Compare fractions to determine which is greatest. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Why Do We Compare Fractions?

People have been working with fractions for thousands of years! Long ago, farmers needed to split land and food fairly. They had to figure out which piece was bigger.

Imagine sharing a pizza with friends. You want to know: is one-third of the pizza bigger than one-fourth? That's comparing fractions! Let's see how this idea grew over time.

1800 BC • Ancient Egypt

Egyptian farmers used fractions to divide land along the Nile River. They mostly used fractions with 1 on top, like ½ and ⅓.

500 BC • Ancient Greece

Greek mathematicians studied how fractions relate to each other. They loved finding patterns in numbers!

600 AD • India

Indian mathematicians wrote fractions the way we do today — with a top number and a bottom number. This made comparing fractions much easier.

Today • The ISEE Test

On the ISEE, you compare fractions to find which is greatest. You can do it quickly with some cool strategies!

So here's the big question: when you see two or more fractions, how do you figure out which one is the biggest? Let's learn how!

SECTION 2

Core Ideas for Comparing Fractions

Before we start comparing, let's review some important ideas. A fraction has two parts. The top number is the numerator (how many pieces you have). The bottom number is the denominator (how many equal pieces the whole is cut into).

Same Denominator? Compare Numerators!

When the bottom numbers match, just look at the top numbers. The bigger numerator means the bigger fraction. Example: ⅗ > ⅖ because 3 > 2.

Same Numerator? Compare Denominators!

When the top numbers match, the smaller denominator means the bigger fraction. Example: ⅓ > ⅕ because thirds are bigger pieces than fifths.

Different Everything? Find a Common Denominator!

When nothing matches, make the bottom numbers the same. Multiply to create equivalent fractions. Then compare the new top numbers.

Cross-Multiply Shortcut

To compare two fractions fast, multiply across in an X pattern. Compare the two products. The bigger product points to the bigger fraction!

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KEY TAKEAWAY

Think of fractions like slices of pizza. If two pizzas are cut into the same number of slices, the one with more slices taken wins. If you take the same number of slices from each pizza, the pizza cut into fewer (bigger) slices gives you more food!

SECTION 3

See It: Fractions on a Number Line

One of the best ways to compare fractions is to picture them on a number line. The farther right a fraction sits, the greater it is. Let's look at ¼, ⅓, and ½ on a number line!

The number line shows ¼ (amber), ⅓ (violet), and ½ (cyan). Since all three fractions have the same numerator of 1, the one with the smallest denominator (½) is the greatest.

See how ½ is the farthest to the right? That means it's the greatest. On the ISEE, picturing a number line in your head is a great strategy. Even a quick sketch on scratch paper can help!

SECTION 4

The Math Behind Comparing Fractions

There are two powerful math methods you can use. Let's learn both!

Method 1: Common Denominators

To compare fractions with different denominators, change them so they have the same bottom number. Then just compare the top numbers!

COMMON DENOMINATOR METHOD

Compare ⅔ and ¾ → Make denominators the same → 8/12 vs. 9/12 → 9/12 wins!

Multiply ⅔ by 4/4 to get 8/12. Multiply ¾ by 3/3 to get 9/12. Since 9 > 8, we know that ¾ > ⅔.

Method 2: Cross-Multiply

Here's a quick shortcut! To compare two fractions, cross-multiply. Multiply the numerator of the first fraction by the denominator of the second. Then multiply the numerator of the second fraction by the denominator of the first. The bigger product tells you the bigger fraction.

CROSS-MULTIPLY METHOD

Compare ⅔ and ¾ → 2 × 4 = 8 vs. 3 × 3 = 9 → 9 > 8 → ¾ is greater!

The product 8 stays next to ⅔. The product 9 stays next to ¾. Since 9 > 8, ¾ is greater.

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ISEE Test Tip

Cross-multiplying is super fast on the ISEE! It works for any two fractions. Remember: each product stays on the side of the numerator you used. The bigger product points to the bigger fraction.

SECTION 5

Which Strategy Should You Use?

On the ISEE, you want to work fast and smart. Different problems call for different strategies. This diagram shows you how to pick the best one!

This decision tree helps you pick the fastest strategy. Start at the top and follow the arrows based on what the fractions look like.

Situation	Example	Best Strategy
Same denominator	⅗ vs. ⅖	Compare numerators: 3 > 2, so ⅗ wins
Same numerator	²⁄₅ vs. ²⁄₇	Smaller denominator wins: ²⁄₅ > ²⁄₇
Comparing to ½	⅜ vs. ⁴⁄₆	⅜ < ½ and ⁴⁄₆ > ½, so ⁴⁄₆ wins
Different everything	⅔ vs. ¾	Cross-multiply: 2×4=8 vs. 3×3=9 → ¾ wins

Quick-reference table for choosing the right strategy

SECTION 6

Worked Example: Step by Step

Let's solve a problem together, just like on the real ISEE. Which fraction is greatest: ⅗, ½, or ⅝?

