Solve fraction word problems involving multiplication or division. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Where Did Fractions Come From?

Have you ever split a pizza with friends? If so, you already know what a fraction is! A fraction is a part of a whole. People have been using fractions for thousands of years to share things fairly and measure things carefully.

Long ago, the ancient Egyptians needed fractions to divide land and food. Over time, people found smarter ways to work with parts of things. Let's look at how fractions grew up!

1800 BC • Ancient Egypt

Egyptians used fractions to divide bread and measure land along the Nile River. They mostly used fractions with 1 on top, like ½ and ¼.

500 BC • Ancient Greece

Greek mathematicians studied how fractions relate to music and shapes. They discovered that fractions help explain patterns everywhere!

600 AD • India

Indian mathematicians invented the way we write fractions today — one number on top and one on the bottom, like ¾.

Today • ISEE and School Math

Fraction word problems show up on the ISEE test. Knowing how to multiply and divide fractions helps you solve them quickly!

On the ISEE, you will see word problems that ask you to find a fraction of something (that means multiply!) or split something into equal fractional parts (that means divide!). Let's learn exactly how to do that.

SECTION 2

Core Ideas You Need to Know

Before we jump into problems, let's review the big ideas. These four rules will help you tackle every fraction word problem on the ISEE.

"Of" Means Multiply

When a problem says "½ of 12," it means ½ × 12. The word "of" is your clue to multiply.

Multiply Straight Across

To multiply fractions, multiply the top numbers (numerators) together and the bottom numbers (denominators) together.

Divide by Flipping

To divide by a fraction, flip the second fraction upside down and then multiply. This flipped fraction is called the reciprocal.

Always Simplify

After you multiply or divide, check if your answer can be made simpler. For example, 4/8 simplifies to ½.

SECTION 3

See It! Fractions in Pictures

Pictures make fractions much easier to understand. Let's look at what ½ × ⅔ really means. We start with a rectangle that represents one whole thing.

This diagram shows how ½ × ⅔ works. First we shade ⅔ of the rectangle (2 out of 3 columns). Then we take ½ of that shaded area (top row only). The purple pieces are 2 out of 6 total pieces, which equals ⅓.

See how the picture matches the math? We multiplied the tops: 1 × 2 = 2. We multiplied the bottoms: 2 × 3 = 6. That gives us 2/6, which simplifies to ⅓. The picture and the math agree!

SECTION 4

The Math Rules

Here are the two key formulas you need. Don't worry — they are simple once you practice them a few times!

MULTIPLYING FRACTIONS

a/b × c/d = (a × c) / (b × d)

Multiply the numerators (top numbers) together. Multiply the denominators (bottom numbers) together. Then simplify if you can.

DIVIDING FRACTIONS (KEEP-CHANGE-FLIP)

a/b ÷ c/d = a/b × d/c

Keep the first fraction. Change the ÷ sign to ×. Flip the second fraction upside down. Then multiply straight across!

MULTIPLYING A FRACTION BY A WHOLE NUMBER

a/b × n = (a × n) / b

Think of the whole number as a fraction: write n as n/1. Then multiply straight across. For example, ¾ × 8 = ¾ × 8/1 = 24/4 = 6.

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ISEE Test Tip

SECTION 5

Types of Fraction Word Problems

On the ISEE, fraction word problems come in a few common types. Learning to spot each type will help you pick the right operation fast. Let's look at them!

This chart shows the four main types of fraction word problems you'll see on the ISEE. Types 1 and 2 use multiplication. Types 3 and 4 use division. Look for the key words to decide!

Here is a quick trick: if the problem asks you to find a smaller part of something, you usually multiply. If the problem asks you to split or share something, you usually divide.

SECTION 6

Let's Solve One Together!

Here's a word problem just like one you might see on the ISEE. Let's work through it step by step.

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THE PROBLEM

Worked example

1
Step 1 — Read and Understand

Maria has ¾ of a pound. She shares it equally among 3 friends. The word "equally" tells us each friend gets the same amount. We need to split ¾ into 3 equal parts.
2
Step 2 — Pick the Operation

"Share equally" means divide. So we do ¾ ÷ 3.

Operation: ¾ ÷ 3
3
Step 3 — Rewrite the Whole Number as a Fraction

Write 3 as a fraction: 3 = 3/1. Now the problem is ¾ ÷ 3/1.
4
Step 4 — Keep, Change, Flip

Keep ¾. Change ÷ to ×. Flip 3/1 to 1/3. Now we have ¾ × ⅓.

¾ × ⅓
5
Step 5 — Multiply Straight Across

Tops: 3 × 1 = 3. Bottoms: 4 × 3 = 12. The answer is 3/12.

3/12
6
Step 6 — Simplify

3 and 12 can both be divided by 3. So 3/12 = 1/4. Each friend gets ¼ of a pound of trail mix!

Answer: ¼ pound

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Key takeaway

CHECK YOUR WORK

SECTION 7

ISEE Test-Taking Strategies

Knowing the math is great, but using smart strategies can help you even more. Here are some tips that work especially well on the ISEE.

