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SECTION GUIDE - AR

ASVAB Arithmetic Reasoning

What it tests, how it is scored, full lessons, and 10 practice questions with answers

What AR Tests

Arithmetic Reasoning (AR) is the word-problem section: rates and distance, percents and interest, ratios, averages, work problems, and everyday money math. You set up the problem from a short story, then do the arithmetic by hand. No calculator is allowed.

Scored questions (CAT-ASVAB)15
Time limit55 minutes (about 220 seconds per question)
Part of the AFQT?Yes

AR is one of the four AFQT sections, so it directly decides whether you can enlist. It also feeds the Army GT, CL, CO, EL, FA, and SC line scores. See ASVAB scores explained for how section scores become your AFQT and line scores.

How to Score Higher

Arithmetic Reasoning Lessons

Word problems requiring mathematical reasoning. Each lesson below is also in the interactive study guide, where you can take a quiz to check yourself.

Lesson 1 Ratios & Proportions

Ratios and proportions are the #1 most tested topic on Arithmetic Reasoning. Master this, and you're already ahead.

What is a Ratio?

A ratio compares two quantities. If there are 3 boys and 5 girls, the ratio is 3:5 (or 3 to 5, or 3/5). It tells you how the quantities relate to each other.

What is a Proportion?

A proportion says two ratios are equal. If 2 apples cost $1, then 6 apples cost $3. We write this as 2/1 = 6/3. Both ratios equal the same thing.

The Cross-Multiply Trick

To solve any proportion, cross-multiply. If a/b = c/d, then a×d = b×c. This is your secret weapon for these problems.

a b = c d a × d = b × c

Worked example

If 4 tickets cost $30, how much do 10 tickets cost?

  1. Set up the proportion: 4 tickets/$30 = 10 tickets/$x
  2. Write as fractions: 4/30 = 10/x
  3. Cross-multiply: 4 × x = 30 × 10
  4. Solve: 4x = 300, so x = 75
  5. Answer: $75

Tip: Always put the same units in the same position (tickets on top, dollars on bottom).

Worked example

A map scale shows 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, what's the actual distance?

  1. Set up: 1 inch/50 miles = 3.5 inches/x miles
  2. Cross-multiply: 1 × x = 50 × 3.5
  3. Solve: x = 175 miles

Tip: Map and scale problems are just proportions in disguise.

Key takeaway: For any ratio problem: (1) Set up the proportion with matching units, (2) Cross-multiply, (3) Solve for x, (4) Check if your answer makes sense.

Lesson 2 Percentages

Percentages are everywhere: shopping, banking, test scores. Once you understand that 'percent' just means 'per 100,' these problems become simple.

What is a Percent?

Percent means 'per 100.' So 25% means 25 out of 100, or 25/100, or 0.25. To convert a percent to a decimal, just move the decimal point 2 places left: 25% → 0.25

Finding a Percent OF Something

To find X% of a number, multiply by the decimal. What is 20% of 80? → 80 × 0.20 = 16

Percent Increase/Decrease

For a 20% increase, multiply by 1.20. For a 20% decrease, multiply by 0.80. This is faster than calculating the change and adding/subtracting.

Finding the Original (Working Backwards)

If a price AFTER a 20% increase is $60, the original is $60 ÷ 1.20 = $50. Don't just subtract 20%. That gives the wrong answer!

Worked example

A $120 jacket is on sale for 25% off. What's the sale price?

  1. 25% off means you pay 75% (100% - 25% = 75%)
  2. Convert: 75% = 0.75
  3. Multiply: $120 × 0.75 = $90
  4. Answer: $90

Tip: Shortcut: For X% off, multiply by (1 - X/100)

Worked example

After a 15% raise, Tom earns $46,000. What was his original salary?

  1. After 15% increase, he has 115% of original
  2. $46,000 = 1.15 × original
  3. Original = $46,000 ÷ 1.15 = $40,000

Tip: WARNING: Don't just subtract 15% from $46,000. That gives $39,100, which is wrong!

Key takeaway: Key formulas: Part = Whole × (Percent/100). For increase: New = Original × (1 + rate). For decrease: New = Original × (1 - rate). To find original: Original = New ÷ (1 ± rate).

