7.2 Fraction Arithmetic & Quantities

5. Adding & Subtracting Fractions and Mixed Numbers

Case 1: Identical Denominators

Add or subtract the numerators directly; leave the denominator untouched:

$$\frac{3}{11} + \frac{5}{11} = \frac{8}{11}, \qquad \frac{9}{14} - \frac{3}{14} = \frac{6}{14} = \frac{3}{7}$$

Case 2: Related denominators, with a general extension

1. Find the LCM of the denominators.
2. Convert each fraction to an equivalent fraction with that common denominator.
3. Add or subtract numerators.
4. Simplify the final fraction or convert to a mixed number if improper.

For a core related-denominator example, $\frac{1}{4}+\frac{3}{8}=\frac{2}{8}+\frac{3}{8}=\frac{5}{8}$. The following denominators are not multiples of each other, so this is a general extension.

Worked Example: Evaluate $\frac{5}{8} + \frac{1}{6}$.

• Denominators are 8 and 6. $\text{LCM}(8, 6) = 24$.
• $\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$
• $\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}$
• Sum: $\frac{15}{24} + \frac{4}{24} = \frac{19}{24}$.

Case 3: Mixed Numbers Addition

When adding mixed numbers, add whole numbers and fractional parts separately:

$$2\frac{3}{5} + 1\frac{1}{2}$$

• Add wholes: $2 + 1 = 3$.
• Add fractions: $\frac{3}{5} + \frac{1}{2} = \frac{6}{10} + \frac{5}{10} = \frac{11}{10} = 1\frac{1}{10}$.
• Combine: $3 + 1\frac{1}{10} = 4\frac{1}{10}$.

Case 4: Mixed Numbers Subtraction (The Regrouping Trap)

A critical hazard occurs when subtracting mixed numbers where the second fraction is larger than the first fraction:

$$4\frac{1}{6} - 1\frac{2}{3}$$

Here, $\frac{2}{3} = \frac{4}{6}$, which is strictly greater than $\frac{1}{6}$. You cannot simply calculate $4 - 1 = 3$ and subtract numerators.

Strategy A: Regrouping (Borrowing from Whole)

1. Express fractions with a common denominator (6): $$4\frac{1}{6} - 1\frac{4}{6}$$
2. Borrow $1$ from the whole number $4$ and convert it into $\frac{6}{6}$: $$4\frac{1}{6} = 3 + 1 + \frac{1}{6} = 3 + \frac{6}{6} + \frac{1}{6} = 3\frac{7}{6}$$
3. Subtract whole numbers and fractions: $$(3 - 1) + \left(\frac{7}{6} - \frac{4}{6}\right) = 2 + \frac{3}{6} = 2\frac{1}{2}$$

Strategy B: Converting to Improper Fractions First

1. Convert both mixed numbers to improper fractions: $$4\frac{1}{6} = \frac{25}{6}, \qquad 1\frac{2}{3} = \frac{5}{3} = \frac{10}{6}$$
2. Subtract directly: $$\frac{25}{6} - \frac{10}{6} = \frac{15}{6}$$
3. Simplify and convert back to a mixed number: $$\frac{15}{6} = \frac{5}{2} = 2\frac{1}{2}$$

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6. Multiplying Fractions

Fraction by an Integer (Whole Number)

To multiply a proper fraction by a whole number, interpret it as repeated addition:

$$5 \times \frac{3}{8} = \frac{5 \times 3}{8} = \frac{15}{8} = 1\frac{7}{8}$$

Key Principle: Only multiply the numerator by the integer. Never multiply the denominator by the integer, as that divides the fraction rather than multiplying it.

Optional extension: fraction by a fraction

Multiply numerators together, and multiply denominators together:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

The Power of Cross-Cancelling

Before multiplying large numerators and denominators, look diagonally to divide out common factors. This prevents dealing with unwieldy products that are difficult to simplify later.

Worked Example: Calculate $\frac{14}{25} \times \frac{15}{21}$.

