Word Problems with Whole Numbers

🔢 Word Problems with Whole Numbers

Part 1 of 7 — Word Problems on the AR Subtest

Topic Overview

What these questions look like

The official description of Arithmetic Reasoning (AR) is short: "Ability to solve basic arithmetic word problems." Most AR items are a little story of two to four sentences (a paycheck, a supply order, a shift schedule, a room that needs new flooring) followed by one question and four answer choices.

The arithmetic is usually just adding, subtracting, multiplying and dividing. What makes AR feel hard is the reading:

• deciding which operation the story is describing,
• doing two or three steps in the right order,
• and answering the question that was actually asked, in the units it asked for.

This topic is about whole numbers and whole-dollar money. Fractions, percents, ratios and averages each have their own topic in this unit.

How AR works on test day

Some questions may be unscored tryout items, and you can't tell which ones, so treat every question as if it counts. If you take a paper version, the pace is much faster (30 questions in 36 minutes), but the skills are the same.

Why AR matters so much

AR is one of the four subtests in your AFQT score: AR + MK + 2 × VE, using standard scores, turned into a percentile from 1 to 99. The AFQT decides whether you can enlist. AR also feeds job line scores. For example, the Army GT score and the Air Force G score both use AR + VE, and the Marine Corps GT score uses AR + MC + VE.

Key Fact: Whole-number word problems are the foundation of AR. Every harder AR topic (fractions, percents, rates) still needs you to read the story, pick the operations and keep track of units.

Five habits that beat AR without a calculator

1. Read the last sentence first. That is where the question lives. Know what you are finding ("total cost", "how many more", "change from a $50 bill") and its unit before you touch a number.
2. Translate the words into a number sentence. Write it down: $6 \times 85 + 4 \times 120$. Setting it up is most of the work.
3. Track units at every step. Write "hours", "sq ft" or "dollars" next to each number on your scratch paper. Many wrong choices are the right number in the wrong unit.
4. Estimate first. Round to friendly numbers to know roughly what the answer should be. An estimate often eliminates two choices before you compute, and it catches slips afterwards.
5. Check against the story and the choices. Change can't be more than the bill you paid with, and a total can't be smaller than one of its parts. If setting up is hard, plug the answer choices back into the story (back-solving).

Mental-math moves you will use all the time

Worked Example: a supply order

A motor pool orders 6 tires at $85 each and 4 batteries at $120 each. What is the total cost of the order?

$205 · $510 · $990 · $2,050

Step 1 — Last sentence. We want the total cost, in dollars, of everything ordered.

Step 2 — Number sentence. Each tire costs the same, so the tires cost $6 \times 85$. The batteries cost $4 \times 120$. The total is

$6 \times 85 + 4 \times 120$

Step 3 — Estimate. $6 \times 85$ is close to $6 \times 90 = 540$, and $4 \times 120 = 480$. The answer should be near $540 + 480 = 1{,}020$, so about $1,000.

Step 4 — Compute.

$6 \times 85 = 480 + 30 = 510 \qquad 4 \times 120 = 480 \qquad 510 + 480 = 990$

The total is $990.

Step 5 — Check. $990 is right next to the estimate of about $1,000. Now look at why the other choices are there:

Key Fact: On AR the wrong choices are not random. Each one is what you get from a specific mistake, which is exactly why an estimate and a final check are worth the few seconds they take.

Warm-up. Use the five habits on two short problems. 🔥

Warm-up, continued. ✍️

A unit orders 18 cases of bottled water. Each case holds 24 bottles.

1) How many bottles are in the order? (number only, in bottles)

Key Takeaways — Part 1

• AR is 15 scored word problems in 55 minutes on the CAT-ASVAB: about 3.5 minutes each, no calculator, scratch paper only
• You can't go back, so work each problem carefully the first time and check it before you answer
• The habits: last sentence first, write a number sentence, track units, estimate, then check against the story and the choices
• Every wrong choice comes from a real mistake: adding instead of multiplying, stopping one step early, the wrong unit

Up Next: Part 2 — turning the words of a problem into the right operation, and why key words can mislead you.

