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Calculating with Fractions and Mixed Numbers

A baking recipe often requires scaling up or cutting down, forcing you to mix halves, quarters, and whole units smoothly.

To add or subtract fractions, you must find a common denominator using the Lowest Common Multiple (LCM) of the denominators.
For all operations involving mixed numbers, AQA examiners recommend converting them to improper fractions first. Multiplying the whole number parts separately is a common mistake and will NOT work.
The general rule for multiplying fractions is:

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Step 4: Add the numerators: $\frac{35 + 21}{15} = \frac{56}{15}$
Step 5: Final Answer: $3\,\frac{11}{15}$

Worked Example: Subtraction

Calculate $5\,\frac{1}{3} - 2\,\frac{6}{7}$

Step 1: Convert to improper fractions: $\frac{16}{3} - \frac{20}{7}$
Step 2: Find a common denominator: The LCM of 3 and 7 is 21.

Worked Example: Multiplication

Calculate $1\,\frac{1}{5} \times \frac{7}{12}$

Step 1: Convert to improper fraction: $\frac{6}{5} \times \frac{7}{12}$
Step 2: Cross-simplify: The 6 and 12 share a factor of 6, simplifying the calculation to

Worked Example: Division

Calculate $2\,\frac{2}{5} \div 1\,\frac{1}{2}$

Step 1: Convert to improper fractions: $\frac{12}{5} \div \frac{3}{2}$
Step 2: Apply the KFC rule: Keep the first fraction (), Flip the second to find its (), and Change to multiplication ().

Formal Operations with Decimals

Why do a £1.50 coffee and a £0.25 snack cost £1.75 instead of £1.525? Because place value dictates that tenths can only be added to tenths, and hundredths to hundredths.

For addition and subtraction, ensure strict decimal alignment so that digits of the same place value share the same column.
Add zero placeholders to numbers with fewer decimal places to prevent alignment errors.
When dividing by a decimal, transform the divisor into an integer by multiplying both values by the same power of 10. Do NOT write division remainders using an 'r' at GCSE level; always calculate to a full decimal or fraction.

Worked Example: Decimal Subtraction

Calculate $16 - 9.4$

Step 1: Align and add placeholders: $16.0 - 9.4$
Step 2: Tenths column: $0 - 4$ cannot be done. Use exchanging to move $1$ from the $6$ (units) to make it $5$ . The becomes .

Operations with Directed Numbers

Understanding how to calculate across zero is essential for navigating real-world contexts, such as when bank balances plunge from credit into overdraft.

Directed numbers require us to track both magnitude and positive/negative signs.
Separate the calculation from the sign logic. Perform formal written methods on the absolute values, then apply the correct sign to the result. Do NOT include negative signs inside your formal column carrying or exchanging steps.
For mixed signs in addition/subtraction, find the difference. For numbers with the same sign (e.g., $-15.2 - 4.3$ ), use column addition on the magnitudes ( $15.2 + 4.3 = 19.5$ ) and keep the original sign ( $-19.5$ ).
For multiplication and division: Same signs result in a positive answer; different signs result in a negative answer.

Worked Example: Formal Subtraction with Directed Integers

Calculate $-673 + 289$

Step 1: Logic: The signs are different, so we find the difference: $-(673 - 289)$ . The larger absolute magnitude is 673, so the final answer will be negative.
Step 2: Units column: $3 - 9$ cannot be done. Use exchanging (or regrouping) to take 1 from the tens. $7$ becomes $6$ ; becomes . .

Worked Example: Long Multiplication with Directed Decimals

Calculate $-8.6 \times 9.4$

Step 1: Sign rule: Different signs mean the answer is negative.
Step 2: Formal integer method: Remove decimals to calculate $86 \times 94$ .
Step 3: Calculate rows: $86 \times 4 = 344$ . Place a zero placeholder for the tens row: $86 \times 90 = 7740$ .

Worked Example: Bus Stop Division with Directed Decimals

Calculate $17.28 \div -3.2$

Step 1: Sign rule: Different signs mean the answer is negative.
Step 2: Transform divisor: Multiply both values by 10 to make the divisor an integer: $172.8 \div 32$ .
Step 3: Calculate using bus stop division: 32 into 172 goes 5 times ( $32 \times 5 = 160$ ), with a remainder of 12.
Step 4: Decimal and remainder: Place the decimal point in the quotient. 32 into 128 goes 4 times ( $32 \times 4 = 128$ ), with no remainder. Magnitude is .

Students often add denominators together when adding fractions (e.g., $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$ ) — this is mathematically incorrect and will score zero marks for your method.

When working with directed decimals, students frequently include negative signs inside their column addition or subtraction steps, which leads to calculation errors; always calculate the magnitude separately from the sign.

In AQA exams, any final answer involving currency must be formatted to exactly two decimal places (e.g., write £1.40 instead of £1.4).

Method marks (M1) are frequently awarded for explicitly showing the conversion of mixed numbers to improper fractions, even if you make an arithmetic error later in the calculation.

Use 1 significant figure rounding to quickly estimate and check if your final answer is mathematically sensible (e.g., $32.87 + 15.12 \approx 30 + 20 = 50$ ).

When squaring negative numbers on your calculator, you must use brackets (e.g., $(-3)^2$ ); entering $-3^2$ without brackets will incorrectly result in $-9$ .

A shared multiple of denominators allowing for the direct addition or subtraction of fractions.

A numerical value consisting of a whole number combined with a proper fraction.

A fraction where the numerator is greater than or equal to the denominator.

The result of dividing 1 by a given number, also known as the multiplicative inverse.

The process of writing numbers vertically with their decimal points perfectly lined up to ensure place values match.

The numerical value that a digit holds by virtue of its specific position within a number.

Zeros added to a number to maintain proper alignment in decimal columns or to hold place value during multiplication.

The process of moving value between place-value columns during formal subtraction to allow a calculation to proceed.

Numbers that have both magnitude and direction, indicating whether they are positive or negative.

Standardised algorithmic layouts, such as column, grid, or bus stop, used to solve mathematical calculations step-by-step.

A formal written method where numbers are aligned vertically by their place value and added column by column.

Another term for exchanging; moving a '10' or '100' to the column to the right to enable subtraction.

A common formal written method used to perform short division.

The final numerical result obtained by dividing one quantity by another.