Year 7 · Unit 2 Percentages · Lesson 4 of 6

If 40% is 64, what is 100%?

A part and its percentage can tell you the whole. Work back through 1%, then ask the other question: what percentage is one amount of another?

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 3 of this unit) Find 10% of 360 and then 30% of 360.
    Answer36 and 108. 10% is a tenth, and 30% is three of them.
  2. (Lesson 2 of this unit) Write 0.45 as a percentage.
    Answer45%. 0.45 is 45 hundredths.
  3. (Unit 1, lesson 7) Estimate 19 × 21 by rounding to the nearest ten.
    Answer20 × 20 = 400.
Words used in this lesson
whole
the full amount; 100%. For example: the whole book, 120 pages.
part
a share of the whole. For example: 48 pages out of 120.
work back
undo a calculation to find the start. For example: from 30% being 45 back to 100%.
1%
one hundredth of the whole. For example: 1% of 120 is 1.2.
inverse
the operation that undoes another. For example: dividing undoes multiplying.
check
test an answer by working in the other direction. For example: 40% of 160 is 64.

Going back to the whole

A farmers' co-op in Ghana sold 45 kg of mangoes. That was 30% of everything it picked. How much did it pick? The percentage tells you how much of the whole the 45 kg is, so the whole must be bigger than 45. Work back: if 30% is 45, then 10% is 15, and 100% is ten times that, 150 kg.

Predict first: the shaded part is 64 and it is 40% of the whole. Is the whole more or less than 100? Why? Then set the percentage to 40, slide until the part reads 64, work out the whole, and press Show the whole.

hidden
40
64the part you are given
1.61% of the whole
not shown yetthe whole, 100%

40% of the whole is 64. So 1% is 64 ÷ 40 = 1.6. Multiply by 100 to find the whole, then press Show the whole to check.

Show me the steps
  1. 10% of the whole is 36, so 1% is 36 ÷ 10 = 3.6 and the whole is 360.
  2. 20% of the whole is 32, so 1% is 1.6 and the whole is 160.
  3. 25% of the whole is 50, so 1% is 2 and the whole is 200.
  4. 40% of the whole is 64, so 1% is 1.6 and the whole is 160.
  5. 75% of the whole is 60, so 1% is 0.8 and the whole is 80.
  6. 100% of the whole is 240: the part is the whole, and 1% is 2.4.

To find the whole, go back through 1%. Divide the part by the percentage to get 1%, then multiply by 100. If 40% is 64, then 1% is 64 ÷ 40 = 1.6 and 100% is 160.

To find what percentage one amount is of another, divide the part by the whole, then multiply by 100: 18 out of 40 is 18 ÷ 40 = 0.45 = 45%.

Check by working forward. 40% of 160 is 64, as given. The whole is bigger than the part when the percentage is less than 100%.

What percentage is one amount of another?

This is the other direction. You know the part and the whole, and you want the percentage. Move both sliders. Make the part larger than the whole and watch where the bar ends.

40
18
0.45part ÷ whole
45%as a percentage

18 out of 40: 18 ÷ 40 = 0.45, which is 45%.

Show me the steps
  1. 18 out of 40: 18 ÷ 40 = 0.45, so 45%.
  2. 30 out of 40: 0.75, so 75%.
  3. 8 out of 80: 0.1, so 10%.
  4. 7 out of 25: 0.28, so 28%.
  5. 5 out of 40: 0.125, so 12.5%.
  6. 30 out of 20: 1.5, so 150%: the part is bigger than the whole.

A common mistake: working forward when the question works back. “30% of the whole is 45” does not mean 30% of 45. That would give 13.5, a number smaller than the part, which cannot be the whole. When the percentage is less than 100%, the whole must be larger than the part.

Worked example: the whole from a part

40% of the water in a tank is 36 litres. How much does the tank hold?

  1. 40% is 36 litres, so 10% is 36 ÷ 4 = 9 litres.
  2. 100% is ten times 10%: 9 × 10 = 90 litres.
  3. Check: 40% of 90 = 4 × 9 = 36.

Your go. The first step is given. 15% of the students in a school take the bus, and that is 21 students. How many students are there?

  1. 15% is 21, so 5% is 21 ÷ 3 = 7.
  2. Now find 100% from 5% yourself, then check.
Check the answer

100% is 20 lots of 5%: 20 × 7 = 140 students. Check: 10% of 140 is 14 and 5% is 7, so 15% is 21.

Check before you go on

1. 20% of a number is 30. What is the number?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 10% of a number is 14. What is the number?

3. 50% of the pages of a book is 85 pages. How many pages does the book have?

Level 2 · Building

4. 25% of a school's students are in Year 7, and that is 60 students. How many students are in the school?

5. What percentage is 18 of 72?

6. Which calculation gives 15 as a percentage of 60?

Level 3 · Stretch

7. A town grew to 130% of its old size and now has 6500 people. How many people did it have before?

8. Priya has read 48 pages, which is 40% of her book. How many pages does she still have to read?

Something to think about

20% of a number A is 8, and 8 is also 40% of a number B. Find A and B. A is what percentage of B?

One way to think about it

20% of A is 8, so 10% is 4 and A = 40. 40% of B is 8, so 10% is 2 and B = 20. A is 40 out of 20, which is 200% of B: twice as big. A percentage can be over 100%.

Exit ticket

9. 40% of the guests are children and there are 18 children. Nneka works out 18 × 40 ÷ 100 = 7.2 guests. What is wrong?