Which is greatest: ⅗, ½, or ⅝?

1
Step 1 — Find a Common Denominator

The denominators are 5, 2, and 8. We need a number that all three divide into evenly. Think: what is the least common multiple of 5, 2, and 8? It's 40!

Common denominator = 40
2
Step 2 — Convert ⅗ to Fortieths

Multiply top and bottom by 8: 3 × 8 = 24 and 5 × 8 = 40. So ⅗ = 24/40.

⅗ = 24/40
3
Step 3 — Convert ½ to Fortieths

Multiply top and bottom by 20: 1 × 20 = 20 and 2 × 20 = 40. So ½ = 20/40.

½ = 20/40
4
Step 4 — Convert ⅝ to Fortieths

Multiply top and bottom by 5: 5 × 5 = 25 and 8 × 5 = 40. So ⅝ = 25/40.

⅝ = 25/40
5
Step 5 — Compare the Numerators

Now we have 24/40, 20/40, and 25/40. Compare the tops: 20, 24, 25. The biggest numerator is 25, which belongs to ⅝.

⅝ is the greatest!

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Check Your Answer

On the ISEE, always check: does your answer make sense? We know ⅝ is bigger than ½ (because 5 > 4, which is half of 8). And ⅝ is bigger than ⅗ because 0.625 > 0.6. It checks out!

SECTION 7

Comparing the Methods

Each method for comparing fractions has its strengths. On the ISEE, time matters! Let's see which method is best for each situation.

Method	Strengths	Limitations
Compare Numerators	Super fast! No math needed.	Only works when denominators match.
Compare Denominators	Quick and easy when tops match.	Only works when numerators match.
Cross-Multiply	Fast for any pair of fractions.	Only compares two fractions at a time.
Common Denominator	Works for any number of fractions.	Takes more steps and bigger numbers.
Benchmark to ½	Great for quick elimination.	Only helps if fractions are on different sides of ½.

Each method has its time and place on the ISEE

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ISEE STRATEGY

Think of these methods like tools in a toolbox. A screwdriver is great for screws, but you need a hammer for nails. Pick the tool that fits the problem. And remember — on the ISEE, look at the answer choices first. Sometimes you can eliminate wrong answers quickly without doing all the math!

SECTION 8

Fractions, Decimals & Beyond

Comparing fractions connects to bigger math ideas you'll see later. Sometimes the ISEE mixes fractions with decimals or mixed numbers. Here's how they all connect!

What You Know Now	What's Coming Next
Comparing fractions like ⅔ and ¾	Comparing decimals like 0.66 and 0.75
Finding common denominators	Adding and subtracting fractions
Comparing two fractions	Ordering three or more fractions from least to greatest
Fractions less than 1	Mixed numbers like 2⅓ and 1¾

Your fraction skills are building blocks for future math!

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Decimal Shortcut

Some fractions are easy to turn into decimals. ½ = 0.5, ¼ = 0.25, ¾ = 0.75, ⅕ = 0.2. If you know these by heart, you can compare even faster on the ISEE!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own before looking at the answer. Remember: on the ISEE, always answer every question — there's no penalty for guessing!

Question 1: CONCEPTUAL

Four students each ate a piece of a pizza that was cut into 7 equal slices. Ana ate ²⁄₇ of the pizza, Ben ate ³⁄₇ of the pizza, Cara ate ⁴⁄₇ of the pizza, and Dan ate ⁵⁄₇ of the pizza. Who ate the most pizza? (A) Ana (B) Ben (C) Cara (D) Dan

Show answer

The correct answer is (D) Dan. Each pizza was cut into 7 equal slices, so all four fractions have the same denominator of 7. To find who ate the most, compare the numerators: Ana ate 2 slices, Ben ate 3 slices, Cara ate 4 slices, and Dan ate 5 slices. Since 5 is the largest numerator, ⁵⁄₇ is the greatest fraction, and Dan ate the most pizza. (A) Ana is wrong because 2 is the smallest numerator. (B) Ben is wrong because 3 < 5. (C) Cara is wrong because 4 < 5.

