Chapter 2 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

M4.2.2 Chapter 2

Finding the original amount

A herd grew by 25% and now has 150 goats. How many goats did it have before?

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

  • Find the original amount after a 20% rise gives 96
  • Explain why a 10% rise then a 10% fall is a loss

Last year a herd had some goats. It grew by 20% and now has 96 goats. We want the number it started with, the original amount.

The start is 100%. After a rise of 20%, the herd is 120% of the start. So 96 goats is 120% of the original amount.

Draw it first

Work back from the new amount

  1. A bar labelled New herd: 96 goats, with the label 120%. A shorter bar labelled Old herd: ? goats, with the label 100%. A box ? joins to a box 96 by an arrow labelled × 1.2, and a return arrow labelled ÷ 1.2 goes back.New herd: 96 goats120%

    1 The new herd is 96 goats. That is 120% of the original.

  2. A bar labelled New herd: 96 goats, with the label 120%. A shorter bar labelled Old herd: ? goats, with the label 100%. A box ? joins to a box 96 by an arrow labelled × 1.2, and a return arrow labelled ÷ 1.2 goes back.New herd: 96 goats120%Old herd: ? goats100%

    2 The old herd is the shorter bar. We do not know it yet.

  3. A bar labelled New herd: 96 goats, with the label 120%. A shorter bar labelled Old herd: ? goats, with the label 100%. A box ? joins to a box 96 by an arrow labelled × 1.2, and a return arrow labelled ÷ 1.2 goes back.New herd: 96 goats120%Old herd: ? goats100%?× 1.296÷ 1.2

    3 The rise multiplied the old herd by 1.2. To go back, divide the new herd by 1.2.

A rise multiplies by 1.2. Going back undoes the multiplying, so divide by 1.2. Divide by the multiplier, whether the change was a rise or a fall. To divide by 1.2, multiply both numbers by 10 first: 96 ÷ 1.2 = 960 ÷ 12 = 80.

Worked example

A herd grew by 20% and now has 96 goats. Find the original amount.

  1. The new amount is 100% + 20% = 120% of the original.
  2. 120% as a decimal is 1.2. So original × 1.2 = 96.
  3. Go back by dividing by the multiplier: 96 ÷ 1.2 = 80.
  4. Check: 80 × 1.2 = 96. So the herd had 80 goats before.

Answer: 96 ÷ 1.2 = 80

A bar labelled Old herd: 80 goats, with the label 100%. A longer bar labelled New herd: 96 goats, with the label 120%, whose extra dotted piece is 16 goats.Old herd: 80 goats100%New herd: 96 goats120%
The rise is 20% of the old herd: 16 goats. It is not 20% of the new herd.

A wrong idea many people have

To undo a rise of 20%, take 20% off 96.

Why it feels right: A rise and a fall of the same percent sound like opposites that cancel.

The test that breaks it: 20% of 96 is 19.2, and 96 − 19.2 = 76.8. Now check by rising 20%: 76.8 × 1.2 = 92.16, not 96.

What is true: The rise was 20% of the old amount, 80, and not 20% of 96. Divide by 1.2 to reach 80.

A fall works the same way. The multiplier is below 1, and you still divide by it.

A bar labelled Start: ? litres, with the label 100%. A shorter bar labelled Now: 140 litres, with the label 70%. The missing 30% is a dashed outline.Start: ? litres100%Now: 140 litres70%
140 litres is 70% of the start.

Worked example

A tank lost 30% of its water and now holds 140 litres. How much did it hold at the start?

  1. A fall of 30% leaves 100% − 30% = 70% of the start.
  2. 70% as a decimal is 0.7. So start × 0.7 = 140.
  3. Divide by the multiplier: 140 ÷ 0.7 = 200.
  4. The tank held 200 litres. Check: 200 × 0.7 = 140.

Answer: 140 ÷ 0.7 = 200

Now a puzzle. A herd of 200 goats grows by 10%. Then it falls by 10%. Is it back where it began?

A box 200 joins to a box 220 by an arrow labelled × 1.1. That box joins to a box 198 by an arrow labelled × 0.9.200× 1.1220× 0.9198
The herd ends at 198, which is 2 below where it began.

It is not back. The rise of 10% was worked out on 200. The fall of 10% was worked out on 220, which is a bigger amount. A bigger amount loses more.

