Addition, subtraction, multiplication, division of fractions: Worked examples

Before a single symbol: recall that a fraction $\frac{a}{b}$ means "cut a whole into $b$ equal pieces, take $a$ of them." The top number $a$ is the numerator (how many pieces), the bottom $b$ is the denominator (size of each piece). A negative fraction like $- \frac{3}{5}$ means "owe" three of those fifths, or "sit three-fifths to the left of zero" on a number line. Look at the figure below and keep that number-line picture in mind — it saves you on every sign question.

Figure — Addition, subtraction, multiplication, division of fractions

On the line above, $0$ sits in the middle. Positive fractions live to the right; negative fractions live to the left. Adding a positive means stepping right; adding a negative (or subtracting a positive) means stepping left.

The scenario matrix

Every fraction problem lands in one of these cells. The worked examples that follow are each tagged with the cell they cover, so together they fill the whole grid.

Cell	What makes it tricky	Example
A — Add/subtract, both positive	need a common denominator	Ex 1
B — Subtraction giving a negative	order matters, sign of the answer	Ex 2
B2 — Subtracting a negative ( $a - (- b)$ )	two minuses become a plus	Ex 2b
C — Mixed numbers	convert to improper first	Ex 3
D — Multiply, one negative	different signs → negative	Ex 4
D2 — Multiply, two negatives	same signs → positive	Ex 4b
E — Zero as an input	$0$ times / plus / divided	Ex 5
F — Whole number in disguise	write $n = \frac{n}{1}$	Ex 6
G — Division, answer >1 and <1	"how many fit?" both directions	Ex 7
H — Divide by a whole number (limiting)	flip the whole number	Ex 8
I — Real-world word problem	choose the right operation	Ex 9
J — Exam twist (order of operations, decimals)	BODMAS + decimal↔fraction	Ex 10

Two sign rules we'll lean on (from the number line, not memory):

Same signs → positive; different signs → negative for $\times$ and $\div$ .
Subtracting $\frac{c}{d}$ is the same as adding $- \frac{c}{d}$ ; and subtracting a negative is adding a positive.

Cell A — plain addition

Worked example Ex 1 · Compute

$\frac{3}{8} + \frac{5}{12}$ Forecast: guess — will the answer be more or less than $1$ ? (Both parts are less than a half, so under $1$ .)

Find the LCM of $8$ and $12$ : it's $24$ . Why this step? We can only add equal-size pieces. The LCM is the smallest denominator both fit into, so the numbers stay small.
Rewrite using the golden rule (defined above — multiply top and bottom by the same number): $\frac{3}{8} = \frac{3 \cdot 3}{8 \cdot 3} = \frac{9}{24}$ , and $\frac{5}{12} = \frac{5 \cdot 2}{12 \cdot 2} = \frac{10}{24}$ . Why this step? Multiplying top and bottom by the same number never changes the value — it just re-cuts the pizza into equal slices.
Add numerators: $\frac{9 + 10}{24} = \frac{19}{24}$ . Why this step? Same-size pieces now — just count them.

Verify: $\frac{19}{24} \approx 0.79 < 1$ ✓ (matched the forecast). Also $\frac{9}{24} < \frac{10}{24}$ so the $\frac{5}{12}$ part is the bigger chunk — sensible.

Cell B — subtraction that goes negative

Cell B2 — subtracting a negative fraction

Cell C — mixed numbers

Cell D — multiplication with one negative + cancelling

Worked example Ex 4 · Compute

$- \frac{9}{10} \times \frac{5}{6}$ Forecast: different signs → answer is negative. Both factors have size under $1$ -ish, so the size shrinks below $\frac{9}{10}$ .

Sign first: one negative, one positive → the result is negative. Work with sizes, restore the sign at the end. Why this step? Handling the sign separately stops silly errors.
Cancel before multiplying, showing every factor. Write each top and bottom in factored form: $\frac{9}{10} \times \frac{5}{6} = \frac{3 \cdot 3}{2 \cdot 5} \times \frac{5}{2 \cdot 3} .$ The factor $5$ on top cancels the $5$ on the bottom; one factor $3$ on top cancels the $3$ on the bottom: $\frac{3 \cdot 3}{2 \cdot 5} \times \frac{5}{2 \cdot 3} = \frac{3}{2} \times \frac{1}{2} .$ Why this step? Cancelling is the golden rule ( $\frac{ak}{bk} = \frac{a}{b}$ ) applied early — factoring first makes the shared numbers obvious, so we shrink big numbers before multiplying.
Multiply straight across: $\frac{3}{2} \times \frac{1}{2} = \frac{3}{4}$ .
Restore the sign: $- \frac{3}{4}$ .