• Without cross-cancelling: $\frac{14 \times 15}{25 \times 21} = \frac{210}{525}$, which requires complex division.
• With cross-cancelling:
  • Examine diagonal 1: $14$ and $21$ share common factor $7$ ($14 \div 7 = 2$, $21 \div 7 = 3$).
  • Examine diagonal 2: $15$ and $25$ share common factor $5$ ($15 \div 5 = 3$, $25 \div 5 = 5$).
• Rewrite the calculation: $$\frac{2}{5} \times \frac{3}{3} = \frac{2}{5} \times 1 = \frac{2}{5}$$
• Result: $\frac{2}{5}$ obtained in seconds with smaller numbers to check.

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7. Fractions of Quantities & Reverse Problems

Forward Problems: Finding a Fraction of an Amount

• Unit Fraction: To find $\frac{1}{d}$ of an amount $N$, divide by the denominator: $N \div d$.
• Non-Unit Fraction: To find $\frac{n}{d}$ of an amount $N$: $$\text{Value} = (N \div d) \times n$$

Worked Example: A school library has 720 books. If $\frac{5}{8}$ of the books are fiction, how many fiction books are there?

• Step 1: Find 1 unit part ($\frac{1}{8}$ of 720): $$720 \div 8 = 90$$
• Step 2: Multiply by numerator (5 parts): $$90 \times 5 = 450$$
• There are 450 fiction books.

Reverse Fraction Problems (Bar Model Approach)

Reverse problems describe the value of a fractional part and ask for the original whole. In these questions, candidates must divide by the numerator first to find one unit part, then multiply by the denominator.

Reverse Problem: In a running club, $\frac{3}{5}$ of the members are juniors. If there are 36 juniors, how many members are in the club altogether?

Total Club Members (5 equal parts):
┌──────────┬──────────┬──────────┬──────────┬──────────┐
│ 1 part   │ 1 part   │ 1 part   │ 1 part   │ 1 part   │
└──────────┴──────────┴──────────┴──────────┴──────────┘
◄────────── 3 parts = 36 ────────► ◄── 2 parts ──►

• Step 1: Establish what 1 part ($\frac{1}{5}$) equals by dividing the given total by the numerator: $$36 \div 3 = 12$$
• Step 2: Multiply the value of 1 part by the total number of parts (the denominator, 5): $$12 \times 5 = 60$$
• Total members = 60.

Two-Step Remainder Problems

A frequent high-difficulty question format on the Pre-Test introduces fractions of a remaining amount:

Worked Example: Mrs Henderson bakes 120 biscuits for a school fete. In the morning, she sells $\frac{2}{5}$ of the biscuits. In the afternoon, she sells $\frac{3}{4}$ of the remaining biscuits. How many biscuits are left unsold at the end of the day?

• Step 1 (Morning Sale): $$\frac{2}{5} \text{ of } 120 = (120 \div 5) \times 2 = 24 \times 2 = 48 \text{ biscuits sold}$$
• Step 2 (Calculate Remainder): $$120 - 48 = 72 \text{ biscuits remaining}$$
• Step 3 (Afternoon Sale): $$\frac{3}{4} \text{ of remainder } (72) = (72 \div 4) \times 3 = 18 \times 3 = 54 \text{ biscuits sold}$$
• Step 4 (Final Unsold Count): $$72 - 54 = 18 \text{ biscuits left}$$
• (Alternative Method using remaining fractions:) Morning leaves $1 - \frac{2}{5} = \frac{3}{5}$. Afternoon leaves $1 - \frac{3}{4} = \frac{1}{4}$ of the remainder. $\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$ of the whole. $\frac{3}{20} \times 120 = 18$ biscuits.

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8. Summary of Common Exam Traps

Multiply a mixed number by a whole number

For $3 \times 1\frac{2}{5}$, think of three groups, each containing one whole and two fifths. Three wholes give 3; six fifths give $1\frac{1}{5}$. Add them to obtain $4\frac{1}{5}$. An improper-fraction check gives $3\times\frac{7}{5}=\frac{21}{5}=4\frac{1}{5}$. This is repeated addition, so the denominator stays 5; multiplying it by 3 would change the size of each part incorrectly.