Chapter 1 of 7

All 7 chapters
  1. 1Percentage increase and decrease, readunderstood
  2. 2Finding the original amount, readunderstood
  3. 3Simple and compound interest, readunderstood
  4. 4Growth and decay, readunderstood
  5. 5Harder ratio problems, readunderstood
  6. 6Proportion as a formula, readunderstood
  7. 7Graphs of proportion, readunderstood

M4.2.1 Chapter 1

Percentage increase and decrease

How do I add 12% to 250 in one step?

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

  • Increase 250 by 12% using the multiplier 1.12
  • Find the percentage change from 80 to 92

A packet holds 250 seeds. A new packet holds 12% more. How many seeds is that?

Percent means out of 100. So 12% is 12/100, which is 0.12. 12% of 250 is 0.12 × 250 = 30. Add that to the start: 250 + 30 = 280.

That took two steps. There is a way to do it in one.

Draw it first

Add the rise to the start

  1. A bar labelled Start: 250 seeds, with the label 100%. A second bar labelled After the rise: 280 seeds, with the label 112%, is longer by a dotted piece. A box 250 joins to a box 280 by an arrow labelled × 1.12.Start: 250 seeds100%

    1 The start is 100% of the seeds. That bar is 250 seeds.

  2. A bar labelled Start: 250 seeds, with the label 100%. A second bar labelled After the rise: 280 seeds, with the label 112%, is longer by a dotted piece. A box 250 joins to a box 280 by an arrow labelled × 1.12.Start: 250 seeds100%After the rise: 280 seeds112%

    2 The rise adds 12% more. The new bar is 112% of the start, and the extra piece is dotted.

  3. A bar labelled Start: 250 seeds, with the label 100%. A second bar labelled After the rise: 280 seeds, with the label 112%, is longer by a dotted piece. A box 250 joins to a box 280 by an arrow labelled × 1.12.Start: 250 seeds100%After the rise: 280 seeds112%250× 1.12280

    3 112% is 1.12 as a decimal. One multiplication takes 250 to 280.

The start is 100%. A rise of 12% makes 100% + 12% = 112%. As a decimal, 112% is 1.12. The number 1.12 is the multiplier. Multiply by it, and the rise is done.

Worked example

Increase 250 by 12% using a multiplier.

  1. A rise of 12% makes 100% + 12% = 112% of the start.
  2. 112% as a decimal is 1.12. This is the multiplier.
  3. Multiply the start by the multiplier: 250 × 1.12 = 280.

Answer: 250 × 1.12 = 280

A fall works the same way. A fall of 35% leaves 100% − 35% = 65% of the start. As a decimal that is 0.65. The multiplier for a fall is below 1.

A bar labelled Start: 240 seeds, with the label 100%. A shorter bar labelled After the fall: 156 seeds, with the label 65%. The 35% that is missing is a dashed outline.Start: 240 seeds100%After the fall: 156 seeds65%
A fall of 35% leaves 65% of the start. The missing 35% is the dashed part.

Worked example

Decrease 240 by 35%.

  1. A fall of 35% leaves 100% − 35% = 65% of the start.
  2. 65% as a decimal is 0.65. This is the multiplier.
  3. Multiply: 240 × 0.65 = 156.

Answer: 240 × 0.65 = 156

A table with the head Change and Multiplier. Rows: Rise of 5%, 1.05; Rise of 12%, 1.12; Fall of 15%, 0.85; Fall of 40%, 0.6.ChangeMultiplierRise of 5%1.05Rise of 12%1.12Fall of 15%0.85Fall of 40%0.6
A rise adds to 1. A fall takes away from 1.

Sometimes you are given the old and the new amount and asked how much it changed. A percentage change says the change as a percent of the amount you started with.

Work out the change: new minus original. Divide it by the original. Multiply by 100. As a formula: percentage change = (new − original) ÷ original × 100.

A bar labelled Original: 80 seeds, with the label 100%. A longer bar labelled New: 92 seeds, with the label 115%. The extra dotted piece is 12 seeds.Original: 80 seeds100%New: 92 seeds115%
The dotted piece is the change, 12 seeds. It is measured against the original, which is 100%.

Worked example

Find the percentage change from 80 to 92.

  1. The change is new minus original: 92 − 80 = 12.
  2. Divide the change by the original: 12 ÷ 80 = 0.15.
  3. Multiply by 100: 0.15 × 100 = 15.
  4. The amount rose by 15%.

Answer: A rise of 15%

See it three ways

With things

Take 80 stones. Put them in 10 piles of 8. Add 12 more stones. That is one more pile and half a pile.

