Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 5 of this unit) What multiplier increases an amount by 30%?
Answer
1.3. 100% + 30% = 130%. - (Lesson 4 of this unit) What percentage is 18 of 72?
Answer
25%, because 18 ÷ 72 = 0.25. - (Unit 1, lesson 7) Round 13.333 to one decimal place.
Answer
13.3. The hundredths digit is 3, which is less than 5.
Words used in this lesson
- percentage change
- the change as a percentage of the original. For example: 80 to 100 is a 25% increase.
- original
- the amount you start from. For example: 80 in 80 to 100.
- new amount
- the amount you end at. For example: 100 in 80 to 100.
- increase, decrease
- a change up, a change down. For example: from 400 to 300 is a decrease.
- multiplier
- the number that applies a percentage change in one step. For example: 0.8 for a fall of 20%.
- discount
- a reduction in price. For example: 20% discount on a jacket.
Compare the change with where you started
A mobile-money agent in Kampala handled 80 transactions on Monday and 100 on Tuesday. The change is 20. Is that big? It depends on where you started: 20 more than 80 is a quarter more, but 20 more than 800 is hardly noticeable. A percentage change compares the change with the original amount, the one you started from.
Predict first: from 80 to 100 is a rise of 25%. Is a fall from 100 back to 80 a fall of 25%? Then set both sliders and see.
From 80 to 100 the change is +20. 20 ÷ 80 = 0.25, so the amount increased by 25%.
Show me the steps
- 80 to 100: change +20, 20 ÷ 80 = 0.25, so +25%.
- 100 to 80: change −20, 20 ÷ 100 = 0.2, so −20%. The same gap, a different original.
- 200 to 100: change −100, 100 ÷ 200 = 0.5, so −50%.
- 100 to 200: change +100, 100 ÷ 100 = 1, so +100%.
- 130 to 150: change +20, 20 ÷ 130 is about 0.154, so about +15.4% to one decimal place.
- 100 to 100: no change, so 0%.
Percentage change = change ÷ original × 100. The original is the amount you start from, the “before”. Say increase or decrease, and give the size.
From 80 to 100: 20 ÷ 80 = 0.25, so a 25% increase. From 100 to 80: 20 ÷ 100 = 0.2, so a 20% decrease. The gap is the same; the original is different.
To work back to the original from a new amount, divide by the multiplier. After a rise of 20% the new amount is original × 1.2, so original = new ÷ 1.2.
Going back to the original
A bag costs 360 rupees after a rise of 20%. What did it cost before? It is tempting to take 20% off 360. Try it in the figure first: take the percentage off the new amount and see where you land.
300 up 20% is 360. Taking 20% off 360 gives 288, not 300. Dividing 360 by 1.2 gives 300.
Show me the steps
- 300 up 20% is 360. Taking 20% off 360 gives 288. Dividing 360 by 1.2 gives 300.
- 300 down 20% is 240. Adding 20% of 240 (48) gives 288. Dividing 240 by 0.8 gives 300.
- 200 up 50% is 300. Taking 50% off gives 150. Dividing 300 by 1.5 gives 200.
- 100 down 40% is 60. Adding 40% of 60 gives 84. Dividing 60 by 0.6 gives 100.
- 500 up 10% is 550. Taking 10% off gives 495. Dividing 550 by 1.1 gives 500.
- 100 up 60% is 160. Taking 60% off gives 64. Dividing 160 by 1.6 gives 100.
A common mistake: taking the percentage off the new amount to go back. 300 up 20% is 360, and 20% of 360 is 72, not 60, so 360 − 72 = 288. The percentage was taken from 300, so it must be undone by dividing by the multiplier, 360 ÷ 1.2 = 300.
Worked example: percentage change
The number of visitors to a stall rose from 250 to 300. Find the percentage increase.
- The change is 300 − 250 = 50.
- Divide by the original: 50 ÷ 250 = 0.2.
- 0.2 × 100 = 20% increase.
After a 20% reduction a jacket costs 360 rupees. What was the original price?
- A fall of 20% leaves 80%, so the multiplier is 0.8: original × 0.8 = 360.
- Divide to undo: 360 ÷ 0.8 = 450 rupees.
- Check: 450 × 0.8 = 360.
Your go. The first step is given. A phone's price falls from 36 000 to 27 000 naira. Find the percentage decrease.
- The change is 27 000 − 36 000 = −9000.
- Now divide by the original and give the percentage, then check.
Check the answer
9000 ÷ 36 000 = 0.25, so the price fell by 25%. Check: 36 000 × 0.75 = 27 000.
With a calculator
A scientific calculator does the division and the multiplication by 100 in one go: 2 ÷ 15 × 100 = 13.333... Round to the place the question asks for: 13.3 to one decimal place. To work back, divide by the multiplier: 276 ÷ 1.15 = 240.
Estimate first. 2 out of 15 is a little less than 2 out of 10, which is 20%, and a little more than 1 out of 10, which is 10%, so an answer near 13% makes sense. Check which number you divided by: it is the original.
Check before you go on
1. A price rises from 50 to 60. Which calculation gives the percentage increase?
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. The number of customers in a shop goes from 60 to 75. What is the percentage increase?
3. A bicycle costs 500 and is reduced to 400. What is the percentage decrease?
Level 2 · Building
4. A bus fare goes from 40 pesos to 46 pesos. What is the percentage increase?
5. After a discount of 25% a coat costs 360 rupees. What was its price before the discount, in rupees?
6. Which is the larger percentage increase: 20 to 30, or 100 to 130?
Level 3 · Stretch
7. The number of visitors to a museum falls from 1350 to 1100 in a month. Use a scientific calculator to find the percentage decrease, to one decimal place.
8. After a rise of 15%, a price is 345 rupees. What was the price before the rise, in rupees?
Something to think about
A town of 5000 people falls to 4500, then rises back to 5000. What is the percentage fall, and what is the percentage rise needed to get back? Are they equal?
One way to think about it
The fall is 500 ÷ 5000 = 10%. The rise is 500 ÷ 4500 = 0.111..., about 11.1%. They are not equal, because the rise is measured against the smaller amount, 4500.
Exit ticket
9. A price rises from 200 to 250 and later falls from 250 back to 200. Rashid says both changes are 25%. Which reply is right?