ASVAB Arithmetic Reasoning study guide

All fourteen procedures this subtest draws on, a four-step method for turning a sentence into arithmetic, fourteen worked examples — each one ending in the trap that costs points — and three answer-choice strategies for when the setup will not come.

What this subtest actually asks of you

Arithmetic Reasoning is one of the mathematics subtests, and it is 15 of the 135 scored questions on the computer ASVAB — 11% of the scored test. It is also one of the four subtests that produce your AFQT score, the percentile that decides whether you can enlist (how the ASVAB is scored). Across those four subtests: Arithmetic Reasoning and Mathematics Knowledge are 15 scored questions each, Word Knowledge 15, and Paragraph Comprehension 10.

Two things follow, and they shape how to study it:

Every question is a word problem. There is no column of bare arithmetic to work through. The operations involved are basic — percents, ratios, rates, averages, and a little simple probability — and nearly all of the difficulty is in turning a sentence into the right operation.
The scratch paper does the work. This subtest is not testing mental arithmetic; it is testing whether you can set a problem up. Write the translation down.

The fourteen procedures

Every Arithmetic Reasoning question is one of these fourteen. Each one is worked out below — the rule, the steps, and the trap that costs points on it. The arithmetic underneath them is small: fractions, decimals and percents converted between; multiplication and division by hand; simple interest; minutes and hours. The rest is multi-step questions that glue two of these together.

A percent of a number — "15% of 200" is 0.15 × 200. The same move computes sales tax, a tip, a discount amount, and simple interest. → example 1
The price after a discount — find the percent of the price, then subtract it from the price. Two steps, and the question asks for the second one. → example 2
A percent increase or decrease — a change is always measured against the value you started from: (new − old) ÷ old. The same rule covers "increase 80 by 30%" (multiply by 1.30). → example 3
Two percent changes in a row — multiply the factors, never add the rates: a 40% rise then a 25% fall is ×1.40 × 0.75, not ×1.15. → example 4
Finding the percent, or the whole, from a part — part ÷ whole gives the percent ("12 of 48 is what percent?"), and when you are given the percent and the part, divide to get the whole. → example 5
A rate per unit — speed, price per item, miles per gallon, pay per hour. Find the rate first (divide), then use it to answer the question, which is usually one step further on. → example 6
Scaling a recipe or a ratio — find the amount per single serving (or per one part of the ratio), then multiply by what the question asks for. A proportion is two ratios set equal — and when the ratios must add to a total, one part is missing. → example 7
A fraction of a quantity — multiply by the fraction, and read carefully whether the question wants the fraction of the amount or what is left after you take it. → example 8
Two rates working together — rates add, times do not: 1/6 + 1/12 of the job in an hour means 4 hours together. → example 9
An average, and a missing value — the average is total ÷ count, so the total is average × count, and a missing value is that total minus the numbers you have. → example 10
Totals, change and counting equal groups — add prices, subtract from what you paid, and multiply when the question says "each" or "per". For a pay question, split the hours into the normal rate and the premium rate. → example 11
A fraction of an hour — 45 minutes is 45/60 = 0.75 hours; 3 hours 20 minutes is 3 + 20/60 hours. Convert before you multiply by a rate. → example 12
The probability of one event — the count the question asks about, over the whole group; the complement is everything else, and the two add to 1. → example 13
Median, mode and range — sort the values first. The median is the middle one (average the two middles when the count is even), the mode is the value that appears most often, and the range is the largest minus the smallest. → example 14

A four-step method

1. Find the question mark. Before any arithmetic, name the one quantity being asked for, and its unit. "The sale price, in dollars." "The percent." "The number of gallons."

This step alone defeats the classic wrong answer on this subtest: correct arithmetic applied to a different quantity — the discount instead of the price, the rate instead of the total.

2. Name it. Write it down: x = the sale price. Naming it is what makes the last step possible.

3. Translate one operation at a time, in the order the story gives it. English hides arithmetic in four small words:

of → multiply ("25% of 40")
per → divide ("miles per gallon", "per hour")
off → subtract from the whole ("25% off" means you pay 75% of it)
than → tells you which way the comparison points ("$3 more than", "$5 less than")

Then chain the operations in the order they appear, without collapsing two of them into one.

4. Check the answer against the story. A sale price must be lower than the original price. Two workers together cannot take longer than the faster worker alone. An average sits between the smallest and largest values. If your answer contradicts the story, the arithmetic is wrong — and you have found that in two seconds instead of re-reading the problem.

