Equivalent fractions, simplifying fractions: Worked examples

The scenario matrix

Every simplifying/equivalence task falls into one of these cells. We will hit each one.

#	Case class	What makes it special	Example we'll do
C1	Ordinary simplify	Share a common factor $> 1$	$\frac{18}{24} \to \frac{3}{4}$
C2	Big / prime-factor simplify	Numbers too big to guess GCD	$\frac{84}{126}$
C3	Already lowest terms	GCD is already $1$ — nothing to do	$\frac{8}{15}$
C4	Zero numerator	$a = 0$ — a degenerate value	$\frac{0}{7}$
C5	Zero denominator	$b = 0$ — undefined, not a fraction	$\frac{5}{0}$
C6	Build an equivalent	Grow, don't shrink, to a target denominator	$\frac{3}{7} \to \frac{?}{28}$
C7	Negative fraction	A minus sign — which sign, and where?	$\frac{- 12}{18}$
C8	Equal-or-not test	Two fractions, no simplifying needed	$\frac{6}{9}$ vs $\frac{10}{15}$
C9	Word problem	Translate real life into a fraction, then simplify	ribbon lengths
C10	Exam twist	Cancel-across-plus trap; solve for a missing part	$\frac{2 + 3}{2 + 5}$ and a proportion

Cells C1–C3 = shrinking. C4–C5 = degenerate inputs. C6 = growing. C7 = signs. C8 = comparison. C9–C10 = real world & traps.

The one tool we'll lean on: prime factorisation

Figure — Equivalent fractions, simplifying fractions

Look at the board above: $84$ and $126$ each become a tower of prime bricks. The bricks they both have (the pale-yellow highlighted $2 \times 3 \times 7$ ) are exactly their GCD.

C2 · Big numbers — prime factorisation earns its keep

C4 · Zero on top (degenerate but legal)

C5 · Zero on the bottom (illegal)

C6 · Build an equivalent fraction (growing)

C7 · Negative fractions — where does the sign live?

Worked example Simplify

$\frac{- 12}{18}$ . Forecast: guess the sign and the size of the answer.

Split off the sign. $\frac{- 12}{18} = - \frac{12}{18}$ . Why? A minus on the numerator makes the whole fraction negative — the sign is a property of the value, not of one number. So handle size and sign separately.
Simplify the size. $g cd (12, 18) = 6$ (since $12 = 2^{2} \times 3$ , $18 = 2 \times 3^{2}$ , shared $= 2 \times 3 = 6$ ). $\frac{12 \div 6}{18 \div 6} = \frac{2}{3}$ . Why? Same rule as always — the minus doesn't affect which factors are shared.
Reattach the sign: $- \frac{2}{3}$ .

Verify: $- 12 \times 3 = - 36$ and $18 \times (- 2) = - 36$ — equal ✓. Note $\frac{- 12}{18} = \frac{12}{- 18} = - \frac{12}{18}$ : the three forms all name the same negative number.

C8 · Equal or not? (comparison, no simplifying required)

C9 · Word problem — translate, then simplify

C10 · Exam twists — the trap and the missing part

Worked example (a) A student writes

$\frac{2 + 3}{2 + 5} = \frac{3}{5}$ . Is this valid? (b) Solve $\frac{3}{5} = \frac{x}{20}$ . Forecast (a): does cancelling the $2$ 's work? (b): what is $x$ ?

Part (a) — the cancel-across-plus trap.

Test the claim numerically. $\frac{2 + 3}{2 + 5} = \frac{5}{7}$ , but the student claims $\frac{3}{5}$ . Why test first? Numbers settle the argument instantly.
Compare: $\frac{5}{7} \approx 0.714$ but $\frac{3}{5} = 0.6$ . Not equal — the cancel is illegal. Why it's illegal: you may only cancel a factor that multiplies the whole top and whole bottom. In $2 + 3$ , the $2$ is a term (added), not a factor (multiplied). See the parent note's mistake box.

Part (b) — solve for the missing numerator.

Find the bottom multiplier: $5 \times k = 20 \Rightarrow k = 4$ . Why? Same as C6 — the denominators tell us how the pieces were re-cut.
Apply to the top: $x = 3 \times 4 = 12$ .

Verify: (a) $\frac{2 + 3}{2 + 5} = \frac{5}{7} \neq = \frac{3}{5}$ ✓ (trap exposed). (b) cross-multiply $3 \times 20 = 60$ , $5 \times 12 = 60$ ✓.

Recall Quick self-test across the whole matrix

Simplify $\frac{84}{126}$ . ::: $\frac{2}{3}$ (divide by $g cd = 42$ ). What is $\frac{0}{7}$ ? ::: $0$ — zero pieces of a whole. What is $\frac{5}{0}$ ? ::: Undefined — nothing satisfies $0 \cdot x = 5$ . Write $\frac{3}{7}$ with denominator $28$ . ::: $\frac{12}{28}$ (multiply top & bottom by $4$ ). Simplify $\frac{- 12}{18}$ . ::: $- \frac{2}{3}$ . Is $\frac{2 + 3}{2 + 5} = \frac{3}{5}$ ? ::: No — $\frac{5}{7} \neq = \frac{3}{5}$ ; you can't cancel across a plus. Fraction of a $75$ cm ribbon that a $45$ cm cut is? ::: $\frac{3}{5}$ .

Connections

Prime factorisation — the brick-tower method used in C2, C7, C9.
Factors, multiples and GCD/HCF — the GCD is the product of shared bricks.
Ratios and proportion — cross-multiplication (C8, C10b) is the same tool.
Decimals and percentages — sanity-checks in C9 ( $\frac{3}{5} = 0.6 = 60%$ ).
Adding and subtracting fractions — building equivalents (C6) is the first step there.
Number Systems – rational numbers — every case here outputs a rational in lowest terms.

Worked examples — Equivalent fractions, simplifying fractions