Fraction Mastery — Gifted & Enrichment | Gifted & Enrichment | Scholars & Parents

What Is This?

Fractions are one of the most important and most misunderstood topics in all of mathematics. True fraction mastery goes well beyond adding simple fractions — it means understanding why the algorithms work, handling all four operations fluently with any combination of fractions and mixed numbers, comparing and ordering complex fractions efficiently, and applying fraction reasoning to multi-step competition problems.

The Meaning of a Fraction

A fraction a/b represents a parts of a whole divided into b equal parts. The denominator tells you the size of each part; the numerator tells you how many parts you have.

Key equivalence principle: Multiplying or dividing both numerator and denominator by the same non-zero number produces an equivalent fraction.

3/4 = 6/8 = 15/20 = 75/100

All represent the same quantity — three quarters of a whole.

Comparing and Ordering Fractions

Method 1 — Common Denominator

Convert both fractions to the same denominator (using the LCM), then compare numerators.

Compare 5/6 and 7/9: LCM(6,9) = 18. 5/6 = 15/18. 7/9 = 14/18. Since 15 > 14, 5/6 > 7/9.

Method 2 — Cross Multiplication

For a/b vs c/d: compare a×d with b×c.

Compare 7/11 and 5/8: 7×8 = 56 vs 11×5 = 55. Since 56 > 55, 7/11 > 5/8.

This works because a/b > c/d ↔ ad > bc (when b, d > 0).

Method 3 — Benchmarking

Use 0, 1/2, and 1 as benchmarks to quickly place fractions.

3/7 < 1/2 (since 3 < 7/2 = 3.5)
5/9 > 1/2 (since 5 > 9/2 = 4.5)
So 5/9 > 3/7 without calculating a common denominator

Ordering Multiple Fractions — Competition Strategy

To order 5/7, 4/5, 7/9, 11/13 efficiently:

Each fraction is close to 1 — find the "gap to 1":

5/7: gap = 2/7
4/5: gap = 1/5
7/9: gap = 2/9
11/13: gap = 2/13

Smaller gap = closer to 1 = larger fraction. Order gaps: 2/7 > 1/5 > 2/9 > 2/13. So fractions ordered least to greatest: 5/7 < 4/5 < 7/9 < 11/13.

Adding and Subtracting Fractions

Unlike Denominators

Find the LCM of the denominators (the LCD), convert, then add/subtract numerators.

Example: 5/6 + 7/8 LCD(6,8) = 24. 5/6 = 20/24. 7/8 = 21/24. 20/24 + 21/24 = 41/24 = 1 and 17/24.

Mixed Numbers

Method A — Convert to improper fractions, then add:

3 and 2/5 + 2 and 3/4 = 17/5 + 11/4 = 68/20 + 55/20 = 123/20 = 6 and 3/20.

Method B — Add whole and fractional parts separately:

3 and 2/5 + 2 and 3/4: Whole: 3+2=5. Fractions: 2/5 + 3/4 = 8/20 + 15/20 = 23/20 = 1 and 3/20. Total: 5 + 1 and 3/20 = 6 and 3/20 ✓.

Subtraction with Borrowing

5 and 1/6 − 2 and 5/8: Convert: 31/6 − 21/8. LCD=24. 124/24 − 63/24 = 61/24 = 2 and 13/24.

Or: Borrow from the whole number. 5 and 1/6 = 4 and 7/6. Now 4 and 7/6 − 2 and 5/8: LCD=24 → 4 and 28/24 − 2 and 15/24 = 2 and 13/24 ✓.

Multiplying Fractions

Fraction × Fraction

Multiply numerators, multiply denominators, simplify.

Example: 5/6 × 9/10

Before multiplying, cross-cancel: 5 and 10 share factor 5; 9 and 6 share factor 3. = (5÷5)/(6÷3) × (9÷3)/(10÷5) = 1/2 × 3/2 = 3/4.

Cross-cancelling before multiplying keeps numbers small and avoids simplifying large products.

Fraction × Whole Number

3/4 × 16 = (3 × 16)/4 = 48/4 = 12.

Or: 16 ÷ 4 × 3 = 4 × 3 = 12 (divide by denominator first, then multiply by numerator).

