Year 7 · Unit 2 Percentages · Lesson 3 of 6

How do you find 35% of anything?

Build any percentage from 10%, 5% and 1%, then see why 20% of a larger amount can beat 30% of a smaller one. A scientific calculator checks the awkward ones.

Number and AlgebraScientific calculator

Warm-up

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

  1. (Lesson 1 of this unit) What percentage is 1/4 of a whole?
    Answer25%, because 4 × 25 = 100.
  2. (Lesson 2 of this unit) Write 0.3 as a percentage.
    Answer30%. 0.3 is 3 tenths, which is 30 hundredths.
  3. (Unit 1, lesson 1) What is 360 ÷ 10?
    Answer36. Each digit moves one place to the right.
Words used in this lesson
percentage of
a share, in hundredths, of an amount. For example: 35% of 360 is 126.
amount
the whole quantity that a share is taken from. For example: 360 naira.
benchmark
a value known by heart, such as 10%, to build others from. For example: 10% of 360 is 36.
multiplier
the number you multiply by to find a share. For example: 35% has multiplier 0.35.
estimate
a quick answer that is close to the real one. For example: 30% of 3000 is about 900.
tenth
one of ten equal parts. For example: 10% is a tenth.

Build it from 10%

A cloth seller in Nairobi takes 15% off a 240-shilling cloth. How much comes off? You do not need a rule for 15%. You need three things you can do in your head: 10% is a tenth, 5% is half of that, and 1% is a hundredth. 10% of 240 is 24, 5% is 12, so 15% is 24 + 12 = 36.

Predict first: 10% of 360 is 36. What is 35% of 360? Which blocks would you add? Then slide and see.

360
35
3610% of the amount
126the percentage of it

10% of 360 is 36. 35% is 3 blocks of 36 and one half block of 18: 108 + 18 = 126.

Show me the steps
  1. Amount 360, 10%: 360 ÷ 10 = 36.
  2. Amount 360, 30%: three blocks of 36 make 108.
  3. Amount 360, 35%: 108 plus 5%, which is half of 36, so 108 + 18 = 126.
  4. Amount 360, 1%: 360 ÷ 100 = 3.6.
  5. Amount 200, 47%: 4 blocks of 20 make 80, one 5% is 10 and two 1% slivers are 2 × 2 = 4. 80 + 10 + 4 = 94.
  6. Amount 800, 100%: ten blocks of 80 make the whole 800.

10% is ÷ 10, 5% is half of that and 1% is ÷ 100. Any percentage is built from these: 35% = 10% + 10% + 10% + 5%. Write the pieces, then add.

The same job in one step is p% of A = A × p ÷ 100, or A times the decimal (35% is 0.35). Use the pieces when the numbers are friendly and the decimal on a calculator when they are not.

Order does not matter: p% of A and A% of p are the same, because multiplication commutes. 20% of 30 = 30% of 20 = 6.

A bigger percentage, a bigger amount?

Not always. A percentage is a share of its own whole, so the whole matters as much as the percentage. Set the two bars to different wholes and try to make the smaller percentage win.

80
30
150
20
24first share
30second share

30% of 80 is 24 and 20% of 150 is 30. The second is greater, by 6, even though its percentage is smaller.

Show me the steps
  1. 30% of 80 is 24 and 20% of 150 is 30. The second is greater by 6.
  2. 50% of 40 is 20 and 10% of 400 is 40. The second is greater by 20.
  3. 25% of 120 is 30 and 50% of 60 is 30. They are equal.
  4. 100% of 20 is 20 and 5% of 400 is 20. They are equal: a small whole at 100% matches a large whole at 5%.
  5. 60% of 200 is 120 and 40% of 100 is 40. The first is greater by 80.

A common mistake: “15% of 80 is 15.” The percentage is a share, not an amount. 15% of 80 is 15 hundredths of 80, which is 12. Another slip is taking the percentage away: 20% of 70 is 14, not 70 − 20 = 50.

Worked example: 35% of an amount

A market seller has 360 naira and gives 35% of it to a charity. How much is that?

  1. 10% of 360 is 360 ÷ 10 = 36.
  2. 30% is 3 × 36 = 108.
  3. 5% is half of 36, which is 18.
  4. 35% = 30% + 5% = 108 + 18 = 126 naira.

Your go. The first step is given. Find 45% of 240 rupees.

  1. 10% of 240 is 24.
  2. Now build 45% from blocks of 10% and one 5%, then check.
Check the answer

40% is 4 × 24 = 96, and 5% is half of 24, which is 12. 96 + 12 = 108 rupees.

With a calculator

A scientific calculator finds a percentage of an amount in one go: 28% of 3150 is 3150 × 0.28 = 882. Type 0.28, or type 28 ÷ 100 first, because 28% means 28 hundredths.

Estimate before you press anything. 30% of 3150 is about 3 × 315 = 945, so an answer near 900 is sensible. If the calculator shows 88.2 or 8820, a decimal place has slipped. The estimate catches it; the calculator does not.

Check before you go on

1. Which is a quick way to find 5% of an amount?

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. Find 10% of 450 rupees.

3. Find 25% of 88.

Level 2 · Building

4. Find 15% of 280 pesos.

5. A school has 1250 students and 4% travel by boat. How many students is that?

6. Which is greater: 40% of 60 or 15% of 180?

Level 3 · Stretch

7. A mill grinds 2450 kg of rice and 37% is sold in the village. Use a scientific calculator. How many kilograms is that? Give your answer to one decimal place.

8. A hall has 240 seats. 35% are for guests and the rest are for students. How many seats are for students?

Something to think about

Is 8% of 50 the same as 50% of 8? Work both out, then say whether this is luck or always true.

One way to think about it

8% of 50 = 4, and 50% of 8 = 4. It is always true: p% of A is p × A ÷ 100, and multiplying can be done in either order. So 4% of 150 is 150% of 4, which is 1.5 × 4 = 6.

Exit ticket

9. Rashid says: “To find 20% of 70, I do 70 − 20 = 50.” Which reply is right?