Chapter 8 of 8

All 8 chapters
  1. 1Equal fractions and simplest form, readunderstood
  2. 2Adding and taking away fractions, readunderstood
  3. 3Multiplying and dividing fractions, readunderstood
  4. 4Tenths, hundredths and thousandths, readunderstood
  5. 5Adding and taking away decimals, readunderstood
  6. 6Multiplying and dividing decimals, readunderstood
  7. 7Fractions and decimals, readunderstood
  8. 8Percentages, readunderstood

M3.3.8 Chapter 8

Percentages

What does 35% mean, and what is 15% of 80?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Write 35% as a fraction and as a decimal
  • Find 15% of 80 by finding 10% and 5%
  • Write one amount as a percentage of another

Percent means out of 100. The sign % stands for it. 35% means 35 out of every 100. As a fraction that is 35/100. As a decimal it is 0.35.

Draw it first

Show 35% on a hundred grid

  1. A square grid of 100 small squares. Step 1: the empty grid, one whole. Step 2: 35 squares are shaded, three rows with lines and 5 squares with dots. Step 3: a note under the grid says 35% = 0.35.

    1 A hundred grid is one whole. That is 100%. Each small square is 1%.

  2. A square grid of 100 small squares. Step 1: the empty grid, one whole. Step 2: 35 squares are shaded, three rows with lines and 5 squares with dots. Step 3: a note under the grid says 35% = 0.35.

    2 Shade 35 squares. That is 35%.

  3. A square grid of 100 small squares. Step 1: the empty grid, one whole. Step 2: 35 squares are shaded, three rows with lines and 5 squares with dots. Step 3: a note under the grid says 35% = 0.35.35% = 0.35

    3 35 out of 100 is 35/100, and 35 ÷ 100 = 0.35. So 35% = 0.35.

Worked example

Write 35% as a fraction and as a decimal.

  1. 35% means 35 out of 100. As a fraction that is 35/100.
  2. Divide the top and the bottom by 5. 35 ÷ 5 = 7 and 100 ÷ 5 = 20.
  3. So 35% = 7/20 in simplest form.
  4. For the decimal, divide 35 by 100. 35 ÷ 100 = 0.35.
  5. So 35% = 7/20 = 0.35.

Answer: 7/20 and 0.35

See it three ways

With things

A tank holds 100 litres. Joseph pours in 75 litres. Each square in the picture stands for 1 litre.

As a picture

A square grid of 100 small squares. Seven whole rows are shaded with lines, and 5 more squares with dots. The caption says 75 litres of 100 litres.75 litres of 100 litres

In symbols

75 litres out of 100 litres is 75% = 3/4 = 0.75.

A number line from 0.00 to 1.00 with a tick every 0.05, labelled 0.00, 0.25, 0.50, 0.75 and 1.00. Three points are marked: 25% at 0.25, 50% at 0.5 and 75% at 0.75.0.000.250.500.751.0025%50%75%
25% = 0.25 = 1/4. 50% = 0.5 = 1/2. 75% = 0.75 = 3/4.

To find a percentage of an amount, build it from parts you know. 10% is a tenth, so divide by 10. 1% is a hundredth, so divide by 100. 5% is half of 10%.

Draw it first

Find 15% of 80 kg

  1. A bar for 80 kg, which is 100%, cut into 10 sections of 10% each. Step 1: the whole bar, 100% over 80. Step 2: the first section is shaded, 10% over 8. Step 3: 15% of the bar is shaded, with 15% over 12. The caption says amounts in kg.100%80amounts in kg

    1 Draw a bar for the whole amount, 80 kg. It is 100%. Cut it into 10 equal sections of 10% each.

  2. A bar for 80 kg, which is 100%, cut into 10 sections of 10% each. Step 1: the whole bar, 100% over 80. Step 2: the first section is shaded, 10% over 8. Step 3: 15% of the bar is shaded, with 15% over 12. The caption says amounts in kg.10%8100%80amounts in kg

    2 One section is 10%. 10% of 80 kg is 80 ÷ 10 = 8 kg.

  3. A bar for 80 kg, which is 100%, cut into 10 sections of 10% each. Step 1: the whole bar, 100% over 80. Step 2: the first section is shaded, 10% over 8. Step 3: 15% of the bar is shaded, with 15% over 12. The caption says amounts in kg.15%12100%80amounts in kg

    3 5% is half of a section, which is 4 kg. So 15% is 8 kg + 4 kg = 12 kg.

