Changing units and scaling up

Measurements come in different units. To compare or combine amounts you must first put them in the same unit.

• Length: 1 m = 100 cm, 1 km = 1000 m
• Mass: 1 kg = 1000 g, 1 tonne = 1000 kg
• Capacity: 1 litre = 1000 ml

Watch out: × when you change to a smaller unit (more of them), ÷ when you change to a bigger unit. 3.5 km = 3.5 × 1000 = 3500 m.

Same units first

Reading a scale

A scale tells you how a drawing or map compares to real life. A scale of 1 : 50 000 means 1 cm on the map = 50 000 cm in real life (which is 500 m).

Tip: the two numbers in a scale are in the same unit: 1 cm : 50 000 cm, or 1 m : 50 000 m.

Your turn — map scale

What a ratio means

A ratio compares parts to each other. Orange squash mixed 1 : 4 means 1 part squash for every 4 parts water.

Common mistake: a ratio is not a fraction of the whole. In 1 : 4, the squash is 1 out of 5 (⅕), not ¼. Add the parts to get the whole.

Ratio vs fraction

Ratios in simplest form

Simplify a ratio like a fraction: divide both sides by the same number (their highest common factor).

You can also write a ratio in the form 1 : n. Divide both parts by the first: 4 : 10 = 1 : 2.5. This is handy for comparing.

Simplest form

Dividing an amount in a ratio

To share an amount in a ratio, use the bar model method:

• Add the parts to find the total number of parts.
• Divide the amount by the total parts to find one part.
• Multiply each share by its number of parts.

Check: the shares must add back to the original amount (£15 + £25 = £40). ✔

Your turn — sharing

Squash mixed 1 to 4: what fraction is squash?

Up 20%, then down 20%: back to £50?

Direct proportion

Two quantities are in direct proportion if, when one doubles, the other doubles too — they go up together at the same rate. The unitary method (find one, then scale) solves these.

Graph clue: direct proportion is a straight line through the origin (0, 0).

Your turn — direct proportion

Inverse proportion

Two quantities are in inverse proportion when one increases as the other decreases, so their product stays the same.

Common mistake: don't treat this like direct proportion. More builders means fewer days, so you divide — you don't scale up.

Direct or inverse?

Direct or inverse proportion?

Tap whether each situation shows direct or inverse proportion.

Percentage of an amount

"Per cent" means "per 100". To find a percentage, turn it into a decimal multiplier and multiply.

Increase & decrease

To increase by a percentage, use a multiplier greater than 1; to decrease, use one less than 1.

• Increase by 20% → × 1.20
• Decrease by 15% → × 0.85
• Two changes in a row → use both multipliers: up 10% then down 10% → × 1.10 × 0.90

Common mistake: percentage change is measured from the original value, not the new one. A price rises £20 → £25: change = £5, so 5 ÷ 20 × 100 = 25% (not 5 ÷ 25).

Your turn — percentage increase

Simple interest

Money in a savings account earns interest. With simple interest, the same amount is added each year, based on the original amount.

With simple interest the yearly amount never changes. (Compound interest, where interest earns interest, comes later at GCSE.)

Your turn — simple interest

Up 20%, then down 20%: back to £50?

Is the bigger box always better value?

Speed, distance & time

Speed is a compound unit — it combines distance and time (e.g. m/s or km/h). Use the formula triangle:

Your turn — speed

More compound units

Density = mass ÷ volume (e.g. g/cm³). Unit pricing compares value for money — find the price for one gram, ml or item.

Tip: to compare "best value", always work out the cost of the same amount (e.g. per 100 g) for each.

Your turn — density

Is the bigger box always better value?

Which strand?

Tap a task, then tap the correct box: is it a Ratio share or a Compound-unit rate?

Best value

The methods to know

You've covered the key ideas in the KS3 Ratio, proportion and rates of change strand. Press Finish to see your score.