As a picture

A bar labelled 80 stones in 10 piles of 8, with the label 100%, cut into 10 equal parts. A second bar labelled 12 stones added, with the label 15%, covers one and a half of the parts.80 stones in 10 piles of 8100%12 stones added15%

In symbols

12 ÷ 80 × 100 = 15%.

Try it

Cut a strip

You need: paper, a ruler, a pencil.

  1. Draw a strip 10 cm long. Let the whole strip stand for 100%. Mark every 1 cm.
  2. Shade the first 6.5 cm from the left. Leave the last 3.5 cm white.

What you will see: The shaded part is 65% of the strip. A fall of 35% leaves 65%, so the multiplier is 0.65.

If you cannot try it, this is what happens.
A bar labelled Whole strip: 10 cm, with the label 100%, cut into 10 equal parts. A second bar labelled Shaded: 6.5 cm, with the label 65%, is hatched and covers six and a half of the parts.Whole strip: 10 cm100%Shaded: 6.5 cm65%

A wrong idea many people have

To increase 250 by 12%, add 12: 250 + 12 = 262.

Why it feels right: The words say "12" and "add", so it feels natural to add the 12 you can see.

The test that breaks it: Work out 12% of 250. It is 0.12 × 250 = 30, not 12. The rise is 30 seeds, so 262 is too low.

What is true: The percent is a fraction of the start, not a number to add. Use the multiplier: 250 × 1.12 = 280.

A wrong idea many people have

The amount falls from 125 to 100, so the fall is 25 out of 100, which is 25%.

Why it feels right: The new amount is the one you are looking at, and 100 makes the sum look like a percent already.

The test that breaks it: A fall of 25% would leave 125 × 0.75 = 93.75, not 100. Check by multiplying.

What is true: Divide by the original amount, 125. The change 25 ÷ 125 = 0.2, so the fall is 20%.

Key words

percent
out of 100. So 12% is 12 out of 100, or 0.12.
multiplier
the number you multiply by to change an amount, such as 1.12 for a rise of 12%.
percentage change
the change divided by the original amount, times 100.

Back to your guess

The answer is B: Multiply 250 by 1.12 to get 280.

12% of 250 is 30, and that is what 0.12 gives. The rise adds 30 to the 250, so the new amount is 280. Multiplying by 1.12 does both steps at once.

Check yourself

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

  1. Which multiplier increases an amount by 8%?

    Tap your answer.

    1. Not this one. This is 8% of the amount, not the amount plus 8%. A rise keeps the whole 100% and adds 8% to it, so 1 + 0.08. Go to M4.2.1. Try another.

    2. Not this one. Read the decimal carefully. 1.8 is 1 + 0.8, which is a rise of 80%. A rise of 8% is 1 + 0.08. Go to M4.2.1. Try another.

    3. Yes, that is right. A rise of 8% makes 100% + 8% = 108% of the start, which is 1.08 as a decimal.

    4. Not this one. This is the multiplier for a fall of 8%. A rise adds to 1. Go to M4.2.1. Try another.

    See the answer

    1.08

    A rise of 8% makes 100% + 8% = 108% of the start, which is 1.08 as a decimal.

    • If you chose A: This is 8% of the amount, not the amount plus 8%. A rise keeps the whole 100% and adds 8% to it, so 1 + 0.08. Go to M4.2.1.
    • If you chose B: Read the decimal carefully. 1.8 is 1 + 0.8, which is a rise of 80%. A rise of 8% is 1 + 0.08. Go to M4.2.1.
    • If you chose D: This is the multiplier for a fall of 8%. A rise adds to 1. Go to M4.2.1.

  2. The picture shows 360 being increased by 25%. What number goes in the empty box?

    A box 360 joins to an empty box ? by an arrow labelled × 1.25.360× 1.25?
    Tap your answer.

    1. Yes, that is right. 360 × 1.25 = 450. The 25% rise is 90, and 360 + 90 = 450.

    2. Not this one. You added 25 to 360. A rise of 25% is a quarter of 360 added on, which is 90. Go to M4.2.1. Try another.

    3. Not this one. This is 25% of 360, the size of the rise. The question asks for the new amount, which is 360 plus 90. Go to M4.2.1. Try another.

    4. Not this one. This is 360 decreased by 25%. The arrow says × 1.25, which is bigger than 1, so the amount grows. Go to M4.2.1. Try another.

    See the answer

    450

    360 × 1.25 = 450. The 25% rise is 90, and 360 + 90 = 450.