Steps 1 and 4 are what earn points here. The arithmetic rarely is.

The fourteen worked examples

Each example is one of the fourteen procedures: the rule in action, the steps, and the trap that example is built to catch.

1. A percent of a number (tax and interest)

A purchase totals $85.00 before tax. If the sales tax is 6%, what is the total?

Change the percent to a decimal: 6% = 0.06.
Tax: 0.06 × 85 = $5.10.
Add it to the price: 85 + 5.10 = $90.10.
Check the size: 6% of 85 has to be a few dollars, and 5.10 is.
The trap: answering $5.10 — the tax, not the total. The same shape covers "15% of 200" (0.15 × 200 = 30) and simple interest ($2,000 at 5% for a year is 0.05 × 2,000 = $100), where the question does or does not want the rate added back. Read which.

2. Percent off — the discount is not the answer

A jacket is priced at $40.00 and is on sale for 25% off. What is the sale price?

The question mark is the price you pay, not the discount.
25% of $40.00: 0.25 × 40 = 10.00.
Subtract it: 40.00 − 10.00 = $30.00.
Check against the story: the sale price has to be lower than $40 and higher than the discount.
The trap: $10.00 is one of the choices, and it is a perfectly executed first step. A $24.00 guide at 25% off is 0.25 × 24 = $6.00, so you pay $18.00.

3. Percent change — measured against the ORIGINAL

A shirt that was $60.00 is on sale for $45.00. What percent has been taken off?

The change is $60.00 − $45.00 = $15.00.
As a percent of the original: 15 ÷ 60 = 0.25 → 25%.
Check by going back: 25% of 60 is 15, and 60 − 15 = 45.
To increase by a percent instead, multiply: increase 80 by 30% is 80 × 1.30 = 104; decrease 250 by 12% is 250 × 0.88 = 220.
The trap: dividing by $45 instead (15 ÷ 45 = 33%). A change — up or down — is always measured against where you started.

4. Two percent changes — multiply, never add

A price is raised 40%, and then the raised price is discounted 25%. The final price is what percent of the original?

Multiply the factors, in order: 1.40 × 0.75 = 1.05.
So the final price is 105% of the original.
Check the direction: a 40% rise and a 25% fall do not cancel, because the fall lands on the raised price.
On a $150.00 item discounted 20% and then a further 10%: 150 × 0.80 × 0.90 = $108.00.
The trap: 115%, from subtracting the rates (1.40 − 0.25). This family is the one most people get wrong, and it is worth drilling deliberately.

5. Finding the percent, and finding the whole

12 is what percent of 48? And: if 15% of a number is 45, what is the number?

Part ÷ whole: 12 ÷ 48 = 0.25 → 25%.
The reverse reads the same equation backwards: 0.15 × n = 45.
Divide: 45 ÷ 0.15 = 300.
Check: 15% of 300 is 45.
The trap: multiplying when the question wants you to divide. "15% of a number is 45" gives you the part; the number asked for is the whole, and it must be bigger than 45.

6. Unit rates — do not stop one step early

A car travels 350 miles on 14 gallons of fuel. How many gallons are needed to travel 500 miles?

Rate: 350 ÷ 14 = 25 miles per gallon.
Gallons: 500 ÷ 25 = 20.
The same shape at speed: 240 miles in 4 hours is 240 ÷ 4 = 60 miles per hour, so 7 hours covers 60 × 7 = 420 miles.
Check the direction: more miles needs more gallons, and 20 is more than 14.
The trap: answering 25 — the rate you calculated on the way, not the gallons you were asked for. Step 1 again.

7. Scaling a recipe or a ratio

A recipe that serves 4 people calls for 2 cups of rice. How many cups are needed to serve 10 people?

Find the amount for one serving: 2 ÷ 4 = 0.5 cups.
Multiply by the servings asked for: 0.5 × 10 = 5 cups.
The same proportion backwards: 3/4 cup per batch for 6 batches is 0.75 × 6 = 4.5 cups.
Two numbers in a ratio that must add to a total: 3 : 5 summing to 64 means 8 parts, so one part is 64 ÷ 8 = 8, and the smaller number is 3 × 8 = 24.
Check the direction: 10 servings is more than 4, so the rice has to be more than 2 cups.

