Year 9 Maths The Thai Guy

Lesson 14.4 · 30 min

Reverse Percentages and Percentage Error

Finding the original amount after a percentage change, reversing repeated changes, and measuring how far an estimate or measurement is from the true value.

In this lesson you will learn to…

  • Find the original amount when you know the amount after a percentage increase or decrease
  • Explain why you divide by the multiplier rather than taking the percentage off again
  • Reverse a repeated percentage change, such as compound interest
  • Calculate a percentage error

Working back to 100%

After a percentage change you have a different percentage of the original — 115%115\% after a 15%15\% rise, 85%85\% after a 15%15\% fall. To get back, find 1%1\% and then 100%100\%, or divide by the multiplier.

Try it

Work back to 100%

0%50%100%150%1380 baht115%
Known
115% = 1380 baht
1%
1380 baht ÷ 115 = 12 baht
115%
115 × 12 baht = 1380 baht

The price after a rise is 1380 baht, so the bar is marked at . Slide to … then to . Compare with — what you get by wrongly taking off.

Increases and decreases

The multiplier method handles both in one step:

original×multiplier=new⟹original=newmultiplier\text{original} \times \text{multiplier} = \text{new} \quad\Longrightarrow\quad \text{original} = \frac{\text{new}}{\text{multiplier}}

Reversing repeated changes

Compound interest multiplies by the same multiplier every year, so undoing nn years means dividing by mnm^n.

Percentage error

Percentage error measures how far an estimate or a measurement is from the true value, as a percentage of the true value.

Key rules and formulas

  • Reverse percentages

    After a increase the multiplier is ; after a decrease it is . Or: new amount , so find and then .

  • Never take the percentage off again

    The percentage was of the original. Taking of the new amount uses the wrong starting value, so it never gets you back.

  • Repeated changes

    For changes by the same multiplier , e.g. compound interest.

  • Percentage error

    Always divide by the true value, not the estimate or measurement.

Worked examples

Worked example 1

After an increase

After a pay rise, Somchai earns 33 600 baht a month. What did he earn before?

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

After a decrease

A car loses of its value in a year and is now worth 574 000 baht. Find its value a year ago.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

VAT included

A price of 856 baht includes VAT. How much of it is VAT?

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

Percentage error

A bag of rice is labelled kg but actually weighs kg. Find the percentage error in the label.

Try it yourself first, then reveal the solution one step at a time.

Practice: reverse percentages and percentage error

12 questions

original = new ÷ multiplier. Never take the percentage off the new amount.
  1. Question 1 of 12Type the answer
    After a increase, a price is 450 baht. Find the original price.
    baht
  2. Question 2 of 12Type the answer
    After a discount, a bag costs 1260 baht. What was the original price?
    baht
  3. Question 3 of 12Multiple choice
    A town's population rose by to . Which calculation gives the population before the rise?
  4. Question 4 of 12Type the answer
    A restaurant bill of 1177 baht includes a service charge. How much was the bill before the service charge?
    baht
  5. Question 5 of 12Type the answer
    A laptop's price includes VAT and is 32 100 baht. How much VAT is included?
    baht
  6. Question 6 of 12Type the answer
    A shop makes a profit by selling a watch for 5400 baht. How much did the shop pay for it?
    baht
  7. Question 7 of 12Type the answerChallenge
    An investment earns compound interest per year. After 4 years it is worth $9724.05. Find the amount invested, to the nearest dollar.
    $
  8. Question 8 of 12Type the answerChallenge
    A car loses of its value each year. After 2 years it is worth 433 500 baht. What was it worth when new?
    baht
  9. Question 9 of 12Step by stepChallenge
    Two changes in a row.
    A price is increased by and then the new price is reduced by . The final price is 2376 baht.
    1. What single multiplier has the same effect as both changes?

  10. Question 10 of 12Type the answer
    Pim estimates there are 450 sweets in a jar. There are really 400. Find her percentage error.
    %
  11. Question 11 of 12Type the answer
    Using instead of the true value , what is the percentage error? Give your answer to 3 significant figures.
    %
  12. Question 12 of 12Explore and match
    A price after a reduction is 600 baht. Slide to the percentage that shows the original price.

    Move the controls until you match the question, then press Check.

    0%50%100%150%600 baht75%
    Known
    75% = 600 baht
    1%
    600 baht ÷ 75 = 8 baht
    75%
    75 × 8 baht = 600 baht