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Foundations · Fractions

Fractions: equivalent, simplified, compared

What a fraction means, equivalent fractions, simplifying with the GCD, comparing fractions, and mixed numbers versus improper fractions.

First, the rules

A fraction is a cut and a count

The bottom number says how many equal pieces the whole is cut into; the top number says how many pieces you have. Everything else — equivalence, simplifying, comparing — follows from that picture.

★ Numerator / denominator
Example 1
34 = 3 pieces of a whole cut into 4
A fraction is also a division: 3/4 = 3 ÷ 4 = 0.75.
★ Equivalent fractions
Example 1
23 = 46 = 1015 — multiply (or divide) top and bottom by the same number; the length shaded does not change.
★ Simplest form
Example 1
Divide top and bottom by their GCD: 1824 → ÷ 6 → 34.
★ Comparing
Example 2
Same denominator: compare the tops. Different: make a common denominator, or cross-multiply — 58 vs 23: 5·3 = 15 < 2·8 = 16, so 58 < 23.
★ Mixed ↔ improper
Example 3
235 = 2·5 + 35 = 135
174 = 17 ÷ 4 = 4 r 1 = 414
Fraction of a quantity
Example 3
34 of 60: divide by 4 (15), multiply by 3 (45).

See it

Two bars, one whole

Set two fractions and compare their shaded lengths. The thin bar underneath recuts both on the common denominator — that is how you compare, add and subtract them.

Step by step

Worked examples

1

Simplify and find equivalents

equivalence

(a) Simplify 4256. (b) Write 38 with denominator 40.

Ask yourself: What is the biggest number that divides both 42 and 56? What was 8 multiplied by to get 40?

Show the solution
  1. (a) gcd(42, 56) = 14: 42 ÷ 1456 ÷ 14 = 34.
  2. (b) 8 × 5 = 40, so multiply the top by 5 too: 1540.

Answer 34 · 1540

2

Which is bigger?

comparing

Put in order from smallest: 710, 23, 34.

Ask yourself: Find one denominator that 10, 3 and 4 all divide.

Show the solution
  1. lcm(10, 3, 4) = 60: 4260, 4060, 4560.
  2. Order the tops: 40 < 42 < 45.
  3. So 23 < 710 < 34 (as decimals 0.667, 0.7, 0.75 — same order).

Answer 23 < 710 < 34

3

Mixed numbers and a share

mixed numbers · fraction of

(a) Write 327 as an improper fraction and 296 as a mixed number. (b) Find 58 of 96.

Ask yourself: Whole × denominator + numerator; then division with remainder; then divide by the bottom, multiply by the top.

Show the solution
  1. (a) 327 = 237; 29 ÷ 6 = 4 r 5, so 296 = 456.
  2. (b) 96 ÷ 8 = 12, and 12 × 5 = 60.

Answer 237 · 456 · 60

Practice: fractions

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Simplifying, equivalent fractions, comparing, and mixed ↔ improper conversions. Your best score is kept in this browser only.

About this lesson

Fractions from the ground up: what the numerator and denominator mean, equivalent fractions, simplifying with the greatest common divisor, comparing fractions with a common denominator, and converting between mixed numbers and improper fractions. Fraction bars you can move, worked examples and practice questions. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.