffee temperature: 84−41=43°C. New soda temperature: −3+41=38°C.
Difference: 43−38=5, so the coffee ends up 5°C warmer than the soda.
This week Dan's Donuts sold 6 times as many cinnamon crumb donuts as maple glazed donuts. If this week 168 maple glazed and cinnamon crumb donuts were sold, how many cinnamon crumb donuts were sold?
- A) 24
- B) 42
- C) 120
- D) 144
Let maple glazed = m and cinnamon crumb = 6m. Together: m+6m=168, so 7m=168, giving m=24.
Cinnamon crumb donuts: 6(24)=144.
On Monday, the temperature at 7:00 am was 52°F and rose at a constant rate of 3/2°F per hour until 11:00 am. Then the temperature rose at a rate of 2°F per hour. The temperature first passed above 65°F between:
- A) 12:00pm and 1:00pm
- B) 1:00pm and 2:00pm
- C) 2:00pm and 3:00pm
- D) 4:00pm and 5:00pm
From 7am to 11am (4 hours) at 1.5°F/hr: temperature rises 4(1.5)=6°F, reaching 52+6=58°F at 11am.
After 11am, it rises 2°F/hr. At 2pm (3 hours later): 58+3(2)=64°F (still under 65). At 3pm (4 hours later): 58+4(2)=66°F (over 65). So it first passes 65°F between 2:00pm and 3:00pm.
Jasmine and Shandra start running laps from the same starting line at the same time and in the same direction on an outdoor track. Jasmine completes a lap in 60 seconds and Shandra completes a lap in 45 seconds. Both continue running at their same respective rates and in the same direction for 8 minutes. What is the fewest number of seconds after starting that they will both be back at the starting line at the same time?
- A) 135
- B) 180
- C) 225
- D) 240
Both runners are back at the start together at any common multiple of their lap times. The fewest seconds is the least common multiple of 60 and 45, which is 180.
Since 180 seconds (3 minutes) is less than the total 8-minute run, this is a valid, achievable time.
Jamie makes pies and cookies. It takes her 20 minutes to make a pie and 30 minutes to make a tray of cookies. This weekend Jamie is going to spend 8 hours making pies and cookies. She will make twice as many trays of cookies as pies. How many trays of cookies will she make?
- A) 6
- B) 10
- C) 12
- D) 14
Let p = number of pies, so cookie trays = 2p. Total time in minutes: 20p + 30(2p) = 480 (8 hours = 480 minutes).
20p+60p=480, so 80p=480, giving p=6. Cookie trays: 2(6)=12.
Alice's construction company is building a new modern house with a fence on the north side of the property with a length of 780 feet. If Alice wants to place a total of 13 cameras that are evenly spaced out with two on the ends of the fence, how many feet apart should each camera be?
- A) 55
- B) 60
- C) 65
- D) 70
13 evenly spaced cameras (with 2 on the ends) create 13−1=12 equal gaps along the fence.
780/12 = 65 feet apart.
Dominique creates a snack mix by mixing 3 3/8 pound of plantain chips with 2 3/4 pounds of pretzels and 1 1/2 pounds of dried fruit. If Dominique sells each 1/8 pound of the snack mix for $1.75, what is the amount of money in sales that Dominique will get if he sells all of his snack mix?
- A) $61.00
- B) $71.50
- C) $87.50
- D) $106.75
Total weight: 3 3/8 + 2 3/4 + 1 1/2 = 3.375+2.75+1.5 = 7.625 pounds.
Number of 1/8-pound portions: 7.625 / 0.125 = 61. Total sales: 61 × $1.75 = $106.75.
Jake and Chandler are both standing at the starting line on a 400-meter track when they begin to run at the same time in the same direction around the track. Jake runs at a constant rate of 40 seconds per lap while Chandler runs at a constant rate of 70 seconds per lap. How many seconds after beginning to run will Jake have run exactly 1.5 more laps than Chandler?
- A) 30
- B) 55
- C) 105
- D) 140
Jake's lap rate is 1/40 laps per second; Chandler's is 1/70 laps per second. Set up the equation for the gap in laps: t/40 − t/70 = 1.5.
Using a common denominator of 280: 7t/280 − 4t/280 = 3t/280 = 1.5, so 3t=420, giving t=140.
Jonathan is working as a delivery driver for a trucking company. He starts and ends all of his trips at the same loading dock. Jonathan began his last trip on Friday at 5:00 am when he left the loading dock. During his driving time, he drove 840 miles at an average speed of 40 miles per hour. His driving time was 3 times as long as his unloading time, and his resting time was 16 hours. When did Jonathan end his last trip?