🔢 Word Problems with Whole Numbers

Part 2 of 7 — Turning Words into Operations

Four actions, four operations

Every one-step word problem does one of four things with its numbers. Before you look for key words, ask: what is the story doing?

Set up the number sentence before you compute

Write the problem as a short number sentence, with a word standing in for the answer. This takes ten seconds and prevents most operation mistakes.

Worked Example 1. Six friends share a $420 cabin rental equally. How much does each one pay?

• The story splits $420 into 6 equal shares.
• Number sentence: $420 \div 6 = \text{one share}$
• Compute: $420 \div 6 = 70$, so each pays $70.
• Check: $6 \times 70 = 420$. ✓

Now flip it: each of 6 friends pays $70; what is the total rent? The story now repeats $70 six times: $6 \times 70 = 420$, so $420. The word "each" appears in both versions, but the operations are opposite. That is why the action matters more than any single word.

When key words mislead you

Key words are a hint, not a rule. Three of them trip up test-takers again and again.

"How many more" means subtract

Worked Example 2. Squad A ran 27 laps and Squad B ran 45 laps. How many more laps did Squad B run?

This compares two amounts, so subtract the smaller from the larger: $45 - 27 = 18$ laps.

Adding ($27 + 45 = 72$) is the classic trap: the word "more" makes it look like addition. Check by adding back: $27 + 18 = 45$. ✓

"Less than" flips the order

"8 less than 30" means start at 30 and take away 8: $30 - 8 = 22$. The number after "less than" is the starting amount, so it is written first.

Worked Example 3. Maria scored 14 points less than twice Leo's score. Leo scored 23. What did Maria score?

• "Twice Leo's score": $2 \times 23 = 46$
• "14 less than" that: $46 - 14 = 32$

Maria scored 32. Writing $14 - 46$ or $2 \times (23 - 14) = 18$ both misread the order.

"Fewer" doesn't always mean subtract

"March had 9 fewer sign-ups than April" tells you April is bigger. If you know March and want April, you add 9. Always ask which amount is bigger? before you pick the operation.

Key Fact: Act the story out in your head. Are things being combined, compared, repeated or split? The action tells you the operation even when the key word points the wrong way.

"What number" problems: work backwards

Sometimes the story gives you the result and asks for the starting amount. Write the number sentence with a blank for the unknown, then undo the operation.

Worked Example 4. After $128 was taken out for taxes, Kim's paycheck was $642. What was her pay before the deduction?

• Number sentence: $\text{pay} - 128 = 642$
• Undo the subtraction by adding: $642 + 128 = 770$, so $770.
• Check: $770 - 128 = 642$. ✓

Reasonableness: pay before a deduction must be more than the check. If you subtracted again ($642 - 128 = 514$), your answer would be smaller than the check, which can't be right.

Worked Example 5. Theo bought 8 identical event tickets for a total of $96. What did one ticket cost?

• Number sentence: $8 \times \text{price} = 96$
• Undo the multiplication by dividing: $96 \div 8 = 12$, so $12.
• Check: $8 \times 12 = 96$. ✓

Choose the operation, then compute. 🎯

Write the number sentence, then solve. ✍️

1) A gym membership costs $29 per month. How much does one year of membership cost? (number only, in dollars)

2) A crew of 7 splits 245 sandbags equally. How many sandbags does each person carry? (number only, in sandbags)

3) A water tank lost 180 gallons during a training exercise and has 465 gallons left. How many gallons were in the tank before the exercise? (number only, in gallons)

Name the operation. Decide what the story is doing before you compute. 🔽

Key Takeaways — Part 2

• Ask what the story does: combine (add), compare or take away (subtract), repeat (multiply), split (divide)
• Write the number sentence before you compute
• "How many more" is subtraction; "8 less than 30" is $30 - 8$; "fewer" can mean you add if you are finding the bigger amount
• "What number" problems: undo the operation, then check that the starting amount makes sense

Up Next: Part 3 — chaining two to four steps together, and the order of operations.

