Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 3 of this unit) Find 15% of 280.
Answer
42. 10% is 28 and 5% is 14. - (Lesson 2 of this unit) Write 7% as a decimal.
Answer
0.07. 7% is 7 hundredths. - (Unit 1, lesson 6) Work out 100 × (1 + 0.15).
Answer
115. Brackets first: 1 + 0.15 = 1.15, then 100 × 1.15.
Words used in this lesson
- increase
- to make an amount bigger. For example: a rise of 20% on 50 gives 60.
- decrease
- to make an amount smaller. For example: a fall of 25% on 640 gives 480.
- multiplier
- the number you multiply by to apply a change in one step. For example: 1.25 for a rise of 25%.
- original
- the amount before a change. For example: the old fare, 50 rupees.
- reduction
- a decrease in a price. For example: a 15% reduction.
- service charge
- an extra percentage added to a bill. For example: 7% added to 400 rupees.
Change in one step
A bus fare in Colombo rises by 25%. The old fare is 40 rupees. Find 25% of 40, which is 10, and add it: 50 rupees. That is two steps, and it works. Here is a faster idea: the new fare is 100% of the old fare plus 25% of it, so it is 125% of the old fare. A fare of 125% is the old fare times 1.25. One multiplication.
Predict first: a price is 80 and it goes up by 25%. What is the multiplier? And the new price? Then slide to check, and try a decrease.
An increase of 25% on 80 adds 20, giving 100. In one step: 80 × 1.25 = 100.
Show me the steps
- 80 up 25%: the change is +20, the new amount is 100, the multiplier is 1.25.
- 80 down 25%: the change is −20, the new amount is 60, the multiplier is 0.75.
- 240 up 10%: +24, 264, multiplier 1.1.
- 240 down 15%: −36, 204, multiplier 0.85.
- 400 up 5%: +20, 420, multiplier 1.05.
- 200 down 50%: −100, 100, multiplier 0.5.
Increase by p% means multiply by 1 + p/100. Decrease by p% means multiply by 1 − p/100. A rise of 7% is × 1.07. A fall of 15% is × 0.85. The multiplier finds the change and applies it in one step.
Check it by hand when you can: 15% off 640 takes off 10% (64) and 5% (32), which is 96, so 640 − 96 = 544, and 640 × 0.85 = 544.
The change is always a percentage of the amount you started that step with.
Up, then down: back where you started?
A price rises by 50% and then falls by 50%. Is it back where it started? Do not guess. Take any starting amount and try the sliders, then try other percentages.
200 rises by 50% to 300, then falls by 50% of 300, which is 150, to 150. That is 25% below the start.
Show me the steps
- 200 up 10% is 220, then down 10% of 220 (22) is 198. That is 1% below the start.
- 200 up 20% is 240, then down 20% is 192. That is 4% below.
- 200 up 30% is 260, then down 30% is 182. That is 9% below.
- 200 up 50% is 300, then down 50% is 150. That is 25% below.
- 400 up 25% is 500, then down 25% is 375. That is 6.25% below.
- 200 up 5% is 210, then down 5% is 199.5. That is 0.25% below: small, but still below.
A common mistake: “up 50% and down 50% cancel.” The rise is a percentage of the starting amount and the fall is a percentage of the larger amount. 200 up 50% is 300, and 50% of 300 is 150, so the price falls to 150, not back to 200. Percentage changes are not added.
Worked example: a reduction
A bag costs 640 naira and is reduced by 15%. What is the new price?
- A fall of 15% leaves 100% − 15% = 85%, so the multiplier is 0.85.
- 640 × 0.85 = 544 naira.
- Check by building: 10% is 64, 5% is 32, so 15% is 96, and 640 − 96 = 544.
Your go. The first step is given. A restaurant in Kathmandu adds a 7% service charge to a 400-rupee bill. What is the total?
- 1% of 400 is 4.
- