Fractions and Decimals in Context

🧩 Fractions and Decimals in Context

Part 1 of 7 — Fractions and Decimals on the AR Subtest

Topic Overview

What These Questions Look Like

The Arithmetic Reasoning (AR) subtest is officially described as the "ability to solve basic arithmetic word problems," and fractions and decimals show up in a big share of them. They rarely say "compute $\frac{3}{4} \times 36$." Instead they tell a short story:

• A fraction of a group: "Three-fourths of the 36 soldiers in a unit passed the swim test..."
• Measurements in mixed numbers: board lengths, hours worked, miles run, cups in a recipe.
• Money with decimals: a paycheck at $14.50 an hour, the change from a $20 bill, the cost of 12 gallons of gas.

What is really being tested is whether you can turn the words into the right operation and carry it out cleanly by hand.

The Conditions You Work Under

If you take a paper version of the test, the pace is much faster, so the shortcuts in this lesson matter even more.

AR is one of the four subtests in your AFQT score, the score that decides whether you can enlist, and it also feeds job line scores such as the Army GT score. Fraction and decimal points are some of the most winnable points on it.

Seven Habits, Applied to Fractions and Decimals

Key Fact: The most common way to lose a fraction point is not a hard calculation. It is answering a slightly different question than the one asked, such as giving the number who passed when the question asked how many did not pass.

A Fully Worked Example

Problem. Three-fourths of the 36 soldiers in a unit passed the swim test on the first try. How many soldiers did not pass on the first try?

Step 1 — Read the last sentence. The question asks for the soldiers who did not pass. Circle "not" on your scratch paper.

Step 2 — Estimate. Three-fourths is most of the group, so the number who did not pass is a small part of 36: well under half, so less than 18.

Step 3 — Translate and compute. If $\frac{3}{4}$ passed, then $1 - \frac{3}{4} = \frac{1}{4}$ did not pass.

$\frac{1}{4} \text{ of } 36 = 36 \div 4 = 9$

Step 4 — Check. Passed: $36 \div 4 \times 3 = 27$. Passed plus not passed: $27 + 9 = 36$. ✓ And $9$ is less than $18$, as the estimate said.

The trap. The answer choices will almost certainly include $27$, the number who did pass. If you skip Step 1, you will compute $27$ correctly and still get the question wrong.

Warm-up. 🎯 Read the last sentence first, then compute.

Warm-up: fraction to decimal. ✍️

1) A water jug is $\frac{3}{4}$ full. Write $\frac{3}{4}$ as a decimal (number only).

2) A recruit has finished $\frac{1}{2}$ of a 6-mile run. How many miles has the recruit run? (number only, in miles)

Key Takeaways — Part 1

• Fraction and decimal questions on AR are word problems: a fraction of a group, mixed-number measurements, or money
• You have about 3.5 minutes per question on the computer test and no calculator, so write each step on scratch paper
• Read the last sentence first: the part that did, or the part that did not? The amount spent, or the amount left?
• Estimate before you compute, and check that your answer fits the story

Up Next: Part 2 — finding a fraction of a quantity, the remaining fraction, and the "fraction of what is left" trap.

🧩 Fractions and Decimals in Context

Part 2 of 7 — Finding a Fraction of a Quantity

"Of" Means Multiply

"Five-sixths of 42" means $\frac{5}{6} \times 42$. Without a calculator, do it in two easy steps:

1. Divide by the denominator (the bottom number) to find the size of one equal part.
2. Multiply by the numerator (the top number) to count how many parts you need.

Example 1. What is $\frac{5}{6}$ of 42?

$42 \div 6 = 7 \qquad 7 \times 5 = 35$

Estimate: $\frac{5}{6}$ is a little less than the whole, so the answer should be a little less than 42. $35$ fits.

Example 2 — a fraction of a population. A town has 2,400 residents, and $\frac{3}{8}$ of them are under 25 years old. How many residents are under 25?

$2{,}400 \div 8 = 300 \qquad 300 \times 3 = 900$

Estimate: $\frac{3}{8}$ is a bit less than half, and half of 2,400 is 1,200. So 900 residents is reasonable.

Key Fact: Divide first, then multiply. Dividing first keeps the numbers small, and in AR problems the division almost always comes out even.

The Fraction That Is Left

Many questions give you the fraction that did something and ask about the rest. The rest is always $1 -$ the fraction.

Example 3. Two-thirds of a 27-member crew work the day shift. The rest work nights. How many work nights?

$1 - \frac{2}{3} = \frac{1}{3} \qquad \frac{1}{3} \text{ of } 27 = 27 \div 3 = 9$

Check: day shift $= 27 \div 3 \times 2 = 18$, and $18 + 9 = 27$. ✓

Two Fractions in a Row: "Of the Rest"

This is the classic AR fraction trap. Read slowly:

Example 4. A recruit's paycheck is $960. She spends $\frac{1}{3}$ of it on rent and then $\frac{1}{4}$ of the rest on food. How much money is left?

Step 1 — First fraction. Rent: $960 \div 3 = 320$, so $320.

Step 2 — Find what is left. $960 - 320 = 640$, so $640 is "the rest."

Step 3 — Second fraction applies to the rest. Food: $640 \div 4 = 160$, so $160.

Step 4 — Answer the question asked. Left: $640 - 160 = 480$, so $480.

The trap: taking $\frac{1}{4}$ of the whole $960 (which is $240) gives $400 left. The words "of the rest" (or "of the remaining," "of what was left") tell you the second fraction applies to the smaller amount.

Shortcut: multiply the fractions that are left. After rent, $\frac{2}{3}$ of the pay is left. After food, $\frac{3}{4}$ of that is left:

$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2} \qquad \frac{1}{2} \text{ of } 960 = 480$

Key Fact: Whenever you see "of the rest" or "of the remaining," write "rest = ___" on your scratch paper and fill it in before you use the second fraction.

Finding the Whole When You Know a Part

Sometimes the question runs backward: it tells you the size of a fraction and asks for the whole.

Example 5. Eighteen trucks make up $\frac{3}{7}$ of a convoy's vehicles. How many vehicles are in the convoy?

Step 1 — Find one part. 3 parts are 18 trucks, so 1 part is $18 \div 3 = 6$ vehicles.

Step 2 — Count all the parts. The whole has 7 parts: $6 \times 7 = 42$ vehicles.

Check: $\frac{3}{7}$ of 42 is $42 \div 7 \times 3 = 18$. ✓

Reasonableness: the whole must be bigger than the part. If your answer is smaller than 18, you multiplied by the fraction instead of working backward.

Back-solving works too. The whole must split into 7 equal parts, so it must be a multiple of 7. If the choices were 42, 54, 63 and 126, you could test each: only $\frac{3}{7}$ of 42 gives 18.

Fraction of a quantity. 🎯

Work it out. ✍️

1) A high school has 840 students, and $\frac{2}{7}$ of them ride the bus. How many students do not ride the bus? (number only)

2) A storage tank holds 480 gallons of fuel. On Monday $\frac{3}{8}$ of the fuel is used. On Tuesday $\frac{1}{2}$ of the remaining fuel is used. How many gallons are left? (number only, in gallons)

Key Takeaways — Part 2

• A fraction of a number: divide by the denominator, multiply by the numerator
• The part that did not is $1 -$ the fraction that did
• "Of the rest" means the second fraction applies to what is left, not to the original amount
• Working backward: part $\div$ numerator = one part; one part $\times$ denominator = the whole, which must be bigger than the part

Up Next: Part 3 — adding and subtracting fractions and mixed numbers in measurements.

🧩 Fractions and Decimals in Context

Part 3 of 7 — Adding and Subtracting Fractions and Mixed Numbers

Common Denominators First

You can only add or subtract fractions when they count the same size pieces. Thirds and fourths are different sizes, so rewrite both as twelfths first.

Example 1. A cook uses $\frac{1}{3}$ cup of oil in the batter and $\frac{1}{4}$ cup in the pan. How much oil is that altogether?

The smallest number both 3 and 4 divide into is 12:

$\frac{1}{3} = \frac{4}{12} \qquad \frac{1}{4} = \frac{3}{12} \qquad \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \text{ cup}$

Add the numerators; keep the denominator.

Never add the tops and the bottoms. $\frac{1}{3} + \frac{1}{4}$ is not $\frac{2}{7}$. A quick check shows why: $\frac{2}{7}$ is less than $\frac{1}{3}$ by itself, and adding more oil cannot leave you with less.

Key Fact: If you cannot spot the common denominator, multiply the two denominators. It always works ($3 \times 4 = 12$); you may just need to simplify at the end.

Mixed Numbers in Measurements

AR loves mixed numbers: $5\frac{1}{2}$ hours, $8\frac{1}{4}$ feet, $2\frac{2}{3}$ cups. To add them, add the whole numbers and the fractions separately, then regroup.

Example 2. A part-time worker logs $5\frac{1}{2}$ hours on Monday and $6\frac{3}{4}$ hours on Tuesday. How many hours is that in all?

Estimate: about $5\frac{1}{2} + 7 = 12\frac{1}{2}$, a little less since $6\frac{3}{4}$ is under 7.

$12\frac{1}{4}$ is a little less than $12\frac{1}{2}$, just as the estimate said. ✓

Don't forget to regroup. Leaving the answer as $11\frac{5}{4}$ is not wrong in value, but the answer choices will show $12\frac{1}{4}$, and $11\frac{1}{4}$ (the carry dropped) is a common wrong choice.

Subtracting: When You Need to Borrow

Example 3. A board is $8\frac{1}{4}$ feet long. A carpenter cuts off $2\frac{5}{8}$ feet. How long is the piece that is left?

Estimate: about $8 - 2\frac{1}{2} = 5\frac{1}{2}$ feet.

Step 1 — Common denominator: $8\frac{2}{8} - 2\frac{5}{8}$

Step 2 — Borrow. You cannot take $\frac{5}{8}$ from $\frac{2}{8}$, so borrow 1 whole from the 8. One whole is $\frac{8}{8}$:

$8\frac{2}{8} = 7 + \frac{8}{8} + \frac{2}{8} = 7\frac{10}{8}$

Step 3 — Subtract: $7\frac{10}{8} - 2\frac{5}{8} = 5\frac{5}{8}$ feet.

Check: $5\frac{5}{8} + 2\frac{5}{8} = 7\frac{10}{8} = 8\frac{2}{8} = 8\frac{1}{4}$. ✓ And $5\frac{5}{8}$ is close to the estimate.

Two borrowing slips to avoid:

• Flipping the subtraction to $\frac{5}{8} - \frac{2}{8}$ (subtracting the smaller fraction from the larger), which gives $6\frac{3}{8}$.
• Borrowing "10" the way you do with whole numbers: the borrowed 1 is $\frac{8}{8}$, so $\frac{2}{8}$ becomes $\frac{10}{8}$, not $\frac{12}{8}$.

Comparing and Ordering Fractions

Some questions ask who ran farther, which tank is fuller, or which fraction is largest.

Method 1 — Cross-multiply. Compare $\frac{5}{8}$ and $\frac{3}{5}$: multiply each numerator by the other denominator.

$5 \times 5 = 25 \qquad 3 \times 8 = 24$

$25 > 24$, so $\frac{5}{8} > \frac{3}{5}$ (the 25 sits on the side of $\frac{5}{8}$).

Method 2 — Benchmark against $\frac{1}{2}$. A fraction is more than half when its numerator is more than half its denominator.

Method 3 — Convert to decimals when you know the benchmarks (Part 5): $\frac{5}{8} = 0.625$ and $\frac{3}{5} = 0.6$.

Key Fact: A bigger denominator means smaller pieces. $\frac{7}{12}$ has the biggest numbers in a list, but that does not make it the biggest fraction.

Add, subtract, compare. 🎯

Measure it. ✍️

1) A carpenter needs two shelves, one $2\frac{3}{8}$ feet long and one $3\frac{5}{8}$ feet long. What is their total length? (number only, in feet)

2) A recruit plans to run 5 miles and has run $3\frac{3}{4}$ miles so far. How many more miles must she run? Give a decimal or a mixed number (number only, in miles).

Set it up right. 🔽

Key Takeaways — Part 3

• Add or subtract only with a common denominator; never add the denominators
• Mixed numbers: wholes with wholes, fractions with fractions, then regroup an improper fraction
• Subtracting a bigger fraction: borrow 1 whole as $\frac{d}{d}$ (for example $\frac{8}{8}$), not as 10
• Compare fractions by cross-multiplying, by the $\frac{1}{2}$ benchmark, or by converting to decimals

Up Next: Part 4 — multiplying and dividing fractions: cutting boards, scaling recipes, and knowing which operation the words call for.

🧩 Fractions and Decimals in Context

Part 4 of 7 — Multiplying and Dividing Fractions in Context

Multiplying: Tops Times Tops, Bottoms Times Bottoms

To multiply fractions, multiply the numerators and multiply the denominators. No common denominator needed.

Example 1. Two-thirds of a field is planted, and $\frac{3}{4}$ of the planted part is corn. What fraction of the whole field is corn?

"Three-fourths of two-thirds" means multiply:

$\frac{3}{4} \times \frac{2}{3}$

Cancel first. The 3 on top cancels the 3 on the bottom, and the 2 on top goes into the 4 on the bottom twice, leaving a 2 on the bottom:

$\frac{3}{4} \times \frac{2}{3} = \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$

(Without canceling: $\frac{6}{12}$, which simplifies to the same $\frac{1}{2}$.)

Half of the field is corn. Reasonableness: a fraction of a fraction must be smaller than both, and $\frac{1}{2}$ is smaller than $\frac{2}{3}$ and $\frac{3}{4}$. ✓

Key Fact: Cancel any top number with any bottom number before you multiply. The numbers stay small and you rarely need to simplify at the end.

Mixed Numbers: Make Them Improper First

Never multiply the whole parts and the fraction parts separately. Convert each mixed number to an improper fraction: whole $\times$ denominator $+$ numerator, over the denominator.

Example 2 — scaling a recipe. A recipe calls for $2\frac{2}{3}$ cups of flour. The mess hall cook makes $1\frac{1}{2}$ times the recipe. How much flour is needed?

Estimate: about $2\frac{1}{2}$ cups times $1\frac{1}{2}$ is about $2\frac{1}{2} + 1\frac{1}{4} = 3\frac{3}{4}$, so 4 cups fits.

The slip to avoid: $2 \times 1 + \frac{2}{3} \times \frac{1}{2} = 2\frac{1}{3}$ cups. That skips the cross terms and is far too small.

Dividing: Invert and Multiply

"How many pieces of length $\frac{3}{4}$ fit into 9?" is a division: $9 \div \frac{3}{4}$. To divide by a fraction, flip it and multiply.

Example 3. How many $\frac{3}{4}$-foot shelf brackets can be cut from a 9-foot board?

$9 \div \frac{3}{4} = 9 \times \frac{4}{3} = \frac{36}{3} = 12 \text{ brackets}$

Reasonableness: each bracket is shorter than 1 foot, so you get more than 9 of them. Dividing by a fraction less than 1 always gives a bigger number.

Example 4. A pot holds $7\frac{1}{2}$ cups of rice. Each serving is $\frac{3}{4}$ cup. How many servings is that?

$7\frac{1}{2} \div \frac{3}{4} = \frac{15}{2} \times \frac{4}{3} = \frac{60}{6} = 10 \text{ servings}$

Check: $10 \times \frac{3}{4} = \frac{30}{4} = 7\frac{1}{2}$. ✓

Multiply or Divide? Let the Words Decide

Key Fact: After you divide, ask: "Should I get more pieces than I started with?" If the piece is smaller than 1 unit, yes.

Multiply or divide? 🎯

Pieces and portions. ✍️

1) A pot holds 10 cups of soup. Each bowl gets $\frac{2}{3}$ cup. How many bowls can be served? (number only)

2) One lap of a track is $\frac{3}{8}$ mile. A recruit runs 6 laps. How many miles is that? Give a decimal or a mixed number (number only, in miles).

Read the words, pick the operation. 🔽

Key Takeaways — Part 4

• Multiply fractions: tops times tops, bottoms times bottoms, and cancel first
• Convert mixed numbers to improper fractions before multiplying or dividing
• Divide by a fraction: invert and multiply; dividing by a fraction less than 1 gives a bigger answer
• "Of" and "times" mean multiply; "how many ___ fit" and "split equally" mean divide

Up Next: Part 5 — decimals and money by hand: paychecks, change, unit costs and gas prices.

🧩 Fractions and Decimals in Context

Part 5 of 7 — Decimals and Money by Hand

Money problems are decimal problems, and on AR you do them without a calculator. Each operation has one rule about the decimal point.

Adding and Subtracting: Line Up the Decimal Points

Stack the numbers so the decimal points sit in one column, and fill empty places with zeros.

Example 1 — making change. A soldier buys a meal for $13.47 and pays with a $20 bill. How much change does he get?

So the change is $6.53.

Count-up check (the way a cashier does it): from $13.47, add 3 cents to reach $13.50, 50 cents to reach $14.00, then $6 to reach $20.00. Total: $0.03 + 0.50 + 6 = 6.53$, so $6.53. ✓

Key Fact: Writing $20 as 20.00 before you subtract prevents the most common change-making slip: forgetting to borrow, which gives $7.53.

Multiplying: Count the Decimal Places

Multiply as if there were no decimal points, then put back as many decimal places as the two factors have in total.

Example 2 — a paycheck. A recruit earns $14.50 an hour and works 36 hours. What is his gross pay?

Estimate: about $15 $\times$ 36 $= 540$, so a bit less than $540.

By hand: $1450 \times 36 = 52{,}200$. The factors have 2 decimal places in total (14.50 has two, 36 has none), so the answer is $522.00$: $522.00.

Faster: break the wage apart. $14 \times 36 = 504$ and $0.50 \times 36 = 18$ (half of 36), so $504 + 18 = 522$, which is $522.00. ✓

Example 3 — a fill-up. Gas costs $3.20 a gallon. How much do 12 gallons cost?

$320 \times 12 = 3{,}840 \;\rightarrow\; 2 \text{ decimal places} \;\rightarrow\; 38.40$

So the fill-up is $38.40. Estimate: about $3 $\times$ 12 $= 36$, so $38.40 is right, while $384.00 or $3.84 has the point in the wrong place.

Dividing: Make the Divisor a Whole Number

When you divide by a decimal, move the decimal point in the divisor to the right until it is a whole number, and move the point in the dividend the same number of places.

Example 4. Snacks cost $0.65 each. How many can you buy with $7.80?

$7.80 \div 0.65 \;\rightarrow\; 780 \div 65 = 12$

(Both points moved 2 places.) Check: $65 \times 12 = 780$. ✓

Example 5 — unit cost. A pack of 6 energy bars costs $4.50. What is the cost per bar?

$4.50 \div 6 = 0.75$

The divisor 6 is already whole, so the decimal point goes straight up into the answer: $0.75 per bar. Reasonableness: 6 bars at under $1 each makes sense for a $4.50 pack.

Fraction and Decimal Benchmarks

Know these cold. They turn many money problems into one quick step.

Use the friendlier form. "0.75 times $36" is easier as "$\frac{3}{4}$ of 36": $36 \div 4 \times 3 = 27$, so $27.

Money by hand. 🎯

Do it on scratch paper. ✍️

1) Gas costs $3.25 a gallon. How many dollars does it cost to buy 14 gallons? (number only, in dollars)

2) A bolt is $\frac{3}{8}$ inch wide. Write its width as a decimal. (number only, in inches)

3) Three roommates split a $157.50 moving-truck bill evenly. How many dollars does each pay? (number only, in dollars)

Key Takeaways — Part 5

• Add or subtract: line up the decimal points and fill in zeros (write $20 as 20.00)
• Multiply: multiply as whole numbers, then count the total decimal places in both factors
• Divide by a decimal: move the point in the divisor and the dividend the same number of places
• Estimate every money answer: a misplaced decimal point is off by a factor of 10 and is easy to spot
• Know the benchmarks: $\frac{1}{8} = 0.125$, $\frac{1}{4} = 0.25$, $\frac{3}{8} = 0.375$, $\frac{1}{2} = 0.5$, $\frac{3}{4} = 0.75$

Up Next: Part 6 — test strategy and the traps that catch the most people on fraction and decimal questions.