🔢 Word Problems with Whole Numbers

Part 3 of 7 — Multi-Step Problems and Order of Operations

A multi-step problem is a chain of one-step problems

Most AR questions take two to four steps. Each step is one of the four actions from Part 2. The answer to one step becomes a number in the next one.

The plan is always the same:

1. What does the question want? (Last sentence, with its unit.)
2. What do I know? (The numbers in the story.)
3. What is missing in between? (The intermediate results you must find first.)

Label every intermediate result

On scratch paper, write a word next to each number you find. When you reach the end, you can see at a glance which number answers the question, and you won't pick a subtotal by mistake.

Worked Example 1 — Buying and making change. A soldier buys 3 T-shirts at $14 each and 2 pairs of socks at $6 each. She pays with a $100 bill. How much change does she get?

Her change is $46. Estimate check: she spent a bit more than $50, so the change should be a bit less than $50. ✓ Notice that $54 (the total spent) is sitting right there on the scratch paper. It answers a different question.

Overtime pay and trips

Worked Example 2 — Overtime. Jamal earns $18 an hour for his first 40 hours each week and $27 an hour for every hour over 40. Last week he worked 46 hours. What was his pay?

His pay was $882. Reasonableness: 46 hours at about $20 an hour is about $920. Most of his hours were at $18, so a little under $920 makes sense. ✓

Watch for the traps: paying all 46 hours at $18 gives $828 and all 46 at $27 gives $1,242. The two rates apply to different hours.

Worked Example 3 — A trip and a refuel. A convoy truck burns 6 gallons of diesel each hour. It drives 7 hours on Monday and 5 hours on Tuesday. Diesel costs $4 a gallon. What does the fuel for both days cost?

The fuel costs $288. The units tell the story: hours times gallons per hour gives gallons, and gallons times dollars per gallon gives dollars.

Order of operations when it's one expression

Sometimes you write the whole setup as one expression, or a choice shows you one. Then the order of operations decides the value.

(PEMDAS: "Please Excuse My Dear Aunt Sally.") Multiplying does not always come before dividing; they are a tie, done left to right. The same goes for adding and subtracting.

Why the parentheses matter

The change problem from Worked Example 1 as one expression:

$100 - (3 \times 14 + 2 \times 6) = 100 - (42 + 12) = 100 - 54 = 46$

Leave out the parentheses and the value changes:

$100 - 3 \times 14 + 2 \times 6 = 100 - 42 + 12 = 70$

That expression subtracts the shirts but adds the socks back. When a total is being subtracted, it needs parentheses.

Key Fact: If an answer choice is an expression, evaluate it with PEMDAS and compare it with the story. A missing pair of parentheses is a favorite wrong choice.

Chain the steps. Label each intermediate result. 🎯

Your turn. ✍️

1) Evaluate $8 + 6 \times 5 - 12 \div 4$. (number only)

2) A unit orders 5 cases of bottled water at $7 a case and 3 boxes of granola bars at $11 a box. It pays with two $50 bills. How much change does it get? (number only, in dollars)

Key Takeaways — Part 3

• A multi-step problem is a chain of one-step problems: want, know, missing in between
• Write a label next to every intermediate result so you never answer with a subtotal
• Overtime: regular hours × regular rate + extra hours × overtime rate
• Order of operations: parentheses, exponents, multiply/divide left to right, add/subtract left to right
• When a whole total is subtracted (change from a bill), it needs parentheses

Up Next: Part 4 — measurement: converting units, elapsed time, perimeter and area (including the square-yard trap).

🔢 Word Problems with Whole Numbers

Part 4 of 7 — Measurement: Units, Perimeter and Area

The conversions you need to know cold

The test expects you to know these. Nobody will give them to you.