Lesson 3 Distance, Rate & Time

These problems appear constantly on the ASVAB. A car travels, a plane flies, someone walks: they all use one simple formula.

The DRT Formula

Distance = Rate × Time. That's it! If you know any two values, you can find the third. D = R×T, R = D÷T, T = D÷R.

Two Objects Moving

If objects move TOWARD each other or AWAY from each other, ADD their speeds. If they move in the SAME direction, SUBTRACT their speeds.

Average Speed Trap

Average speed is NOT the average of two speeds! It's always Total Distance ÷ Total Time. This is a common ASVAB trick.

Worked example

Two cars leave the same point traveling in opposite directions, one at 50 mph and one at 70 mph. After how many hours are they 360 miles apart?

  1. They're moving apart, so ADD speeds: 50 + 70 = 120 mph
  2. Use D = R × T, solving for T: 360 = 120 × T
  3. T = 360 ÷ 120 = 3 hours

Tip: Think of it as the gap between them growing at 120 mph.

Worked example

A hiker walks 10 miles at 2 mph, then 10 miles at 5 mph. What's the average speed for the whole trip?

  1. Time for first 10 miles: 10 ÷ 2 = 5 hours
  2. Time for second 10 miles: 10 ÷ 5 = 2 hours
  3. Total: 20 miles in 7 hours
  4. Average speed = 20 ÷ 7 ≈ 2.86 mph

Tip: NOT (2+5)÷2 = 3.5 mph! That's wrong. Use Total Distance ÷ Total Time.

Key takeaway: D = R × T (memorize this!). Add speeds when moving apart; subtract when moving same direction. Average speed = Total Distance ÷ Total Time.

Lesson 4 Work Problems

Work problems ask how long it takes multiple people (or machines) to complete a job together. The trick is to think about work RATES, not time.

Work Rate

If someone can do a job in 6 hours, their work rate is 1/6 of the job per hour. Always express rate as: 1 ÷ (time to complete whole job).

Combining Work Rates

When people work together, ADD their rates. If A does 1/6 per hour and B does 1/4 per hour, together they do 1/6 + 1/4 = 5/12 per hour.

Finding Total Time

Once you have the combined rate, total time = 1 ÷ (combined rate). So if combined rate is 5/12 per hour, time = 1 ÷ (5/12) = 12/5 = 2.4 hours.

Worked example

Alex can paint a room in 6 hours. Beth can paint it in 4 hours. How long if they work together?

  1. Alex's rate: 1/6 room per hour
  2. Beth's rate: 1/4 room per hour
  3. Combined rate: 1/6 + 1/4 = 2/12 + 3/12 = 5/12 room per hour
  4. Time = 1 ÷ (5/12) = 12/5 = 2.4 hours

Tip: The answer must be less than either person's time alone!

Key takeaway: Work rate = 1/time. Add rates when working together. Total time = 1 ÷ combined rate. The combined time is always less than any individual time.

Lesson 5 Statistics Basics

Statistics problems are straightforward once you know the definitions. These appear regularly on the ASVAB.

Mean (Average)

Add all values, then divide by how many there are. Mean of 3, 5, 7 = (3+5+7) ÷ 3 = 15 ÷ 3 = 5.

Median (Middle)

Arrange values in order, then find the middle one. For 2, 5, 9, the median is 5. For even counts, average the two middle values.

Mode (Most Common)

The value that appears most often. In 2, 3, 3, 5, 7, the mode is 3. There can be multiple modes or no mode.

Finding a Missing Value

If you need a certain average, use: Sum = Average × Count. Then subtract the known values to find the missing one.

Worked example

A student scored 78, 82, 91, and 85 on four tests. What score is needed on the fifth test to achieve an 85 average?

  1. Target sum for 5 tests: 85 × 5 = 425
  2. Current sum: 78 + 82 + 91 + 85 = 336
  3. Needed score: 425 - 336 = 89

Tip: Use Sum = Average × Count to work backwards.

Key takeaway: Mean = Sum ÷ Count. Median = middle value (arrange first!). Mode = most frequent. To find missing value: Sum needed - Sum known.

AR Practice Questions

Try each question before opening the answer. On the real CAT-ASVAB the questions adapt to you, so a timed, adaptive practice test is the best way to check your level.

