Finding fractions and percentages of shapes and quantities, sharing in a given ratio, solving direct proportion problems, and calculating a rate such as miles per hour or cost per unit.

What this dot point is asking

The SQA wants you to find a fraction or percentage of a quantity, share an amount in a given ratio, solve direct proportion problems where one quantity scales with another, and calculate a rate such as speed, cost per unit or texts per month.

Fractions and percentages of a quantity

To find a fraction of an amount, divide by the bottom number (the denominator) and multiply by the top number (the numerator). To find a percentage of an amount on Paper 1, the quick non-calculator route is to build the answer from $10\%$ (divide by $10$) and $1\%$ (divide by $100$).

Sharing in a ratio

A ratio splits an amount into shares. Add the ratio numbers to find the total number of shares, divide the amount by that total to find the value of one share, then multiply each ratio number by it.

Direct proportion and rate

Two quantities are in direct proportion when doubling one doubles the other. The reliable method is the unitary method: find the value of a single unit, then scale up to the amount you need.

A rate measures how one quantity changes with another, such as speed (distance per time), a wage (pounds per hour) or a phone plan (texts per month). Calculate it by dividing one quantity by the other, and always state the units.

Examples in context

Proportion runs through the whole course. Recipes scale by direct proportion, currency converts at a fixed rate, and the best-deal questions in the finance area compare unit costs. A rate such as fuel consumption (miles per gallon) or a heart rate (beats per minute) turns two measurements into a single comparable number, which is exactly the skill the SQA tests. Fractions and percentages also feed the finance area, where VAT, discounts and interest are percentages of an amount, and the statistics area, where a frequency is often expressed as a fraction or percentage of a total. Mastering these here makes those later topics far quicker.

A useful check on any proportion answer is to ask whether it is the right size. If $6$ pens cost $\pounds 4.50$, then $10$ pens must cost more, so an answer below $\pounds 4.50$ is wrong. If a recipe for $4$ people is scaled to $2$, every ingredient should halve. This quick sense-check catches the most common slips before they cost marks.

Try this

Q1. Find $\tfrac{2}{3}$ of $\pounds 45$. [2 marks]

• Cue. $45 \div 3 \times 2 = \pounds 30$.

Q2. Share $\pounds 100$ in the ratio $3 : 1$. [2 marks]

• Cue. $4$ shares, one share $\pounds 25$, so $\pounds 75$ and $\pounds 25$.

Q3. A printer prints $90$ pages in $3$ minutes. Find the rate in pages per minute. [2 marks]

• Cue. $90 \div 3 = 30$ pages per minute.