Mixed Number × Mixed Number

Convert to improper fractions first:

2 and 2/3 × 1 and 7/8 = 8/3 × 15/8

Cross-cancel: 8s cancel, 3 and 15 share factor 3. = 1/1 × 5/1 = 5.

Dividing Fractions

Dividing by a fraction = multiplying by its reciprocal.

a/b ÷ c/d = a/b × d/c

Why? Because dividing by c/d is the same as asking "how many times does c/d fit into a/b?" which equals a/b × d/c.

Example: 5/6 ÷ 5/9 = 5/6 × 9/5

Cross-cancel: 5s cancel, 9 and 6 share factor 3. = 1/2 × 3/1 = 3/2 = 1 and 1/2.

Mixed Number Division

3 and 1/2 ÷ 1 and 3/4 = 7/2 ÷ 7/4 = 7/2 × 4/7

Cross-cancel: 7s cancel, 4 and 2 share factor 2. = 1/1 × 2/1 = 2.

Complex Fractions

A complex fraction has a fraction in the numerator, denominator, or both.

(3/4) / (9/16) = 3/4 × 16/9 = (3×16)/(4×9) = 48/36 = 4/3 = 1 and 1/3

Cross-cancel: 3 and 9 share factor 3; 16 and 4 share factor 4. = 1/1 × 4/3 = 4/3 ✓.

Fraction of a Fraction (Nested Problems)

"What is 2/3 of 3/4 of 60?"

Work from the inside out:

3/4 of 60 = 45
2/3 of 45 = 30

Or combine: 2/3 × 3/4 × 60 = (2×3×60)/(3×4) = 360/12 = 30.

Fraction Word Problems — Competition Strategies

The Whole is Unknown

"After spending 2/5 of his money and giving away 1/4 of the remainder, Jake had $36 left. How much did he start with?"

Work backwards:

$36 = the money after giving away 1/4 of the remainder → $36 is 3/4 of the remainder
Remainder = 36 × 4/3 = $48
$48 = 3/5 of the original (since 2/5 was spent)
Original = 48 × 5/3 = $80

Bar Model Approach

Draw a bar representing the total. Shade sections to represent each fraction. The unshaded section represents what remains.

Fraction of a Group

"A jar contains red and blue marbles. 3/8 are red. If there are 15 red marbles, how many total?"

15 = 3/8 of total → total = 15 × 8/3 = 40 marbles.

Key Fraction Relationships to Know

Unit fractions: 1/n. Their sum: 1/2 + 1/3 + 1/6 = 1. This decomposition appears in competition problems.

Egyptian fraction decomposition: Any fraction can be written as a sum of distinct unit fractions. 3/4 = 1/2 + 1/4. 2/5 = 1/3 + 1/15.

The mediant: The mediant of a/b and c/d is (a+c)/(b+d). The mediant always lies between two fractions (a useful property in Farey sequences and competition problems).

Fraction and its complement: a/b + (b−a)/b = 1. If you know one part of a whole, the other is immediate.

Competition Corner

Finding the Missing Numerator or Denominator

If n/7 + 2/3 = 5/3, then n/7 = 5/3 − 2/3 = 3/3 = 1 → n = 7.

The "n/(n+1) family"

Fractions of the form n/(n+1) are just below 1. They are easy to compare: The larger n is, the closer to 1 and the larger the fraction. 17/18 > 15/16 > 11/12 > 7/8.

Fraction Sequences

Find the next term: 1/2, 2/3, 3/4, 4/5, ... Pattern: n/(n+1). Next: 5/6.

Simplifying Before Computing

In competition problems, always look for cancellation opportunities before multiplying. Carrying out the full multiplication and then simplifying wastes time and increases error risk.

Key Takeaways

Cross multiplication is the fastest way to compare two fractions
Use the "gap to 1" strategy to order fractions close to 1
Cross-cancel before multiplying fractions to keep numbers small
Dividing by a fraction = multiplying by its reciprocal
For word problems with unknown totals, work backwards using inverse operations
Mixed numbers: convert to improper fractions for multiplication and division; add/subtract whole and fractional parts separately for addition/subtraction
The mediant (a+c)/(b+d) always lies strictly between a/b and c/d
Know key fraction sums: 1/2 + 1/3 + 1/6 = 1; 1/3 + 1/4 + 1/6 + 1/4 = 1