Worked example

A herd of 200 goats grows by 10%, then falls by 10%. How does the end compare with the start?

  1. A rise of 10% has multiplier 1.1: 200 × 1.1 = 220.
  2. A fall of 10% has multiplier 0.9: 220 × 0.9 = 198.
  3. One multiplier does both: 1.1 × 0.9 = 0.99.
  4. The herd ends at 0.99 of the start, which is 1% less.

Answer: Less, by 1%

See it three ways

With things

Take 100 stones. Add 10%: that is 10 more, so 110 stones. Now take away 10% of the 110: that is 11 stones. 99 are left.

As a picture

A bar labelled Start: 100 stones, with the label 100%. A longer bar labelled After the rise: 110 stones, with the label 110%. A third bar labelled After the fall: 99 stones, with the label 99%, with a short dashed outline for the 1% missing.Start: 100 stones100%After the rise: 110 stones110%After the fall: 99 stones99%

In symbols

100 × 1.1 × 0.9 = 99.

Try it

Try any number

You need: paper, a pencil.

  1. Write any whole number you like. Here we use 300.
  2. Raise it by 10%. Write the answer.
  3. Lower that answer by 10%. Write the new answer.

What you will see: You end below where you began. With 300 you reach 297. Try other numbers: the end is always 0.99 of the start.

If you cannot try it, this is what happens.
A box 300 joins to a box 330 by an arrow labelled × 1.1. That box joins to a box 297 by an arrow labelled × 0.9.300× 1.1330× 0.9297

A wrong idea many people have

A rise of 10% and then a fall of 10% cancel, so you end where you began.

Why it feels right: Plus 10 and minus 10 do cancel. The same number appears twice, so it feels the same here.

The test that breaks it: Take 100. A rise of 10% gives 110. A fall of 10% takes off 10% of 110, which is 11. That leaves 99, not 100.

What is true: Each percent is of a different amount. Multiply the multipliers: 1.1 × 0.9 = 0.99.

Key words

original amount
the amount before the change. It is 100%.

Back to your guess

The answer is C: Divide 150 by 1.25 to get 120.

The 25% rise was worked out on the old herd, not on 150. The new herd is 125% of the old herd, so divide 150 by 1.25. That gives 120. Check: 120 × 1.25 = 150.

Check yourself

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

  1. After a 40% rise an amount is 210. What was the original amount?

    Tap your answer.

    1. Not this one. You took 40% off 210. The 40% rise was worked out on the original amount, which is smaller than 210, so divide by 1.4. Go to M4.2.2. Try another.

    2. Not this one. You took 40 away from 210. The rise is 40% of the original, not the number 40. Go to M4.2.2. Try another.

    3. Not this one. You multiplied by 1.4 again. That makes the amount bigger, but the original was smaller than 210. Go to M4.2.2. Try another.

    4. Yes, that is right. A rise of 40% is a multiplier of 1.4. Divide to go back: 210 ÷ 1.4 = 150.

    See the answer

    150

    A rise of 40% is a multiplier of 1.4. Divide to go back: 210 ÷ 1.4 = 150.

    • If you chose A: You took 40% off 210. The 40% rise was worked out on the original amount, which is smaller than 210, so divide by 1.4. Go to M4.2.2.
    • If you chose B: You took 40 away from 210. The rise is 40% of the original, not the number 40. Go to M4.2.2.
    • If you chose C: You multiplied by 1.4 again. That makes the amount bigger, but the original was smaller than 210. Go to M4.2.2.

  2. After a 20% fall an amount is 64. What was it before?

    Tap your answer.

    1. Not this one. You added 20% of 64. The fall was 20% of the original, which is more than 64, so divide by 0.8. Go to M4.2.2. Try another.

    2. Yes, that is right. A fall of 20% leaves 80%, a multiplier of 0.8. Divide to go back: 64 ÷ 0.8 = 80.

    3. Not this one. You added 20 to 64. The fall is 20% of the original, not the number 20. Go to M4.2.2. Try another.

    4. Not this one. You multiplied by 0.8 again, which is a second fall. The original was bigger than 64. Go to M4.2.2. Try another.

    See the answer

    80

    A fall of 20% leaves 80%, a multiplier of 0.8. Divide to go back: 64 ÷ 0.8 = 80.