Verify: without cancelling, $\frac{9 \cdot 5}{10 \cdot 6} = \frac{45}{60} = \frac{3}{4}$ ✓, and the sign is negative → $- \frac{3}{4}$ ✓.

Cell D2 — multiplication with two negatives

Worked example Ex 4b · Compute

$(- \frac{4}{7}) \times (- \frac{7}{8})$ Forecast: same signs (both negative) → the product must be positive. Guess the size: a bit over $\frac{1}{2}$ .

Sign first: negative $\times$ negative $=$ positive. Why this step? Taking the opposite of an opposite lands you back on the positive side — "the enemy of my enemy is my friend." Handle the sign now, then just work with sizes.
Cancel with explicit factoring: $\frac{4}{7} \times \frac{7}{8} = \frac{4}{7} \times \frac{7}{4 \cdot 2}$ . The $7$ on top cancels the $7$ on the bottom, and the $4$ on top cancels the factor $4$ on the bottom: $\frac{4}{7} \times \frac{7}{4 \cdot 2} = \frac{1}{1} \times \frac{1}{2} = \frac{1}{2} .$ Why this step? Golden rule again — cancel shared factors before multiplying.
Attach the sign from step 1: the answer is $+ \frac{1}{2}$ .

Verify: without cancelling, $\frac{4 \cdot 7}{7 \cdot 8} = \frac{28}{56} = \frac{1}{2}$ ✓, and negative $\times$ negative gives $+ \frac{1}{2}$ ✓ — positive, just over "a bit," matching the forecast.

Cell E — zero as an input

Worked example Ex 5 · Compute (a)

$0 \times \frac{7}{9}$ , (b) $\frac{2}{5} + 0$ , (c) $0 \div \frac{2}{5}$ , (d) $\frac{2}{5} \div 0$ Forecast: three of these are easy; one is a trap.

(a) $0 \times \frac{7}{9} = \frac{0 \cdot 7}{1 \cdot 9} = \frac{0}{9} = 0$ . Why? Zero of anything is nothing.
(b) $\frac{2}{5} + 0 = \frac{2}{5}$ . Why? Adding "no extra pieces" changes nothing — $0$ is the additive identity.
(c) $0 \div \frac{2}{5} = 0 \times \frac{5}{2} = \frac{0}{2} = 0$ . Why? "How many $\frac{2}{5}$ -chunks fit inside nothing?" None.
(d) $\frac{2}{5} \div 0$ = undefined. Why? This asks "how many chunks of size $0$ fit into $\frac{2}{5}$ ?" — no number of zeros ever builds up $\frac{2}{5}$ . Division by zero is not allowed.

Verify: parts (a)(b)(c) give $0, \frac{2}{5}, 0$ ; part (d) has no value. Reason: a fraction's denominator can never be zero — every fraction $\frac{a}{b}$ requires $b \neq = 0$ , because you cannot cut a whole into "zero equal pieces." Flipping-and-multiplying would need the reciprocal of $0$ , which doesn't exist, so the operation is genuinely undefined ✓.

Cell F — a whole number in disguise

Cell G — division both directions

Cell H — dividing by a whole number (limiting case)

Cell I — a word problem

Cell J — exam twist: order of operations + decimals

Active Recall

Recall Which cell, and what's the first move?

A) $\frac{5}{7} - \frac{6}{7}$ ::: Cell B (negative subtraction). Same denominator already → $\frac{5 - 6}{7} = - \frac{1}{7}$ . B) $\frac{3}{8} \div 6$ ::: Cell H. Write $6 = \frac{6}{1}$ , flip → $\frac{3}{8} \times \frac{1}{6} = \frac{1}{16}$ . C) $0 \div \frac{4}{9}$ ::: Cell E. Equals $0$ . D) $\frac{4}{9} \div 0$ ::: Cell E trap. Undefined. E) $(- \frac{2}{5}) \times (- \frac{5}{4})$ ::: Cell D2. Same signs → positive; $= \frac{1}{2}$ . F) $\frac{1}{3} - (- \frac{1}{6})$ ::: Cell B2. Two minuses → add; $= \frac{1}{2}$ .

Connections

Addition, subtraction, multiplication, division of fractions — the parent rules
LCM and HCF — smallest common denominator
Reciprocals and multiplicative inverse — the flip in division
Decimals and fractions — used in Ex 10
Rational numbers — where the negatives and zero live
Ratios and proportion — word-problem cousins