Which way to convert

• Big unit → small unit: multiply. There are more of the small units. 3 hours 25 minutes is $3 \times 60 + 25 = 205$ minutes.
• Small unit → big unit: divide. There are fewer of the big units. 78 inches is $78 \div 12 = 6$ remainder 6, so 6 feet 6 inches.

Reasonableness: after converting to a smaller unit, the number should get bigger. If 3 hours became 20 "minutes", you divided when you should have multiplied.

Key Fact: Before you compute, circle the unit in the last sentence. If the question asks for minutes and the story gives hours, the conversion is one of the steps.

Elapsed time across noon or midnight

Don't subtract clock times directly when the period crosses 12:00. Count up in pieces instead: to the next whole hour, then whole hours, then the leftover minutes.

Worked Example 1. A shift runs from 7:45 a.m. to 4:15 p.m., with a 30-minute unpaid lunch. How many paid hours is the shift?

Take away the 30-minute lunch: 8 paid hours.

Subtracting the clock numbers ($7:45 - 4:15$) gives 3 h 30 min, which is the trap: the clock restarted at noon.

Overnight works the same way. 11:30 p.m. to 7:00 a.m. is 30 min to midnight plus 7 h, so 7 h 30 min.

Perimeter vs. area

A yard 30 ft by 18 ft needs $2 \times 30 + 2 \times 18 = 96$ feet of fence around it.

Area times a unit cost, and the square-yard trap

Worked Example 2. A room is 15 ft by 12 ft. Carpet costs $27 per square yard. What does it cost to carpet the room?

• Area in square feet: $15 \times 12 = 180$ sq ft
• Convert: 1 square yard is a square 3 ft on each side, so $1 \text{ sq yd} = 3 \times 3 = 9 \text{ sq ft}$
• $180 \div 9 = 20$ sq yd
• Cost: $20 \times 27 = 540$, so $540

The traps:

Shortcut: convert the sides first. 15 ft = 5 yd and 12 ft = 4 yd, so the area is $5 \times 4 = 20$ sq yd. Same answer.

Key Fact: A square yard is 9 square feet, not 3. Feet to yards divides by 3; square feet to square yards divides by 9.

Worked Example 3: an L-shaped floor

Tile costs $4 per square foot. What does it cost to tile the floor in Figure 1?

Step 1 — Find the missing sides. The two inner sides are not labelled.

• Inner horizontal side: $20 - 8 = 12$ ft
• Inner vertical side: $14 - 6 = 8$ ft

Step 2 — Split into rectangles and add.

• Rectangle A (tall, left): $8 \times 14 = 112$ sq ft
• Rectangle B (short, right): $12 \times 6 = 72$ sq ft
• Total area: $112 + 72 = 184$ sq ft

Check by splitting the other way: a bottom strip $20 \times 6 = 120$ plus a top block $8 \times 8 = 64$ is also 184 sq ft. ✓

Step 3 — Cost. $184 \times 4 = 736$, so $736.

Trap: multiplying the two longest sides ($20 \times 14 = 280$ sq ft) prices the whole rectangle around the L, including the missing corner, and gives $1,120.

Use Figure 1. 📐

Units, time and area. 🎯

Measure it. ✍️

1) A rectangular garden is 26 ft long and 14 ft wide. How many feet of fencing go all the way around it? (number only, in feet)

2) A mess hall uses 7 gallons of milk at breakfast. How many quarts is that? (number only, in quarts)

3) A shift runs from 6:45 a.m. to 3:15 p.m. How many minutes long is the shift? (number only, in minutes)

Key Takeaways — Part 4

• Know the conversions: 60 min/h, 7 days/week, 12 in/ft, 3 ft/yd, 16 oz/lb, 4 qt/gal
• Big unit to small unit: multiply; small unit to big unit: divide
• Elapsed time across noon or midnight: count up in pieces
• Perimeter is the distance around (fence, trim); area is length × width in square units (carpet, tile)
• 1 sq yd = 9 sq ft; for an L-shape, find the missing sides and split into rectangles

Up Next: Part 5 — division with remainders: when to round up, when to round down, and when the remainder is the answer.

