1. Ratios, units and successive percentages
By the end
- Identify the base of a percentage.
- Convert units before comparing rates.
A percentage describes a fraction of a named base. A 20% decrease multiplies that base by 0.80; a 10% increase multiplies the current value by 1.10. Successive changes usually have different bases, so adding their percentages can give the wrong result.
For a rate, write the units as part of the fraction. If 84 labels are printed in 6 minutes, the rate is 84 labels / 6 minutes = 14 labels per minute. Multiplying by 2.5 minutes cancels minutes and leaves 35 labels.
Before calculating, predict the direction. A discount followed by a smaller percentage increase may still leave a lower final value. After calculating, compare with the original base and state whether you found an amount or a percentage change.
Worked example
A workshop fee of $80 is discounted by 20%, then a 10% service charge is applied to the discounted fee. Find the final fee and the net percentage change.
- After discount: 80 × (1 − 0.20) = 64 dollars.
- After service charge: 64 × 1.10 = 70.40 dollars.
- Relative change: (70.40 − 80) / 80 = −0.12.
The final fee is $70.40: a 12% decrease, not a 10% decrease.
Try it yourself
A 50-litre tank is filled to 60% of its capacity. Then 20% of the water currently in the tank is removed. How many litres remain, and what percentage of capacity is that?
- Use 30 litres as the base of the removal.
- Report both litres and percentage of the original 50-litre capacity.
Reveal the practice solution
Start with 50 × 0.60 = 30 L. Remove 30 × 0.20 = 6 L, leaving 24 L. Then 24/50 = 0.48, so the tank is 48% full.
Watch for this mistake: A 20% decrease and a 20% increase do not cancel: 0.8 × 1.2 = 0.96. The second change acts on a smaller base.
Exam topic alignment
Official category names are shown in English. This lesson covers only the skills named below.
SAT: Problem-Solving and Data Analysis: ratios, rates, units, percentages
ACT: Integrating Essential Skills: rates, percentages and proportional relationships
Sources and review date
- College Board: SAT Problem-Solving and Data Analysis
Official topic alignment for rates, percentages, center and conditional probability. Checked: .
- ACT: Mathematics Test Description
Official reporting-category names and broad skill alignment. No score conversions or exam questions are reproduced. Checked: .
Explanations, examples and quiz questions are original KitForma material. These links support technical facts and curriculum alignment.