    • If you chose A: You added 20% of 64. The fall was 20% of the original, which is more than 64, so divide by 0.8. Go to M4.2.2.
    • If you chose C: You added 20 to 64. The fall is 20% of the original, not the number 20. Go to M4.2.2.
    • If you chose D: You multiplied by 0.8 again, which is a second fall. The original was bigger than 64. Go to M4.2.2.

  3. After a 25% fall an amount is 90. What was it before?

    Tap your answer.

    1. Not this one. You added 25% of 90. The fall was 25% of the original, which is more than 90, so divide by 0.75. Go to M4.2.2. Try another.

    2. Not this one. You added 25 to 90. The fall is 25% of the original, not the number 25. Go to M4.2.2. Try another.

    3. Yes, that is right. A fall of 25% leaves 75%, a multiplier of 0.75. 90 ÷ 0.75 = 120.

    4. Not this one. You took 25% off 90, which is a second fall. The original was bigger than 90. Go to M4.2.2. Try another.

    See the answer

    120

    A fall of 25% leaves 75%, a multiplier of 0.75. 90 ÷ 0.75 = 120.

    • If you chose A: You added 25% of 90. The fall was 25% of the original, which is more than 90, so divide by 0.75. Go to M4.2.2.
    • If you chose B: You added 25 to 90. The fall is 25% of the original, not the number 25. Go to M4.2.2.
    • If you chose D: You took 25% off 90, which is a second fall. The original was bigger than 90. Go to M4.2.2.

  4. A rise of 20% is followed by a fall of 20%. Which single multiplier does the same job?

    A box Start joins to a box Up by an arrow labelled × 1.2. That box joins to a box Down by an arrow labelled × 0.8.Start× 1.2Up× 0.8Down
    Tap your answer.

    1. Yes, that is right. Multiply the two multipliers: 1.2 × 0.8 = 0.96.

    2. Not this one. A rise and a fall of the same percent do not cancel. The fall is a percent of a bigger amount than the rise. Go to M4.2.2. Try another.

    3. Not this one. This is the multiplier of the fall alone. The rise happened first, so multiply by 1.2 as well. Go to M4.2.2. Try another.

    See the answer

    0.96

    Multiply the two multipliers: 1.2 × 0.8 = 0.96.

    • If you chose B: A rise and a fall of the same percent do not cancel. The fall is a percent of a bigger amount than the rise. Go to M4.2.2.
    • If you chose C: This is the multiplier of the fall alone. The rise happened first, so multiply by 1.2 as well. Go to M4.2.2.

  5. An amount rises by 20% and then falls by 20%. Compared with the start, the end amount is:

    Tap your answer.

    1. Not this one. They do not cancel. The fall of 20% is taken from the bigger, risen amount, so it takes off more than the rise added. Go to M4.2.2. Try another.

    2. Not this one. The end multiplier is 0.96, which is below 1. An amount that has been multiplied by less than 1 is smaller. Go to M4.2.2. Try another.

    3. Not this one. The rise comes first and makes up for most of the fall. The end is much nearer the start than 20% less. Go to M4.2.2. Try another.

    4. Yes, that is right. The end is 1.2 × 0.8 = 0.96 of the start. That is 4% less.

    See the answer

    4% less than the start

    The end is 1.2 × 0.8 = 0.96 of the start. That is 4% less.

    • If you chose A: They do not cancel. The fall of 20% is taken from the bigger, risen amount, so it takes off more than the rise added. Go to M4.2.2.
    • If you chose B: The end multiplier is 0.96, which is below 1. An amount that has been multiplied by less than 1 is smaller. Go to M4.2.2.
    • If you chose C: The rise comes first and makes up for most of the fall. The end is much nearer the start than 20% less. Go to M4.2.2.

  6. From an earlier chapter

    Find the percentage change from 60 to 75. Type the number only.

    See the answer

    25

    The change is 75 − 60 = 15. Then 15 ÷ 60 × 100 = 25.

  7. After a 15% rise an amount is 345. Find the original amount.

    See the answer

    300

    A rise of 15% is a multiplier of 1.15. Divide to go back: 345 ÷ 1.15 = 300.

Keep it

Copy this into your exercise book.

To go back from a new amount, divide by the multiplier. Never take the same percent off the new amount. A rise then a fall of 10% gives 0.99.

Draw: Two boxes joined by an arrow labelled × 1.2 above and a return arrow labelled ÷ 1.2 below.

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

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