Compare fractions to determine which is greater. - SSAT Middle Level Quantitative

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.

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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!

Section 5

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

Worked example

1
Step 1: Check if same denominator

5 ≠ 12, so no. Can't compare numerators directly.

Different denominators
2
Step 2: Cross-multiply

Top-left × bottom-right: 3 × 12 = 36. Bottom-left × top-right: 5 × 7 = 35.

36 > 35
3
Step 3: Decide

Since 36 > 35, 3/5 > 7/12. You got it!

3/5 is greater

SECTION 7

Strengths of Each Method

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

Pick the fastest method for SSAT speed!

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

Question 1: CONCEPTUAL

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

Show answer

D. Same denominator (8), so biggest numerator (4) is greatest. A=3/8 smaller numerator. B=2/8 even smaller. C=1/8 tiny. E=0 nothing. Great start!

Question 2: BASIC CALCULATION

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

Show answer

E. Same numerators (1), smallest denominator (2) is biggest. A=1/4 bigger denom smaller. B=1/6 even smaller. C=1/3 medium. D=1/5 close but bigger denom.

Question 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

Show answer

B. Cross-multiply: 2×4=8, 3×3=9; 8<9 so 2/3 < 3/4. A wrong, 2/3 smaller. C not equal (8≠9). D opposite. E you can tell!

Question 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

Show answer

A. 5/6 vs 3/4: 5×4=20, 6×3=18; 20>18 so 5/6 > 3/4. B wrong. C 20≠18. D/Tim smaller. E not by a lot, but Sara wins!

Question 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

Show answer

E. Check pairs: 3/5 vs 5/9: 3×9=27>5×5=25, so 3/5>5/9. 5/9 vs 4/7:5×7=35<4×9=36, 5/9<4/7. So 5/9 < 3/5 < 4/7? Wait no: 3/5=0.6, 5/9≈0.556, 4/7≈0.571. Yes 5/9 < 4/7 < 3/5? Cross: 5/9 & 4/7: 5*7=35, 9*4=36, 35<36 so 5/9<4/7. 4/7 & 3/5:4*5=20,7*3=21,20<21 so 4/7<3/5. Thus 5/9 < 4/7 < 3/5, but options: E is 3/5,5/9,4/7 no. Least to greatest: 5/9, 4/7, 3/5. That's A. Yes A. B wrong order. C swaps. D weird. E reverse-ish.

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