Worked example

Find 15% of 80 kg.

  1. Find 10% first. 10% is a tenth, so divide 80 by 10. 80 ÷ 10 = 8.
  2. 5% is half of 10%. Half of 8 is 4.
  3. Add them. 15% = 10% + 5% = 8 + 4 = 12.
  4. So 15% of 80 kg is 12 kg.

Answer: 12 kg

Now the other way. To write one amount as a percentage of another, write it as a fraction first. Then change the fraction into hundredths. The bottom number must become 100. When the bottom does not go into 100, divide the part by the whole. Then multiply by 100. For 12 out of 40, 12 ÷ 40 = 0.3, which is 30%.

A bar for 50 marks, which is 100%. 36% of the bar is shaded. The percentage scale above the bar shows 0%, 36% and 100%. The amounts below show 0, 18 and 50. The caption says amounts in marks.0%036%18100%50amounts in marks
18 marks out of 50 marks is 36% of the bar.

Worked example

Mei scores 18 marks out of 50. Write 18 as a percentage of 50.

  1. Write it as a fraction: 18/50.
  2. To get 100 on the bottom, multiply by 100 ÷ 50 = 2.
  3. Multiply the top by 2 as well. 18 × 2 = 36.
  4. Check by dividing. 18 ÷ 50 = 0.36.
  5. So 18/50 = 36/100, which is 36%.

Answer: 36%

Try it

Find 30% of 40 beans

You need: 40 beans or small stones, a table to lay them on.

  1. Count out 40 beans. That is 100%.
  2. Share them into 10 equal heaps. Each heap is 10% of the beans.
  3. Count the beans in one heap.
  4. Take 3 heaps. Count the beans in them.

What you will see: Each heap has 4 beans. 3 heaps have 12 beans. So 30% of 40 beans is 12 beans.

If you cannot try it, this is what happens.
A bar for 40 beans, which is 100%, cut into 10 heaps. 30% of the bar is shaded. The percentage scale shows 0%, 30% and 100%. The amounts below show 0, 12 and 40. The caption says amounts in beans.0%030%12100%40amounts in beans

A wrong idea many people have

To find 5% of an amount, divide it by 5.

Why it feels right: The number in 5% is 5, and dividing by a number seems to find a part.

The test that breaks it: Find 5% of 80. Half of 10% is half of 8, which is 4. Now divide 80 by 5. You get 16, which is 20% of 80.

What is true: 5% is 5 out of 100, which is 1/20. So divide by 20. 80 ÷ 20 = 4.

Key words

percent
out of 100. The sign % stands for it. 35% is 35 out of 100.
percentage
an amount written as a part of 100, such as 36% of the marks.

Back to your guess

The answer is B: 12. 10% of 80 is 8, and 5% is half of that, 4.

15% of 80 is 12. Percent means out of 100. 10% of 80 is a tenth of it, 8. 5% is half of 10%, which is 4. And 8 + 4 = 12.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Write 60% as a fraction in simplest form.

    Tap your answer.

    1. Yes, that is right. 60% is 60 out of 100, which is 60/100. The HCF of 60 and 100 is 20. 60 ÷ 20 = 3 and 100 ÷ 20 = 5.

    2. Not this one. That is the right amount, but it is not in simplest form. 6 and 10 are both in the 2 times table, so divide both by 2. Go to M3.3.8. Try another.

    3. Not this one. 60% is 60 hundredths, not 6 hundredths. Write 60/100 and simplify. Go to M3.3.8. Try another.

    See the answer

    3/5

    60% is 60 out of 100, which is 60/100. The HCF of 60 and 100 is 20. 60 ÷ 20 = 3 and 100 ÷ 20 = 5.

    • If you chose B: That is the right amount, but it is not in simplest form. 6 and 10 are both in the 2 times table, so divide both by 2. Go to M3.3.8.
    • If you chose C: 60% is 60 hundredths, not 6 hundredths. Write 60/100 and simplify. Go to M3.3.8.

  2. Write 8% as a decimal.

    Tap your answer.

    1. Not this one. 0.008 is 8 thousandths, which is 0.8%. Divide 8 by 100, not by 1 000. Go to M3.3.8. Try another.

    2. Not this one. 0.8 is 80%. 8% is 8 hundredths, which is 8 ÷ 100. Go to M3.3.8. Try another.

    3. Yes, that is right. 8% is 8 out of 100. 8 ÷ 100 = 0.08.

    See the answer

    0.08

    8% is 8 out of 100. 8 ÷ 100 = 0.08.