Question 1

A carpenter needs to cut a board into pieces that are each 2¾ feet long. If the original board is 16½ feet long, how many complete pieces can be cut?

  1. 4 pieces
  2. 6 pieces
  3. 7 pieces
  4. 8 pieces
Show answer

Answer: B. 6 pieces

Divide total length by piece length: 16½ ÷ 2¾ = 16.5 ÷ 2.75 = 6 complete pieces. Convert the mixed numbers to decimals first.

Question 2

A bicycle wheel has a diameter of 26 inches. Approximately how far does the bicycle travel in one complete rotation? (Use π ≈ 3.14)

  1. 41 inches
  2. 52 inches
  3. 82 inches
  4. 163 inches
Show answer

Answer: C. 82 inches

One rotation equals the circumference: π × diameter = 3.14 × 26 ≈ 81.6 ≈ 82 inches. Use circumference, not radius or area.

Question 3

A supply room has 420 items. After using 1/4 of the items and receiving a shipment of 80 more, how many items are there?

  1. 315
  2. 395
  3. 420
  4. 500
Show answer

Answer: B. 395

Used 1/4 of 420 = 105, leaving 315. Add the shipment: 315 + 80 = 395. Subtract first, then add.

Question 4

A cleaning company charged $600 for cleaning 4,800 square feet. What would they charge for 12,000 square feet?

  1. $950
  2. $1,200
  3. $1,400
  4. $1,500
Show answer

Answer: D. $1,500

Rate is $600 ÷ 4,800 = $0.125 per square foot. For 12,000: 12,000 × 0.125 = $1,500. Or note 12,000 is 2.5× 4,800.

Question 5

Two pizza ovens exist. Oven #1 burns three times as many pizzas as Oven #2. If a total of 12 pizzas were burned on Saturday, how many did Oven #2 burn?

  1. 2
  2. 3
  3. 4
  4. 5
Show answer

Answer: B. 3

Oven 1 burns 3× Oven 2, so 4 equal parts total 12: 12 ÷ 4 = 3 for Oven 2. A trap is picking Oven 1's 9.

Question 6

Aisha wants to paint a room's walls. Each gallon covers 110 square feet. The walls are: two at 10x10 feet and two at 10x15 feet. How many 1-gallon buckets are needed?

  1. 4
  2. 5
  3. 9
  4. 10
Show answer

Answer: B. 5

Wall area = 2×(10×10) + 2×(10×15) = 200 + 300 = 500 sq ft. 500 ÷ 110 ≈ 4.5, so round up to 5 buckets.

Question 7

A server receives an 18% tip on a $65 bill. How much is the tip?

  1. $9.75
  2. $10.40
  3. $11.70
  4. $13.00
Show answer

Answer: C. $11.70

Tip = 18% of $65 = 65 × 0.18 = $11.70. Multiply the bill by 0.18.

Question 8

Pipe A fills a tank in 8 hours. Pipe B fills it in 12 hours. How long to fill the tank with both pipes open?

  1. 4 hours
  2. 4.8 hours
  3. 5 hours
  4. 6 hours
Show answer

Answer: B. 4.8 hours

Rates: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 per hour. Time = 24/5 = 4.8 hours. Add the fill rates and invert.

Question 9

Tickets cost $12 for adults and $7 for children. A family buys 2 adult and 4 children tickets. What is the total?

  1. $48
  2. $52
  3. $56
  4. $60
Show answer

Answer: B. $52

2 adults × $12 + 4 children × $7 = 24 + 28 = $52. Multiply each ticket type, then add.

Question 10

How much must be invested at 4% simple interest to earn $240 in interest over 3 years?

  1. $1,800
  2. $2,000
  3. $2,200
  4. $2,400
Show answer

Answer: B. $2,000

P × 0.04 × 3 = 240, so 0.12P = 240, P = $2,000. Divide the interest by rate × time.

Test Yourself Under Real Conditions

Take a timed, adaptive Arithmetic Reasoning section, the 20-question AFQT Predictor, or a full AFQT practice test. Free, no account needed.

Take a Practice Test

Other section guides: Word Knowledge · Paragraph Comprehension · Mathematics Knowledge · General Science · Electronics Information · Auto & Shop Information · Mechanical Comprehension