8. A fraction of a quantity — read what the change is

A tank is 3/5 full. After 12 gallons are added it is 4/5 full. What is the tank's capacity?

Going from 3/5 to 4/5 means those 12 gallons are the difference: 4/5 − 3/5 = 1/5 of the tank.
If 1/5 is 12 gallons, the full tank is 12 × 5 = 60 gallons.
Check: 3/5 of 60 is 36, and 36 + 12 = 48, which is 4/5 of 60.
Watch what the question asks for: two-thirds of a class of 27 students passed, so 27 × 2/3 = 18 passed and 27 − 18 = 9 did not.
The trap: treating the added 12 gallons as 2/5 of the tank, which gives 30. The sentence describes the change, not the amount at the end.

9. Combined work — rates add, times do not

One worker can finish a job in 6 hours, and a second can finish the same job in 12 hours. How long does the job take with both working?

In one hour they do 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 of the job.
The whole job takes 1 ÷ 1/4 = 4 hours.
A second flavor of the same rule: a pipe fills a pool in 8 hours and a second in 12 hours, so together they add 1/8 + 1/12 = 5/24 per hour, and the pool fills in 24/5 = 4.8 hours.
Check against the story: 4 hours is faster than the faster worker alone, which is the only possible answer.
The trap: 9 hours, the average of 6 and 12. Step 4 kills this instantly, because a second worker cannot make the job slower.

10. An average, and the missing value

The average of five numbers is 12. Four of them are 10, 14, 8 and 15. What is the fifth?

Total = average × count = 12 × 5 = 60.
Add what you have: 10 + 14 + 8 + 15 = 47.
Subtract: 60 − 47 = 13.
Check: 10 + 14 + 8 + 15 + 13 = 60, and 60 ÷ 5 = 12.
The trap: dividing by the number of values you were given. The total is the average × the number of items, and the count is usually the bigger number.

11. Totals, change and counting equal groups

You pay for a $13.75 purchase with a $20 bill. How much change do you receive?

The question mark is the change, not the total.
Subtract: 20.00 − 13.75 = $6.25.
When the question says "each" or "per", multiply: 24 cans per box for 8 boxes is 24 × 8 = 192 cans.
For pay with a premium, split the hours: 40 hours at $18.00 is $720, and 6 overtime hours at time and a half ($27.00) is $162, so the week is $882.
The trap: adding when the story says "each". "4 notebooks at $3.25 and a pen at $1.75" is 4 × 3.25 + 1.75 = $14.75, not 3.25 + 1.75 × 4.

12. A fraction of an hour

How far does a car traveling at a steady 60 miles per hour go in 45 minutes?

Convert the time to hours first: 45 ÷ 60 = 0.75 hours.
Distance = rate × time = 60 × 0.75 = 45 miles.
The other direction converts the same way: 3 hours 20 minutes is 3 + 20/60 = 3.33 hours.
Check the size: 45 minutes is three-quarters of an hour, so the car covers three-quarters of 60 miles.
The trap: multiplying 60 by 45 and answering 2,700. The rate is per hour, so the minutes have to become hours first.

13. Probability — the whole bag is the denominator

A bag holds 5 red, 10 blue and 15 green counters. One counter is drawn at random. What is the probability that it is blue?

Add the group first: 5 + 10 + 15 = 30 counters, and that total is the denominator.
Count what the question names: 10 blue.
Probability = 10/30 = 1/3.
Check the size: blue is neither the biggest group nor the smallest, so the answer should sit between them — and 1/3 does.
The complement is what is left: not blue is 20/30 = 2/3, and 1/3 + 2/3 = 1.
The trap: 10/20 — dividing by the counters that are not blue. The denominator is always the whole group, never the part you are comparing against.

14. Median, mode and range — sort before you read anything

Over six days, a shop sold 4, 9, 7, 11, 11 and 12 items. What are the median, the mode and the range of the daily sales?

Sort first: 4, 7, 9, 11, 11, 12. Every one of these three measures needs the sort.
Six values, so there is no single middle one — the median averages the two middle values: (9 + 11) ÷ 2 = 10.
The mode is the value that appears most often: 11, the only value that appears twice.
The range is the largest minus the smallest: 12 − 4 = 8.
The mean is a fourth number and it is not asked for: 54 ÷ 6 = 9.
The trap: answering with the middle of the list as written. Unsorted, 9 and 11 are not the middle pair — and reporting the mean, 9, is a different question than the one asked.

