Compare fractions to determine which is greater.

SECTION 1

A Quick History of Fractions

Long ago, people needed to share things like food or land fairly. Fractions (parts of a whole) helped them do that. Ancient Egyptians used special fractions around 1800 BC.

1800 BC

Egyptians

Used unit fractions like 1/3 to divide loaves of bread.

300 BC

Greeks

Euclid wrote about comparing fractions in his math books.

1202

Fibonacci

Brought Hindu-Arabic numbers to Europe, making fractions easier.

1600s

Modern Methods

Math books taught cross-multiplication to compare any fractions.

Today on the SSAT, you compare fractions fast. These old ideas make it simple and fun!

SECTION 2

Core Ideas for Comparing Fractions

A fraction shows part of a whole, like 3/4 means 3 parts out of 4 equal parts. The top number is the numerator (pieces you have). The bottom is the denominator (total pieces).

Same Denominator

Bigger numerator wins! Like 3/5 > 2/5.

Same Numerator

Smaller denominator wins! Like 2/3 > 2/5.

Cross-Multiply

For a/b and c/d, check if a×d > b×c.

Decimals

Turn to decimals: 1/2=0.5 > 1/3≈0.33.

✦ KEY TAKEAWAY

Pizza Analogy

SECTION 3

See It with Fraction Bars

Same-size bars show shaded parts. The longer shaded bar is greater.

Look at the bars above. The 1/2 shaded part covers more than 1/3. You can picture this for any fractions!

SECTION 4

The Math Behind Comparisons

The best trick is cross-multiplication. It works for any positive fractions.

CROSS-MULTIPLY RULE

a/b > c/d if a × d > b × c

All positive fractions. Switch > to < if needed.

EXAMPLE

3/4 ? 2/5 → 3×5=15 4×2=8 15>8 so 3/4 > 2/5

Multiply corner to corner and compare products.

This is like a balance scale in your mind. Bigger product on top means bigger fraction. Practice it—you'll ace SSAT questions!

5 · Breakdown

Number Line Breakdown

Dots show where fractions land between 0 and 1. Rightmost is greatest.

The number line helps when picturing sizes. 1/2 is right of 1/3, so it's bigger. Great for tricky ones!

SECTION 6

Step-by-Step: Compare 3/5 and 7/12

Step 1: Check if same denominator5 ≠ 12, so no. Can't compare numerators directly.

Different denominators

Step 2: Cross-multiplyTop-left × bottom-right: 3 × 12 = 36. Bottom-left × top-right: 5 × 7 = 35.

36 > 35

Step 3: DecideSince 36 > 35, 3/5 > 7/12. You got it!

3/5 is greater

SECTION 7

Strengths of Each Method

Pick the fastest method for SSAT speed!

Method	Best For	Why It Works
Same Denominator	Quick check	Direct size compare
Cross-Multiply	Any fractions	Math proof of size
Decimals	Easy mental math	Familiar numbers

✦ KEY TAKEAWAY

SECTION 8

Links to Bigger Math Ideas

This Lesson	Advanced Version
Compare two fractions	Order many: 1/4, 5/8, 2/3
Cross-multiply	Common denominator method
Proper fractions	Mixed numbers, add/subtract

Master this now, and adding fractions later will be easy. You're building super math powers!

SECTION 9

Test Your Skills

PROBLEM 1 — CONCEPTUAL

Which is greater? A. 3/8 B. 2/8 C. 1/8 D. 4/8 E. 0/8

PROBLEM 2 — BASIC CALCULATION

Which is greater? A. 1/4 B. 1/6 C. 1/3 D. 1/5 E. 1/2

PROBLEM 3 — INTERMEDIATE

Compare 2/3 and 3/4. Which is greater? A. 2/3 B. 3/4 C. They are equal D. 3/4 is smaller E. Can't tell

PROBLEM 4 — APPLIED

Sara has 5/6 of her game left. Tim has 3/4. Who has more left? A. Sara B. Tim C. Equal D. Tim by a little E. Sara by a lot

PROBLEM 5 — CRITICAL THINKING

Order from least to greatest: 4/7, 5/9, 3/5. Which is correct? A. 5/9, 4/7, 3/5 B. 3/5, 4/7, 5/9 C. 5/9, 3/5, 4/7 D. 4/7, 5/9, 3/5 E. 3/5, 5/9, 4/7

Tip