ASVAB word problem types and the setup for each
Most ASVAB Arithmetic Reasoning questions fall into about ten repeating types: percent, ratio, rate, work, age, mixture, coins, consecutive numbers, unit conversion and averages. If you can name the type in the first read, you already know the setup. This page gives the signal words, the equation for each type and a worked example.
What kinds of word problems are on ASVAB Arithmetic Reasoning?
Arithmetic Reasoning is a word problem test. The official description is the "ability to solve arithmetic word problems" (What to Expect). On the CAT-ASVAB, Arithmetic Reasoning has 15 scored questions with a 55 minute limit when no tryout questions are added (Official ASVAB: CAT-ASVAB). The paper version has 30 questions in 36 minutes (What to Expect). The stories change (soldiers, recipes, paychecks, trucks), but the math underneath repeats.
Use this table as a sorting guide. When you read a problem, find the signal words and jump to the matching setup.
| Type | Signal words | Setup |
|---|---|---|
| Percent | discount, tax, tip, raise, % of | part = percent × whole |
| Ratio and proportion | ratio, for every, per, scale | set two fractions equal and cross multiply |
| Distance and rate | mph, speed, how long, how far | d = r × t |
| Work together | together, both working, fill, empty | 1/a + 1/b = 1/total time |
| Age | years ago, in X years, twice as old | one variable per person, shift both by the same years |
| Mixture | solution, %, blend, mix | amount × concentration, add the parts |
| Coins and tickets | nickels, dimes, adult and child tickets | count equation plus value equation |
| Consecutive numbers | consecutive, next, in a row | n, n + 1, n + 2 (or n + 2, n + 4 for even or odd) |
| Unit conversion | feet, yards, ounces, hours | multiply by a conversion fraction equal to 1 |
| Average | mean, average, per person | total = average × count |
A general four-step method (find what is asked, list facts, set up, check) is in the word problem lesson. This page goes type by type.
How do you solve age problems?
Give each person a variable for their age now, then add or subtract the same number of years for everyone when the problem moves in time. Most errors come from shifting only one person's age.
Example: A mother is 3 times as old as her son. In 10 years she will be twice as old as he is. How old is the son now?
- Let s be the son's age now. The mother is 3s.
- In 10 years: son s + 10, mother 3s + 10.
- Set up "twice as old": 3s + 10 = 2(s + 10) = 2s + 20.
- Subtract 2s from both sides: s + 10 = 20, so s = 10.
- Check: son 10, mother 30. In 10 years, 20 and 40. 40 is twice 20.
Answer: the son is 10 (the mother is 30)
How do you solve consecutive number problems?
Call the first number n and write the others in terms of n. Consecutive integers go n, n + 1, n + 2. Consecutive even or consecutive odd numbers both go n, n + 2, n + 4.
Example: The sum of three consecutive even numbers is 78. What are they?
- Write them as n, n + 2, n + 4.
- Add: 3n + 6 = 78.
- 3n = 72, so n = 24.
- The numbers are 24, 26 and 28. Check: 24 + 26 + 28 = 78.
Answer: 24, 26 and 28
Shortcut for an odd count of consecutive numbers: the middle one equals the sum divided by the count. 78 ÷ 3 = 26, so the set is 24, 26, 28.
How do you solve "working together" problems?
Convert each worker's time into a rate (jobs per hour), add the rates, then flip the total. A pump that fills a tank in 6 hours fills 1/6 of the tank each hour.
Example: One pump fills a tank in 6 hours. A second pump fills it in 3 hours. How long do they take together?
- Rates: 1/6 and 1/3 of a tank per hour.
- Add with a common denominator: 1/6 + 2/6 = 3/6 = 1/2 tank per hour.
- Flip the rate: 1 tank ÷ 1/2 per hour = 2 hours.
Answer: 2 hours
Sanity check: working together must be faster than the faster worker alone. If your answer is more than 3 hours here, something went wrong. A common wrong move is to average the times (4.5 hours), which ignores that both pumps run at once.
How do you solve coin and ticket problems?
Write two equations: one counts the items and one adds up their value. Work in cents to avoid decimals.