The traps, collected

The part instead of the whole — the discount instead of the sale price; the tax instead of the total; the interest instead of the balance.
The wrong base in a percent change — dividing by the new value instead of the original.
Two percents added instead of multiplied — 40% up then 25% off is ×1.05, not ×1.15.
Multiplying when the question wants division — "15% of a number is 45" asks for the whole, which is bigger than 45.
Averaging what should be added — workers, pipes, printers: rates add, times do not.
The wrong count in an average — the total is the average × the number of items, and the wrong count breaks it.
Minutes left as minutes — a rate per hour needs hours, so 45 minutes is 0.75, not 45.
Adding when the story says "each" — four items at a price is a multiplication.
The fraction of the amount instead of what is left — two-thirds passed means the question may want the one-third that did not.
One step short — reporting the rate, the discount or the change instead of the quantity the question asked for.
The wrong denominator in a probability — dividing by the part you are comparing against instead of by the whole group: 10 of 20, where the bag holds 30.
Reading a list before sorting it — the median and the range both need the values in order, and the middle of the list as written is a different number.

None of these are arithmetic mistakes. They are reading mistakes, which is why the method spends two of its four steps on reading.

Answer-choice strategies: three moves for when the setup will not come

The method above is what earns points, and these three moves are not a substitute for it. They are what to do when you have read a question twice and the setup still will not come — because a question you never start is a question you cannot get right, and a question you finish in four minutes is three questions you did not reach.

Use them in this order. Each one narrows the choices rather than guessing at them.

1. Estimate the size before you compute it

The choices on a word problem are almost never close together. Round the numbers to something workable, get the rough size, and let the choices do the rest.

A jacket is reduced by 15% from $89.00. About what do you pay?

15% of $89 is a little over 15% of $80, which is $12 — so the discount is around $13.
89 − 13 is about $76.
If the choices are $13.35, $75.65, $88.85 and $102.35, the estimate has already eliminated two of them.

Two cautions. Rounding tells you where the answer is, never what it is — finish the arithmetic. And when two choices sit close together (37.5% and 38%), the estimate has done its job and the exact work is the only thing that decides between them.

2. Backsolve — put the choices back into the question

When the question gives you a rule and asks for a value, the choices are candidate answers: test them. This turns a question you cannot set up into four small arithmetic checks.

A number is multiplied by 4, and then 7 is subtracted from the result. If the final result is 61, what is the number?

Take the middle choice first. If 15 gives 4(15) − 7 = 53, the answer has to be bigger, so both choices below 15 are gone in one step.
Try 17: 4(17) − 7 = 61. ✓

Starting in the middle is the whole trick — a wrong choice still tells you which way to move, so one check usually halves the field.

3. Pick a number for the variable

When a choice is an expression rather than a number ("the total cost, in terms of n"), choose a convenient value for the variable, work the question with it, and match the shape of your answer to the choices.

A shirt costs n dollars. Its price is raised by 20%, and then 25% is taken off the new price. What is the final price in terms of n?

Let n = 100, because percentages are easy on 100.
Raised: 100 × 1.20 = 120. Discounted: 120 × 0.75 = 90.
So the final price is 90% of n — and only one choice will say 0.9n or 9n/10.

Three cautions, and they are the ones that cost points. Never pick 0 or 1 — they hide errors in the arithmetic you are trying to check. Avoid a number that makes two choices come out equal; if that happens, pick again. And once you have an answer, put it back into the story — step 4 of the method is what catches a choice you tested against the wrong quantity.

How to practice this

Work the free Arithmetic Reasoning practice set with a pencil and scrap paper rather than in your head — the subtest is not a mental-arithmetic test.
Drill the procedures you get wrong, not the ones you already know. Sorting your errors by procedure is more useful than counting them.
Read the explanation on every question, including the ones you answered correctly. A right answer reached for the wrong reason comes back on test day.
Re-attempt what you missed a day later, not immediately. Immediate repetition measures short-term memory; a day later measures whether the method stuck.

How to study for the ASVAB has the rest of the method, and what the pass adds.

Where to go next

Next topic — Word Knowledge
The exam itself — how the ASVAB is scored, how many questions, and what makes the ASVAB hard.

Arithmetic Reasoning carries 15 of the 135 scored questions on the computer test — the published subtest table's count — and it is one of the four subtests the AFQT score is computed from.

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