🔢 Word Problems with Whole Numbers

Part 5 of 7 — Remainders, Rounding Up and Estimation

One division, three different answers

When a division doesn't come out even, the remainder forces a decision. The story decides, not the arithmetic.

A supply clerk has 230 bandage rolls. Every first-aid kit uses 12 rolls, and the rolls ship in boxes of 12.

$230 \div 12 = 19 \text{ remainder } 2 \qquad (12 \times 19 = 228, \; 230 - 228 = 2)$

Same numbers, three correct answers to three different questions. That is why you read the last sentence first.

How to decide

Worked Example 1 — Bus capacity. 140 recruits are going to a training site. Each bus holds 44 people. How many buses are needed?

• $140 \div 44$: $3 \times 44 = 132$, and $140 - 132 = 8$, so 3 remainder 8.
• 3 buses would leave 8 recruits behind. You need 4 buses.
• Check: $4 \times 44 = 176$, which is enough room for 140. ✓

Worked Example 2 — Spending money. You have $75 to spend on notebooks that cost $8 each. How many can you buy?

• $75 \div 8$: $9 \times 8 = 72$, and $75 - 72 = 3$, so 9 remainder 3.
• The leftover $3 can't buy another notebook. You can buy 9, with $3 left.

Key Fact: A remainder never shows up as a decimal or a fraction in these answers. You can't rent part of a bus. Ask: would the leftovers be stranded? If yes, round up.

Estimate to eliminate, then check by multiplying back

Compatible numbers

Round to numbers that are easy to work with together.

Worked Example 3. A unit buys 48 folding cots at $61 each. What is the total cost?

$1,928 · $2,928 · $3,928 · $29,280

Estimate: $50 \times 60 = 3{,}000$. Only $2,928 is anywhere close. You can answer without the full multiplication, though it's quick: $48 \times 61 = 48 \times 60 + 48 = 2{,}880 + 48 = 2{,}928$.

Front-end estimation

Round each number to its leading digit's place, then add. For $412 + 389 + 276$: about $400 + 400 + 300 = 1{,}100$. The exact sum is 1,077, so any choice near 700 or 1,500 is out.

For division, pick a nearby number the divisor goes into evenly. $4{,}182 \div 6$ is about $4{,}200 \div 6 = 700$. The exact answer is 697.

Multiply back to check

Every division can be checked with multiplication:

$\text{quotient} \times \text{divisor} + \text{remainder} = \text{number you started with}$

For the bandages: $19 \times 12 + 2 = 228 + 2 = 230$. ✓ For the cots: $697 \times 6 = 4{,}182$. ✓ If the check fails, the slip is in your division, and you find it before you answer, not after.

Decide what the remainder means. 🎯

Remainders in context. ✍️

1) A food drive collected 340 cans. Each box holds 24 cans. How many boxes are needed to pack every can? (number only, in boxes)

2) A trainer has 100 resistance bands to share equally among 8 squads, giving each squad the same whole number of bands. How many bands are left over? (number only, in bands)

3) Movie tickets cost $11 each. How many tickets can you buy with $90? (number only, in tickets)

Round up, round down, or use the remainder? 🔽

Key Takeaways — Part 5

• The story decides what to do with a remainder
• Containers or trips needed to hold all of them: round up
• Complete groups, or how many you can buy: round down
• Left over or remain: the remainder is the answer
• Estimate with compatible numbers to eliminate choices, and multiply back to check a division

Up Next: Part 6 — the traps the test sets on whole-number word problems, and how to manage your time.

🔢 Word Problems with Whole Numbers

Part 6 of 7 — Test Strategy and Common Traps

The six traps the wrong answers are built from

AR answer choices are designed around predictable mistakes. Once you know them, you'll recognize them on sight.