Strategy	How It Helps	Example
Look for "of"	The word "of" means multiply. Circle it when you see it!	"⅓ of 15" → ⅓ × 15 = 5
Estimate first	Round the fraction to ½ or 1 to check which answer choice is close.	⅜ of 24 ≈ ½ of 24 = 12, so answer is near 9.
Eliminate wrong answers	Cross out answers that are obviously too big or too small.	½ of 10 can't be 15 (too big!) — cross it out.
Check with a picture	Draw a quick picture or number line to test your answer.	Draw a rectangle, split it, and shade to see the fraction.

ISEE Test Strategies for Fraction Word Problems

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Key takeaway

REMEMBER

SECTION 8

Connecting to Bigger Ideas

The skills you're learning now will keep helping you as math gets bigger. Multiplying and dividing fractions connects to many topics you'll study later.

What You Know Now	What Comes Next
Multiplying fractions like ½ × ⅓	Multiplying decimals like 0.5 × 0.33
Dividing fractions using Keep-Change-Flip	Dividing decimals and solving equations
Finding ¾ of a number	Finding 75% of a number (percents!)
Fraction word problems	Ratio and proportion problems

From Fractions to Future Math

Did you know that fractions, decimals, and percents are all related? When you master fraction multiplication and division, you're also building skills for decimals and percents. You're getting a head start on future math!

SECTION 9

Practice Problems

Time to practice! These problems go from easier to harder, just like the real ISEE. Try each one, then check the answer. You've got this!

Question 1: Conceptual

A pizza has 8 slices. Maria eats 1/4 of the pizza. How many slices does Maria eat?

A. 1
B. 2
C. 3
D. 4

Show answer (B)

To find 1/4 of 8, multiply: 1/4 × 8 = 8 ÷ 4 = 2 slices. The correct answer is B.A (1): This results from dividing 8 by 8 instead of by 4, as if finding 1/8 of the pizza.C (3): This results from subtracting the denominator from the total slices and then halving (8 − 4 = 4, then guessing 3), a computational error with no valid reasoning.D (4): This results from dividing 8 by 2 instead of by 4, as if finding 1/2 of the pizza rather than 1/4.

Question 2: Applied

A class has 30 students. If 2/5 of the students play soccer, how many students play soccer?

A. 6
B. 10
C. 12
D. 15

Show answer (C)

To find 2/5 of 30, multiply: 2/5 × 30 = (30 ÷ 5) × 2 = 6 × 2 = 12 students. The correct answer is C.A (6): This results from finding only 1/5 of 30 (30 ÷ 5 = 6) and forgetting to multiply by the numerator 2.B (10): This results from dividing 30 by 3 instead of 5, as if computing 1/3 of 30.D (15): This results from finding 1/2 of 30 instead of 2/5 of 30.

Question 3: Applied

A baker made 24 muffins. He sold 3/4 of them in the morning. How many muffins did he have left?

A. 4
B. 6
C. 8
D. 18

Show answer (B)

First find 3/4 of 24: (24 ÷ 4) × 3 = 6 × 3 = 18 muffins sold. Then subtract: 24 − 18 = 6 muffins left. The correct answer is B.A (4): This results from computing 1/4 of 24 (which is 6) and then subtracting 2, confusing the numerator and making an arithmetic slip to get 4.C (8): This results from finding 1/3 of 24 instead of 1/4 of 24, treating the denominator as 3 rather than 4.D (18): This is the number of muffins sold (3/4 of 24 = 18), not the number left over. The problem asks how many remain.

Question 4: Applied

Sam has a rope that is 3/4 of a yard long. He cuts it into 3 equal pieces. How long is each piece?

A. 1/4 of a yard
B. 1/3 of a yard
C. 1/2 of a yard
D. 3/4 of a yard

Show answer (A)

Divide 3/4 by 3: (3/4) ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4 of a yard. The correct answer is A.B (1/3): This results from dividing 1 yard by 3 rather than dividing 3/4 of a yard by 3, ignoring that the rope is only 3/4 of a yard.C (1/2): This results from subtracting 1/4 from 3/4 (getting 2/4 = 1/2) instead of dividing 3/4 by 3.D (3/4): This is the original length of the rope before cutting, not the length of each piece.

Question 5: Challenge

A farmer has 36 apples. He gives 1/3 of them to his neighbor and then gives 1/4 of the remaining apples to his daughter. How many apples does the farmer have left?

A. 15
B. 18
C. 21
D. 24

Show answer (B)

Step 1: The farmer gives away 1/3 of 36 = 12 apples to his neighbor. He has 36 − 12 = 24 apples remaining.Step 2: He gives 1/4 of the remaining 24 apples to his daughter: 24 ÷ 4 = 6 apples. He has 24 − 6 = 18 apples left. The correct answer is B.A (15): This results from taking 1/4 of the original 36 (which is 9) and giving that to the daughter: 24 − 9 = 15. The error is finding 1/4 of 36 instead of 1/4 of the remaining 24.C (21): This results from computing 1/4 of 36 = 9, then subtracting only the daughter's share from the original: 36 − 9 − 6 = 21, confusing which amounts to subtract and mixing up the steps.D (24): This is the number remaining after giving 1/3 to the neighbor but before giving any to the daughter. This answer skips the second step entirely.