Question 2: BASIC CALCULATION

Four friends each received one equal piece from a same-sized cake. Mia's cake was cut into 8 equal pieces, Leo's cake was cut into 5 equal pieces, Ava's cake was cut into 3 equal pieces, and Sam's cake was cut into 6 equal pieces. Which friend received the largest piece? (A) Mia, who received ¹⁄₈ of her cake (B) Leo, who received ¹⁄₅ of his cake (C) Ava, who received ¹⁄₃ of her cake (D) Sam, who received ¹⁄₆ of his cake

Show answer

The correct answer is (C) ¹⁄₃. Four friends each received one piece from a same-sized cake. Mia's cake was cut into 8 equal pieces, Leo's into 5, Ava's into 3, and Sam's into 6. All four pieces are unit fractions — fractions with 1 in the numerator. When numerators are the same, the fraction with the smallest denominator is the greatest, because the cake is cut into fewer pieces, making each piece bigger. The denominators are 8, 5, 3, and 6. Since 3 is the smallest denominator, ¹⁄₃ is the greatest fraction. Ava received the largest piece. (A) ¹⁄₈ is the smallest piece because that cake was cut into the most pieces. (B) ¹⁄₅ and (D) ¹⁄₆ have larger denominators than 3, so those pieces are smaller than Ava's.

Question 3: INTERMEDIATE

Four friends each received their own pizza of the same size. Lena ate ¼ of her pizza, Carlos ate ⅜ of his pizza, Priya ate ⅓ of her pizza, and Maya ate ½ of her pizza. Which friend ate the greatest amount of pizza? (A) Lena, who ate ¼ (B) Carlos, who ate ⅜ (C) Priya, who ate ⅓ (D) Maya, who ate ½

Show answer

The correct answer is (D) ½. Four friends each ate part of their own pizza. To find who ate the most, compare all four fractions: ¼, ⅓, ⅜, and ½. Convert each fraction to 24ths, since 24 is the smallest number that ¼, ⅓, ⅜, and ½ all divide into evenly: ¼ = 6/24, ⅓ = 8/24, ⅜ = 9/24, and ½ = 12/24. Now it is easy to compare: 6/24 < 8/24 < 9/24 < 12/24, which means ¼ < ⅓ < ⅜ < ½. The greatest fraction is ½, so Maya ate the most pizza.

Question 4: APPLIED

Maria ate ⅖ of her sandwich. Tom ate ⅜ of his sandwich. Nina ate ⅓ of her sandwich. Sam ate ¼ of his sandwich. All sandwiches were the same size. Who ate the most? (A) Maria (B) Nina (C) Tom (D) Sam

Show answer

The correct answer is (A) Maria. Compare ⅖, ⅜, ⅓, and ¼ by finding a common denominator. The LCM of 5, 8, 3, and 4 is 120. Convert each fraction: ⅖ = 48/120, ⅜ = 45/120, ⅓ = 40/120, ¼ = 30/120. Rank them: 48 > 45 > 40 > 30, so Maria (48/120) ate the most. (B) Nina ate ⅓ = 40/120, the second smallest share. (C) Tom ate ⅜ = 45/120, the second largest share. (D) Sam ate ¼ = 30/120, the smallest share of all four.

Question 5: CRITICAL THINKING

A teacher measured how full four fish tanks were in the classroom: one tank was ⅙ full, one was ⅓ full, one was ⅝ full, and one was ¾ full. Which list correctly orders the tank levels from least full to most full? (A) ⅙, ⅓, ⅝, ¾ (B) ¾, ⅝, ⅓, ⅙ (C) ⅓, ⅙, ⅝, ¾ (D) ⅙, ⅝, ⅓, ¾

Show answer

The correct answer is (A) ⅙, ⅓, ⅝, ¾. Convert each fraction to a common denominator of 24: ⅙ = 4/24, ⅓ = 8/24, ⅝ = 15/24, ¾ = 18/24. Ordering from least to greatest: 4/24, 8/24, 15/24, 18/24 — which is ⅙, ⅓, ⅝, ¾. (B) lists the tanks from most full to least full, which is the opposite of what was asked. (C) places ⅓ before ⅙, but ⅓ > ⅙, so ⅙ must come first. (D) places ⅝ before ⅓, but ⅝ > ⅓, so ⅓ must come before ⅝.

SUMMARY

Let's Review!

Summary

To compare fractions on the ISEE, use the right tool for the job. When denominators are the same, compare the numerators — bigger top means bigger fraction. When numerators are the same, compare the denominators — smaller bottom means bigger fraction. When everything is different, use cross-multiplication for two fractions, or find a common denominator for three or more.

Remember these ISEE tips: use benchmarks like ½ to quickly eliminate wrong answers. Always answer every question because there's no penalty for guessing. And picture fractions on a number line — the farther right, the greater the fraction. You've got this!