A word problem is easier to solve once you know what the answer must measure. Start with a worked discount-and-tax problem, learn how to check the setup, then apply that habit to rates, ratios and other arithmetic questions.
What you’ll learn
Learn the method
A jacket costs $80. A store takes 25% off, then adds 5% sales tax to the discounted price. What is the final cost?
The discount uses the original $80 price. The tax uses the sale price, which you have not found yet. Write the order before doing arithmetic.
25% is one-fourth. One-fourth of $80 is $20, so the discounted price is $80 − $20 = $60.
10% of $60 is $6, so 5% is $3. The final cost is $60 + $3 = $63.
$63. Each percentage uses the amount specified at that stage.
The tempting $64 answer taxes the original $80: 5% of $80 is $4. But the problem says to tax the discounted price. Subtracting 25% and adding 5% as if both use the same base makes the same mistake.
If the discount stayed the same but the tax doubled to 10%, would the final cost double?
No. The sale price stays $60. Only the tax doubles, from $3 to $6, making the final cost $66. The question changes one part of the total.
The example teaches one method. Open a concept below when you need a rule or a different type of example during practice.
Write what the answer must describe: dollars per person, miles traveled, hours required or a remaining amount. Then label the quantities you know. A rate of 48 miles per hour multiplied by 2.5 hours gives 120 miles; dividing distance by that rate would give hours instead. The units help you decide which operation makes sense.
A 20% discount on an $80 item is $16, leaving $64 to pay. The discount and the sale price answer different questions. Percentage change uses the original value as the denominator: a change from 40 to 50 is 10 ÷ 40 = 25%. A later 25% decrease starts from 50, so it does not reverse that increase.
If red and blue parts are in a 2:3 ratio, there are five equal shares altogether. For 35 parts, each share is seven, giving 14 red and 21 blue. For averages, use total = average × number of values. Five scores averaging 18 total 90; subtract the known scores to recover a missing one.
A machine completing a batch in six hours works at one-sixth of a batch per hour. Another taking three hours works at one-third per hour. At constant rates together, they complete one-half of a batch per hour, so one batch takes two hours. This model assumes they can work independently on the same task; use the conditions the question gives.
Now apply what you know
Try a few questions without opening the solutions, then explain why each answer works. Use scratch paper for calculations and work without a calculator. These original, untimed questions sample the subject; the result is not an official score.
Work through one question at a time. Explain your approach, check the worked answer, then move on. Use scratch paper for calculations.
0 of 12 checked · 0 correct.
Progress is saved in this browser. These original study questions do not produce an official AFQT score.
Answer: D. 15 pounds
The total weight is 11 + 14 + 15 + 20 = 60 pounds. Dividing by the four parcels gives 60 ÷ 4 = 15 pounds. Dividing by five would give 12, but there are only four observations. Fourteen is one parcel’s weight, not the mean.
Answer: C. $564
The commission applies to $3,900 − $1,500 = $2,400. Six percent of $2,400 is $144, so total pay is $420 + $144 = $564. The $654 choice applies the commission to all $3,900 instead of only the sales above the threshold.
Answer: A. 40 liters
The tank initially contains 96 × 3/4 = 72 liters. The removed amount is 96 × 1/3 = 32 liters because the fraction refers to capacity. The remainder is 72 − 32 = 40 liters. Removing one third of the current 72 liters would leave 48, but that uses the wrong base.
Answer: B. 6 bags
The garden has 80 square meters. Adding the two path areas gives 20 + 16, but their 2-by-2 intersection is counted twice, so path area is 20 + 16 − 4 = 32. The remaining area is 48 square meters. Five bags cover 45, which is too little; six cover 54, enough. Subtracting both path strips without restoring their overlap gives the incorrect smaller area 44 and would wrongly suggest that five bags suffice.
Answer: B. 10
The first week has 0.12 × 200 = 24 returns, and the second has 0.07 × 200 = 14. The decrease is 24 − 14 = 10 tickets. The difference between 12% and 7% is 5 percentage points; 5 is not the number of tickets.
Answer: C. $90
The workers contributed 3 + 5 + 4 = 12 hours altogether. The five-hour worker receives 5/12 of the bonus: $216 × 5/12 = $90. An equal three-way split would be $72, but their agreement weights the payment by hours.
Answer: C. 90 minutes
The cyclist's head start is 12 × 45/60 = 9 miles. Traveling in the same direction, the scooter closes that gap at 18 − 12 = 6 miles per hour. Catch-up takes 9/6 = 1.5 hours, or 90 minutes. Adding the 45-minute head start gives 135 minutes from the cyclist's departure, but the question measures time from the scooter's departure.
Answer: A. 390
Line A produces 12 × 18 = 216 parts and accepts 216 × 5/6 = 180. Line B produces 15 × 16 = 240 parts and accepts 240 × 7/8 = 210. Together they accept 180 + 210 = 390 parts. The 420 and 426 choices account for rejections on only one line, while 456 includes every rejected part.
Answer: D. 12 beads
The total remains 6 × 12 = 72 beads, and there are still 6 jars. Therefore the average remains 72/6 = 12 beads per jar. Moving beads changes two individual jar counts but not the total. The 11 1/6 choice would result from removing five beads from the collection, which does not happen.
Answer: C. 10
Each visit saves $5 − $3 = $2, so nine visits save exactly the $18 pass fee. At nine visits both methods cost $45; the pass is not yet cheaper. The first whole number beyond nine is ten, when the pass costs $48 and paying without it costs $50.
Answer: B. 3/4 liter
The three pitchers contain 3 × 1 3/4 = 5 1/4 liters. Reserving 3/4 liter leaves 4 1/2 liters. Dividing by six gives 3/4 liter per container. The 7/8-liter choice divides all 5 1/4 liters and forgets the reserved portion.
Answer: C. 5
The perimeter is 2(30 + 20) = 100 feet. Subtracting the two openings leaves 94 feet. Joining n rolls produces 24n − (n − 1) = 23n + 1 feet. Four rolls provide only 93 feet, one foot short; five provide 116 feet and are sufficient. Simply dividing 94 by 24 and rounding up ignores the required overlaps.
Before moving on, choose one error you can explain. Rework that question with the solution closed, then revisit the skill in a later session to check what you remember.
Make a rough prediction before working out the exact answer. A discounted price must be lower than the original; travel time should increase when distance increases at the same speed. Use that prediction to catch reversed operations or misplaced decimals.
After each question, identify whether the mistake came from reading, choosing a relationship or calculating. Rework the problem without the answer in view. On the actual computer test, an answer cannot be revisited after you move on, so build a brief units-and-reasonableness check into your routine.
Choose a focused lesson in fractions, percentages, ratios, algebra or geometry →
Arithmetic Reasoning is one of the four subtests used to calculate the AFQT. It also contributes to service-specific job composites. A practice percentage is not an official AFQT percentile: the actual test uses scaled scores.
The computer test includes unscored tryout questions in selected subtests. The longer time shown is the total for a section with tryouts, not extra time to add. These figures describe the proctored CAT-ASVAB, not untimed PiCAT or our practice sets. See the official What to Expect page's questions and timing table and the guide to test versions.
Sources and materials. Subtest descriptions and timing were checked on September 29, 2026 against the Department of Defense’s What to Expect guide, sample questions and score guide on officialasvab.com. Teaching examples, diagrams and practice questions are independently authored Test Ninjas materials. Test Ninjas is not affiliated with or endorsed by the Department of Defense.