Example: A jar holds 25 coins, all dimes and quarters, worth $4.30. How many of each are there?
- Let q be quarters. Then dimes = 25 − q.
- Value in cents: 10(25 − q) + 25q = 430.
- 250 − 10q + 25q = 430, so 250 + 15q = 430.
- 15q = 180, so q = 12. Dimes = 25 − 12 = 13.
- Check: 12 × 25 = 300 cents and 13 × 10 = 130 cents. 300 + 130 = 430.
Answer: 12 quarters and 13 dimes
Ticket problems (adult and child prices) use the same two equations with ticket prices in place of coin values.
How do you solve ratio and proportion problems?
Set up two equal fractions with the same units in the same positions, then cross multiply. Ratio problems that give a total use "parts": add the ratio numbers to get the number of parts.
Example: Five gallons of paint cover 1,750 square feet. How many gallons are needed for 2,450 square feet?
- Proportion: 5 / 1,750 = g / 2,450.
- Cross multiply: 1,750g = 5 × 2,450 = 12,250.
- g = 12,250 ÷ 1,750 = 7.
- Check with a unit rate: 1,750 ÷ 5 = 350 square feet per gallon. 2,450 ÷ 350 = 7.
Answer: 7 gallons
More ratio practice, including part-to-whole totals, is in the ratios and rates lesson.
How do you solve mixture problems?
Multiply each amount by its concentration to find how much of the pure ingredient it holds, add those, then divide by the total amount. Averaging the two percentages only works when the amounts are equal.
Example: You mix 10 liters of a 20% solution with 30 liters of a 40% solution. What is the concentration of the mix?
- Pure part in the first: 10 × 0.20 = 2 liters.
- Pure part in the second: 30 × 0.40 = 12 liters.
- Total pure part: 2 + 12 = 14 liters. Total mix: 10 + 30 = 40 liters.
- 14 ÷ 40 = 0.35.
Answer: 35%
How do you solve percent and unit conversion word problems?
Translate the words into an equation: "of" means multiply, "is" means equals, and a percent becomes a decimal. For unit conversions, multiply by a fraction that equals 1, such as 12 inches over 1 foot, and cancel the units you do not want.
Example: A shop buys a jacket for $40 and marks it up 30%. What is the selling price?
- 30% of 40: 10% is 4, so 30% is 12.
- Selling price: 40 + 12 = 52.
- Faster: 40 × 1.30 = 52.
Answer: $52
Percent word problems come in three forms. You may need the part (what is 30% of 40), the whole (12 is 30% of what) or the percent (12 is what percent of 40). Write "part = percent × whole" and fill in the two numbers you know. The percentages lesson drills all three.
Example: A crew lays 36 inches of pipe every 4 minutes. How many feet of pipe does it lay per hour?
- Convert inches to feet: 36 ÷ 12 = 3 feet every 4 minutes.
- Per minute: 3 ÷ 4 = 0.75 feet.
- Per hour: 0.75 × 60 = 45 feet.
- Check another way: an hour has 60 ÷ 4 = 15 four-minute blocks, and 15 × 3 = 45.
Answer: 45 feet per hour
Conversions to memorize: 12 inches in a foot, 3 feet in a yard, 5,280 feet in a mile, 16 ounces in a pound, 2,000 pounds in a ton, 4 quarts in a gallon, 2 pints in a quart, 60 minutes in an hour. Metric conversions go by powers of 10: 100 centimeters in a meter and 1,000 meters in a kilometer. Most conversion errors come from multiplying when you should divide. Ask yourself whether the answer should be a bigger or smaller number: there are more inches than feet, so inches to feet means dividing.
What should you do when a word problem does not fit any type?
Go back to the basic plan: write down exactly what the question asks, list the numbers with their units, and draw a quick sketch or table. Many "new" problems are two familiar types joined together, such as a unit conversion followed by a percent.
- Check units first. If the answer must be in yards and the data is in feet, convert early.
- Estimate before you compute so you can reject choices that are far off.
- Plug in the answer choices when setting up an equation feels slow. Start with a middle value.