🧩 Fractions and Decimals in Context

Part 6 of 7 — Test Strategy and Common Traps

The wrong answer choices on AR are not random. Each one is the result of a specific, common mistake. If you know the mistakes, you can see the traps coming.

The Six Big Traps

Key Fact: Each of these traps produces a number that appears among the answer choices. Finding your answer in the list does not prove it is right. Your check does.

Timing, Estimation and Back-Solving

Timing

On the computer test you have 55 minutes for 15 questions, about 3.5 minutes each. A one-step "fraction of a group" question should take about a minute. That banked time pays for the multi-step ones: two fractions in a row, or a paycheck plus a deduction. Remember that you cannot go back, so spend the extra seconds checking before you answer.

Estimate to eliminate choices

Round mixed numbers and prices to friendly values before you compute.

Example. Each of 4 tent poles is $5\frac{7}{8}$ feet long. Round $5\frac{7}{8}$ up to 6: the total is a bit less than $4 \times 6 = 24$ feet. Any answer near 20 or near 30 is out immediately. (Exactly: $4 \times 5\frac{7}{8} = 23\frac{1}{2}$ feet.)

Back-solve "find the whole" questions

When a question gives a part and asks for the whole, the answer must split evenly by the denominator.

Example. "Five-ninths of a group is 35 people. How many are in the group?" Choices: 45, 50, 63, 70. The whole must split into ninths, so it must be a multiple of 9: only 45 and 63 qualify. Test 63: $63 \div 9 \times 5 = 35$. ✓ The group has 63 people.

Reasonableness checklist

Before you click, ask:

• Is a part smaller than the whole, and is a whole bigger than a known part?
• Did dividing by a fraction less than 1 give a bigger number?
• Is a fraction of a fraction smaller than both?
• Does the money answer match my estimate to the nearest power of 10?
• Did I answer the question in the last sentence (left, not used; did not, not did)?

Spot the trap. 🎯

Name the mistake. 🔽

Key Takeaways — Part 6

• The wrong choices come from six traps: second fraction on the original, adding tops and bottoms, wrong operation, misplaced decimal point, fraction used vs fraction left, mixed numbers split apart
• Bank time on one-step items (about a minute) for multi-step items; you cannot go back, so check before you click
• Estimate to throw out choices; back-solve "find the whole" questions with multiples of the denominator
• Run the reasonableness checklist on every answer

Up Next: Part 7 — a mixed mastery set in true ASVAB style.

🧩 Fractions and Decimals in Context

Part 7 of 7 — Mixed Mastery Set

This set mixes everything from Parts 2 through 5 the way the real AR subtest does: a few quick ones, several medium ones, and a few multi-step problems. Treat it like test day:

• No calculator. Use scratch paper for every step.
• Read the last sentence first and circle words like left, not and rest.
• Estimate before you compute, and check before you answer, because on the real test you cannot go back.
• Aim for about 3.5 minutes per question or less.

Set A: warm-up to medium. 🎯

Set B: medium to hard. 🎯

One to type. ✍️

In a survey of 1,600 residents, $\frac{5}{16}$ said they ride the bus to work. How many of the residents surveyed ride the bus to work? (number only)

Key Takeaways — Fractions and Decimals in Context

• Fraction of a quantity: divide by the denominator, multiply by the numerator; the rest is $1 -$ the fraction
• Of the rest: apply the second fraction to what is left, not to the original amount
• Find the whole: part $\div$ numerator $\times$ denominator, and the whole must be bigger than the part
• Add and subtract: common denominators, regroup mixed numbers, borrow 1 whole as $\frac{d}{d}$
• Multiply and divide: cancel first, convert mixed numbers to improper fractions, invert and multiply to divide
• Decimals and money: line up points to add or subtract, count decimal places to multiply, move both points to divide, and always estimate

What to Do Next

1. Take the exit quiz for this topic to lock in your score.
2. Review the flashcards for the fraction methods and decimal benchmarks until they are automatic.
3. Move on to the next topic in the AR unit: Percents, Discounts and Interest. Percents are fractions out of 100, so everything you practiced here carries straight over.