Key Fact: If your answer matches a choice on the first try, that's good, but it doesn't prove you're right. The trap answers match choices too. The final check is what separates them.

Timing: 3.5 minutes is an average, not a limit per question

On the CAT-ASVAB you have 55 minutes for 15 scored questions, about 3.5 minutes each. That is generous, so use it wisely:

• One-step problems (one operation, one unit) should take about 1 minute. Answer, check, move on.
• The time you bank on easy items pays for the multi-step ones (overtime, carpet in square yards, an L-shaped room), which can take 4 or 5 minutes.
• You can't skip or go back. You must answer to move on, so give each problem real effort the first time.
• Stuck after a few minutes? Eliminate what you can with an estimate and choose among what's left. A thoughtful guess beats a blank.
• Don't rush at the end. Unanswered questions are scored as random guesses, and a run of wrong answers late in a subtest is penalized heavily. Keep genuinely working each problem; the time limit is generous enough that this almost never comes up.

Back-solving: let the choices do the work

When the setup is hard to write, plug the choices into the story.

Worked Example. Ella buys some $9 shirts and pays with a $50 bill. Her change is $14. How many shirts did she buy? Choices: 3, 4, 5, 6.

• Try 4: $4 \times 9 = 36$, and $50 - 36 = 14$. ✓ Done.

Start with a middle choice. If it gives too much change, you need a bigger number; too little, a smaller one.

The 20-second reasonableness checklist

Before you click, run through these:

1. Did I answer the last sentence? Not a subtotal, not a different quantity.
2. Is it in the right unit? Hours or minutes, feet or yards, square feet or square yards.
3. Does the size make sense? Change is less than the bill paid with. A total is bigger than each of its parts. Pay before a deduction is more than the check.
4. Is it close to my estimate? If you estimated about 3,000 and got 29,280, a place value slipped.
5. Did I handle the remainder for this story? Buses round up; full kits round down.

Estimation knocks out choices fast

A unit buys 52 tents at $198 each. Which is the total?

$1,296 · $10,296 · $12,960 · $102,960

$50 \times 200 = 10{,}000$, so the answer is about $10,000: only $10,296 fits. Exact: $52 \times 198 = 52 \times 200 - 52 \times 2 = 10{,}400 - 104 = 10{,}296$.

Spot the trap. Each problem has a trap answer waiting. Find the real one. 🕵️

Key Takeaways — Part 6

• The six traps: stopping early, wrong unit, remainder handled wrong, key-word misread, "how many more" added, starting amount forgotten
• Bank time on one-step items (about 1 minute) to spend on multi-step ones
• You can't go back, so check before you click: right question, right unit, sensible size, close to the estimate
• When the setup is hard, back-solve: plug a middle choice into the story

Up Next: Part 7 — a mixed mastery set in ASVAB difficulty order.

🔢 Word Problems with Whole Numbers

Part 7 of 7 — Mixed Mastery Set

Put it all together

This set mixes every skill from the topic, in roughly the order a real AR subtest would build: a few quick ones, a solid middle, then a few that take more steps.

Work each one the way you would at MEPS: last sentence first, a number sentence, units on every line, an estimate, and a final check. Aim for about 2 to 3 minutes per problem, faster on Set A.

Set A — Warm-up. 🟢

Set B — Test level. 🟡

Set C — Stretch. 🔴

One more, with two answers. ✍️

A food bank packs 1,000 cans into boxes that each hold 24 cans. Only full boxes are shipped.

1) How many full boxes are shipped? (number only, in boxes)

2) How many cans are left over? (number only, in cans)

Topic Summary — Word Problems with Whole Numbers

What to do next

1. Take the exit quiz for this topic. If you score 80% or better, the topic is done.
2. Review the flashcards for this topic. They focus on the conversions, the remainder rules and the order of operations.
3. Move on to the next AR topic. Fractions, percents, ratios and averages all use the same habits you practiced here: last sentence first, a number sentence, units, an estimate and a check.