Fraction Equivalence & Comparison

4.F.1Equivalent fractions01/04

Equivalent fractions.

Step01of 04

Two fractions are equivalent if they represent the SAME amount, even though the numerator and denominator are different.

\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{50}{100}

Multiply (or divide) BOTH the numerator AND denominator by the same number to get an equivalent fraction.

4.F.1 — Equivalent fractions.

Step 01 of 04

Two fractions are equivalent if they represent the SAME amount, even though the numerator and denominator are different.

\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{50}{100}

Multiply (or divide) BOTH the numerator AND denominator by the same number to get an equivalent fraction.

Step 02 of 04

Generate equivalent fractions. Start with $3/4$ :

× 2 top and bottom	→6/8
× 3	→9/12
× 5	→15/20
× 10	→30/40

All equal to $3/4$ . Same point on a number line, same area shaded.

Step 03 of 04

Reduce / simplify by dividing top and bottom by a common factor.

\frac{18}{24} \div \frac{6}{6} = \frac{3}{4}

"Simplest form" or "lowest terms" means the numerator and denominator have no common factors other than 1.

Step 04 of 04

Visual model. Imagine cutting a pizza into 2 pieces vs. 4 pieces. Half the pizza = 1/2 (one of two big pieces) = 2/4 (two of four smaller pieces). Same amount of pizza, different number of slices.

Key insight

Multiply (or divide) numerator AND denominator by the same nonzero number — same fraction, different form. Reduce to simplest form by dividing by the greatest common factor.

4.F.2 — Compare fractions with unlike denominators.

Step 01 of 04

Comparing fractions is easy when denominators MATCH — just compare numerators. The challenge is when denominators are DIFFERENT.

Three strategies: common denominator, common numerator, or compare to $1/2$ as a benchmark.

Step 02 of 04

Common denominator. Convert both fractions to have the same denominator (the LCD), then compare numerators.

Compare $2/3$ and $3/5$ . LCD = 15.

\frac{2}{3} = \frac{10}{15}, \frac{3}{5} = \frac{9}{15}

$10 > 9$ , so $2/3 > 3/5$ .

Step 03 of 04

Benchmark fractions. Use easy reference points like 0, $1/2$ , and 1 to compare quickly.

$5/8$ vs. $3/7$ : 5/8 is a bit more than 1/2 (since 4/8 = 1/2). 3/7 is a bit less than 1/2 (since 3.5/7 = 1/2). So 5/8 > 3/7 — no calculation needed.

Step 04 of 04

Common numerator. If two fractions have the SAME numerator, the one with the SMALLER denominator is bigger (because it's divided into fewer pieces).

\frac{3}{5} > \frac{3}{8} (3 of 5 pieces > 3 of 8 smaller pieces)

Key insight

Common denominator = compare numerators. Common numerator = SMALLER denominator wins. Benchmarks (0, 1/2, 1) often let you decide without full conversion.

4.F.3 — Order a set of fractions.

Step 01 of 04

Order three or more fractions by converting them all to a common denominator. Then sort by numerator.

Step 02 of 04

Worked example. Order from least to greatest: $5/6, 3/4, 11/12, 7/8$ .

LCD of 6, 4, 12, 8 is 24.

5/6 = 20/24

3/4 = 18/24

11/12 = 22/24

7/8 = 21/24

Step 03 of 04

Sort numerators.18 < 20 < 21 < 22, so:

\frac{3}{4} < \frac{5}{6} < \frac{7}{8} < \frac{11}{12}

Step 04 of 04

Mixed numbers. If you have mixed numbers in the set, convert each to an improper fraction (or compare whole-number parts first).

$1 \frac{2}{3}$ vs. $1 \frac{3}{5}$ : same whole number 1, so compare the fractions. Common denominator 15: $2/3 = 10/15$ vs. $3/5 = 9/15$ . So $1 \frac{2}{3} > 1 \frac{3}{5}$ .

Key insight

Common denominator for everyone, then sort by numerator. For mixed numbers, compare whole-number parts first; only fall back to the fractions if the wholes are equal.

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