    • If you chose B: You added 25 to 360. A rise of 25% is a quarter of 360 added on, which is 90. Go to M4.2.1.
    • If you chose C: This is 25% of 360, the size of the rise. The question asks for the new amount, which is 360 plus 90. Go to M4.2.1.
    • If you chose D: This is 360 decreased by 25%. The arrow says × 1.25, which is bigger than 1, so the amount grows. Go to M4.2.1.

  3. Decrease 200 by 35%. The picture shows the bars.

    A bar labelled Start: 200, with the label 100%. A shorter bar labelled After the fall, with the label 65%. The missing 35% is a dashed outline.Start: 200100%After the fall65%
    Tap your answer.

    1. Not this one. You took 35 away from 200. A fall of 35% takes away 35 out of every 100, which is 70 from 200. Go to M4.2.1. Try another.

    2. Not this one. This is 35% of 200, the size of the fall. The question asks for what is left: 200 minus 70. Go to M4.2.1. Try another.

    3. Not this one. This adds 35% instead of taking it away. A fall uses a multiplier below 1. Go to M4.2.1. Try another.

    4. Yes, that is right. A fall of 35% leaves 65%, so the multiplier is 0.65. 200 × 0.65 = 130.

    See the answer

    130

    A fall of 35% leaves 65%, so the multiplier is 0.65. 200 × 0.65 = 130.

    • If you chose A: You took 35 away from 200. A fall of 35% takes away 35 out of every 100, which is 70 from 200. Go to M4.2.1.
    • If you chose B: This is 35% of 200, the size of the fall. The question asks for what is left: 200 minus 70. Go to M4.2.1.
    • If you chose C: This adds 35% instead of taking it away. A fall uses a multiplier below 1. Go to M4.2.1.

  4. What is the percentage change from 50 to 56?

    Tap your answer.

    1. Not this one. The change is 6, but 6 is not yet a percent. Divide 6 by the original, 50, and multiply by 100. Go to M4.2.1. Try another.

    2. Yes, that is right. The change is 56 − 50 = 6. Then 6 ÷ 50 × 100 = 12.

    3. Not this one. You divided the change by the new amount, 56. A percentage change is measured against the original, 50. Go to M4.2.1. Try another.

    4. Not this one. The new amount is 112% of the original, but that is the whole amount. The change is only the 12% above the 100%. Go to M4.2.1. Try another.

    See the answer

    12%

    The change is 56 − 50 = 6. Then 6 ÷ 50 × 100 = 12.

    • If you chose A: The change is 6, but 6 is not yet a percent. Divide 6 by the original, 50, and multiply by 100. Go to M4.2.1.
    • If you chose C: You divided the change by the new amount, 56. A percentage change is measured against the original, 50. Go to M4.2.1.
    • If you chose D: The new amount is 112% of the original, but that is the whole amount. The change is only the 12% above the 100%. Go to M4.2.1.

  5. An amount goes from 160 down to 120. What is the percentage change?

    Tap your answer.

    1. Not this one. You divided the change, 40, by the new amount, 120. Divide by the original amount, 160. Go to M4.2.1. Try another.

    2. Not this one. 120 is 75% of 160, so 75% is what is left. The fall is the other 25%. Go to M4.2.1. Try another.

    3. Not this one. The new amount is smaller than the original, so the amount fell. A rise needs a bigger new amount. Go to M4.2.1. Try another.

    4. Yes, that is right. The change is 160 − 120 = 40. Then 40 ÷ 160 × 100 = 25. The amount fell.

    See the answer

    A fall of 25%

    The change is 160 − 120 = 40. Then 40 ÷ 160 × 100 = 25. The amount fell.

    • If you chose A: You divided the change, 40, by the new amount, 120. Divide by the original amount, 160. Go to M4.2.1.
    • If you chose B: 120 is 75% of 160, so 75% is what is left. The fall is the other 25%. Go to M4.2.1.
    • If you chose C: The new amount is smaller than the original, so the amount fell. A rise needs a bigger new amount. Go to M4.2.1.

  6. Increase 160 by 15%.

    See the answer

    184

    A rise of 15% makes 115% of the start, so the multiplier is 1.15. 160 × 1.15 = 184.

  7. A plant grows from 40 cm to 46 cm. What is the percentage increase? Type the number only.

    See the answer

    15

    The change is 46 − 40 = 6 cm. Then 6 ÷ 40 × 100 = 15. The plant grew by 15%.

Keep it

Copy this into your exercise book.

Rise: multiply by 1 + the rate. Fall: multiply by 1 − the rate. Percentage change = change ÷ original × 100. Always divide by the original.

Draw: A bar of 100% and a longer bar of 112%, with an arrow labelled × 1.12 between them.

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

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