    • If you chose A: 0.008 is 8 thousandths, which is 0.8%. Divide 8 by 100, not by 1 000. Go to M3.3.8.
    • If you chose B: 0.8 is 80%. 8% is 8 hundredths, which is 8 ÷ 100. Go to M3.3.8.

  3. The bar stands for an amount of 60 kg. How much is 25% of it?

    A bar for 60 kg, which is 100%, cut into 4 sections. 25% of the bar is shaded. The percentage scale above the bar shows 0%, 50% and 100%. The amounts below show 0, 30 and 60. The caption says amounts in kg.0%050%30100%60amounts in kg
    Tap your answer.

    1. Not this one. 25 is the percentage, not the amount. 25% of the bar is a quarter of 60 kg. Go to M3.3.8. Try another.

    2. Yes, that is right. 25% is a quarter. 60 ÷ 4 = 15, so 25% of 60 kg is 15 kg. The scale shows that 50% is 30 kg, and half of that is 15 kg.

    3. Not this one. You took 25 away from 60. A percentage of an amount is a part of it. 25% is a quarter, so divide 60 by 4. Go to M3.3.8. Try another.

    See the answer

    15 kg

    25% is a quarter. 60 ÷ 4 = 15, so 25% of 60 kg is 15 kg. The scale shows that 50% is 30 kg, and half of that is 15 kg.

    • If you chose A: 25 is the percentage, not the amount. 25% of the bar is a quarter of 60 kg. Go to M3.3.8.
    • If you chose C: You took 25 away from 60. A percentage of an amount is a part of it. 25% is a quarter, so divide 60 by 4. Go to M3.3.8.

  4. What is 15% of 60?

    Tap your answer.

    1. Yes, that is right. 10% of 60 is 6. 5% is half of that, 3. 6 + 3 = 9.

    2. Not this one. You divided 60 by 15. 15% is 15 out of every 100. So build it: 10% of 60 is 6, and 5% is 3. Go to M3.3.8. Try another.

    3. Not this one. You added 15 to 60. A percentage of an amount is a part of that amount, so the answer is less than 60. Go to M3.3.8. Try another.

    See the answer

    9

    10% of 60 is 6. 5% is half of that, 3. 6 + 3 = 9.

    • If you chose B: You divided 60 by 15. 15% is 15 out of every 100. So build it: 10% of 60 is 6, and 5% is 3. Go to M3.3.8.
    • If you chose C: You added 15 to 60. A percentage of an amount is a part of that amount, so the answer is less than 60. Go to M3.3.8.

  5. Write 12 out of 40 as a percentage.

    Tap your answer.

    1. Not this one. You divided 40 by 12, which gives 3.333... Divide the part by the whole: 12 ÷ 40 = 0.3, which is 30%. Go to M3.3.8. Try another.

    2. Not this one. 12% is 12 out of 100. Here the whole is 40, so write 12/40 and change it into hundredths. Go to M3.3.8. Try another.

    3. Yes, that is right. 12/40 = 0.3, because 12 ÷ 40 = 0.3. 0.3 is 30 hundredths, which is 30%.

    See the answer

    30%

    12/40 = 0.3, because 12 ÷ 40 = 0.3. 0.3 is 30 hundredths, which is 30%.

    • If you chose A: You divided 40 by 12, which gives 3.333... Divide the part by the whole: 12 ÷ 40 = 0.3, which is 30%. Go to M3.3.8.
    • If you chose B: 12% is 12 out of 100. Here the whole is 40, so write 12/40 and change it into hundredths. Go to M3.3.8.

  6. Find 15% of 120 kg.

    See the answer

    18 kg

    10% of 120 is 12. 5% is half of that, 6. 12 + 6 = 18.

  7. Write 21 out of 50 as a percentage.

    See the answer

    42%

    21/50 = 42/100, which is 42%.

  8. From an earlier chapter

    Change 1/8 into a decimal.

    See the answer

    0.125

    1 ÷ 8: 10 ÷ 8 = 1 left 2, 20 ÷ 8 = 2 left 4, 40 ÷ 8 = 5. So 0.125.

Keep it

Copy this into your exercise book.

Percent means out of 100. 35% = 35/100 = 0.35. To find 15%, find 10% by dividing by 10, then 5% as half of that, and add.

Draw: A hundred grid with 35 squares shaded. Write 35% = 0.35 under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (828 KB). It opens on any phone.