Distance and speed questions have their own guide: distance, rate and time problems. Percent and interest questions are covered in interest and percent change.
Practice questions
Work each one on paper first, then open the explanation. These are original AptiStep items, not real ASVAB questions.
The sum of two consecutive odd numbers is 56. What is the larger number?
- 27
- 28
- 29
- 31
Show answer and explanation
Answer: C. 29
Write the numbers as n and n + 2. Then 2n + 2 = 56, so 2n = 54 and n = 27. The larger number is 27 + 2 = 29. Check: 27 + 29 = 56. The choice 28 is the average, which is even and cannot be odd.
One painter can paint a room in 4 hours. A second painter needs 12 hours. How long will they take working together?
- 8 hours
- 3 hours
- 16 hours
- 6 hours
Show answer and explanation
Answer: B. 3 hours
Rates are 1/4 and 1/12 room per hour. 1/4 = 3/12, so together 3/12 + 1/12 = 4/12 = 1/3 room per hour. Flip it: 3 hours. Averaging the times (8 hours) is the common mistake, and together must be faster than 4 hours.
Tom is 6 years older than Ana. The sum of their ages is 40. How old is Ana?
- 23
- 20
- 14
- 17
Show answer and explanation
Answer: D. 17
Let Ana be a. Tom is a + 6. Then a + a + 6 = 40, so 2a = 34 and a = 17. Tom is 23. Check: 17 + 23 = 40. Picking 23 gives Tom's age instead of Ana's.
A motor pool has trucks and cars in a ratio of 2 to 5. There are 63 vehicles. How many are cars?
- 45
- 18
- 35
- 25
Show answer and explanation
Answer: A. 45
The ratio has 2 + 5 = 7 parts. Each part is 63 ÷ 7 = 9 vehicles. Cars are 5 parts: 5 × 9 = 45. Trucks are 2 × 9 = 18, which is the distractor for reading the wrong part.
You have 20 coins, all nickels and dimes, worth $1.55. How many dimes do you have?
- 9
- 10
- 11
- 15
Show answer and explanation
Answer: C. 11
Let d be dimes, so nickels = 20 − d. Value in cents: 5(20 − d) + 10d = 155. That gives 100 + 5d = 155, so 5d = 55 and d = 11. Check: 11 dimes = 110 cents, 9 nickels = 45 cents, total 155.
Fencing costs $8 per yard. How much does fencing for a 54 foot side cost?
- $432
- $144
- $48
- $216
Show answer and explanation
Answer: B. $144
Convert feet to yards first: 3 feet = 1 yard, so 54 ÷ 3 = 18 yards. Cost = 18 × 8 = $144. The $432 choice multiplies feet by the price per yard without converting.
You mix 10 liters of a 20% salt solution with 30 liters of a 40% salt solution. What percent salt is the mix?
- 30%
- 60%
- 25%
- 35%
Show answer and explanation
Answer: D. 35%
Salt in the first: 10 × 0.2 = 2 liters. Salt in the second: 30 × 0.4 = 12 liters. Total salt is 14 liters in 40 liters of mix. 14 ÷ 40 = 0.35, or 35%. Averaging 20% and 40% gives 30%, which ignores that there is more of the 40% solution.
Frequently asked questions
How many word problem types do I really need?
Percent, ratio, rate and average problems make up a large share of what you will see. Learn those four setups first, then add work, age, mixture and coin problems.
Should I write equations or plug in answer choices?
Both work. Writing an equation is faster once you recognize the type. Plugging in choices is a good backup when you are stuck, since one choice must work.
Are the word problems on the CAT-ASVAB easier or harder?
The CAT-ASVAB is adaptive: the official page explains that after a wrong answer "an easier item is selected" next (CAT-ASVAB). The difficulty you see depends on how you answer. See CAT-ASVAB vs paper.
What if I run out of time on Arithmetic Reasoning?
The official CAT-ASVAB page says unfinished items are scored "as though they were answered at random" (CAT-ASVAB). Pace yourself so you reach every question.
Sources
Checked November 14, 2026. AptiStep is an independent study site; confirm anything that affects your enlistment with a recruiter.