Parts of a wholeGrades 5–6
A price falls by 20%, and then by 20% again. Has it fallen by 40%? In these tasks a percent is a hundredth of something, and the whole difficulty is the something: the old price or the new one, all the birds or only the hens. A child who asks “percent of what?” before each step avoids most of the mistakes.
100% = 50 10% = 5 30% = 15
One percent is one hundredth of an amount. 100% is the whole amount, and 50% is half of it.
A percentage is always of something: the old price or the new one, all the guests or only the girls. The same 10% of different amounts are different numbers.
10% is the amount divided by 10. From it, 20%, 30% or 5% are one step away: 30% of 80 is 3 × 8 = 24.
Each percentage is of the amount at that moment. When a price changes twice, the second change is of the new price.
| The step | Taken of | The price after it |
|---|---|---|
| 20% off | 200, so 40 off | 160 |
| 20% off again | 160, so 32 off | 128 |
| both together | 200, so 72 off | 36% off, not 40% |
Before each step, say which amount is 100%. Find 1% or 10% of it, then the percentage the task needs. When the amount changes, start again from the new 100%. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
A school has 300 pupils, and 35% of them go to a sports club. How many pupils do not go to a sports club?
Answer 195 pupils.
Example 2 · Grades 5–6
A bike costs 400 coins. In spring its price goes up by 25%, and in autumn the new price goes down by 20%. What does the bike cost now?
400 → 500 → 400 +25% −20%
Answer 400 coins.
Example 3 · Grades 5–6
A farmer has 200 kg of watermelons that are 99% water. A week later they have dried a little and are 98% water. How much do they weigh now?
Answer 100 kg.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The percentage is taken of the wrong amount: the autumn's 20% of the old price instead of the new one, or the 98% of the watermelons' old weight.
What to say
“Which amount is 100% right now?”
A part is taken for the whole, or the whole for a part: the 2 kg that are not water are given as the weight of the watermelons, or the whole school is counted where the question asks about a part of it.
What to say
“Is this number a part, or the whole?”
A number found on the way becomes the answer: the 105 pupils who go to a club when the question asks about those who do not, or the 100 coins the price changed by instead of the price.
What to say
“What exactly does the question ask for?”
In a shop, find a price with a discount and work it out together: what is 20% off 150? Start from 10%.
Practise finding 10% of anything: of 80, of 350, of 1200. From 10%, most other percentages are one step away.
When you hear a percentage on the news or in an advert, ask together: percent of what? Often nobody says.