- A) Saturday at 1:00 am
- B) Sunday at 1:00 am
- C) Sunday at 9:00 am
- D) Monday at 1:00 am
Driving time: 840/40 = 21 hours. Unloading time: 21/3 = 7 hours. Resting time: 16 hours.
Total trip time: 21+7+16 = 44 hours. Starting Friday 5:00am + 44 hours = Friday 5am + 24h (Saturday 5am) + 20h = Sunday 1:00am.
Julia and Caroline plan to attend a concert in Indio. Julia will drive 225 miles at a constant speed of 45 miles per hour, stopping one time for a 15-minute break. Caroline will drive 360 miles and for the first 4 hours at 65 miles per hour and then take a 30-minute break before driving at a constant speed of 50 miles per hour for the rest of her trip. How much earlier, in minutes, must Caroline leave before Julia so they arrive at the same time?
- A) 60
- B) 60.25
- C) 75
- D) 125
Julia's total time: driving 225/45=5 hours, plus a 0.25-hour (15-min) break = 5.25 hours.
Caroline's total time: first 4 hours cover 4(65)=260 miles, leaving 360−260=100 miles at 50 mph = 2 hours, plus a 0.5-hour (30-min) break. Total: 4+2+0.5 = 6.5 hours. Difference: 6.5−5.25 = 1.25 hours = 75 minutes.
A t-shirt company sells their signature t-shirt for $18. At this price, 24 t-shirts are sold each day. For every $1 decrease in price, the company will sell 2 extra t-shirts per day. The company will adjust the price to maximize revenue. What is the maximum possible revenue for 1 day?
- A) $432.00
- B) $440.00
- C) $450.00
- D) $880.00
Let x = number of $1 price decreases. Price = 18−x, and quantity sold = 24+2x. Revenue = (18−x)(24+2x) = 432+12x−2x2.
This is a downward parabola; its maximum occurs at x=−b/(2a)=−12/(2(−2))=3. At x=3: price=$15, quantity=30, revenue=15×30=$450.
There are a total of 1,920 calories in a batch of cookies made by John and 230 of those calories are from fat. When making a batch of cookies, John includes 2 sticks of butter. If each stick is equal to 8 tablespoons of butter and if 80% of the calories from fat in the batch of cookies are from the 2 sticks of butter, which of the following is closest to the number of calories from fat per tablespoon of butter?
- A) 12
- B) 23
- C) 46
- D) 96
Fat calories from butter: 80% of 230 = 184. Total tablespoons: 2 sticks × 8 tbsp/stick = 16 tablespoons.
Calories per tablespoon: 184/16 = 11.5, which rounds to about 12.
On the first day of the month, Otto does 5 pushups. On each day after that, Otto does 4 additional pushups than the previous day. How many pushups does Otto do on the 27th day of the month?
- A) 108
- B) 109
- C) 113
- D) 114
This is an arithmetic sequence with first term 5 and common difference 4. Using an=a1+(n−1)d: a27=5+4(27−1)=5+4(26)=5+104=109.
To accommodate the guests at a concert, the stage manager, Frank, tried 3 different seating configurations. One configuration was to have only rows of 9, one was to have only rows of 15, and one was to have only rows of 20. None of these configurations work because for each, the last row had 2 fewer people than the other rows. What is the lowest possible numbers of guests at the concert?
- A) 88
- B) 133
- C) 178
- D) 358
If the total guests N leave a remainder of −2 (equivalently, N+2 is evenly divisible) when divided by 9, 15, and 20, then N+2 must be a common multiple of all three numbers.
The least common multiple of 9, 15, and 20 is 180. The smallest valid total is N=180−2=178.
Gary's watersport shop rents out 1-person, 2-person, and 4-person kayaks for a day. 1-person kayaks cost $75 per day, 2-person kayaks cost $125 per day, and 4-person kayaks cost $175 per day. Yesterday, Gary made $12,250 by renting 120 kayaks. He rented out twice as many 1-person kayaks as 2-person kayaks. How many 4-person kayaks were rented out?
- A) 15
- B) 30
- C) 35
- D) 55
Let 2-person kayaks = x, so 1-person = 2x. Let 4-person = y. Total kayaks: 2x+x+y=120, so 3x+y=120.
Total revenue: 75(2x)+125(x)+175y=12,250, which simplifies to 275x+175y=12,250. Substitute y=120−3x: 275x+175(120−3x)=12,250, giving 275x+21,000−525x=12,250, so −250x=−8,750, and x=35. Then y=120−3(35)=120−105=15.
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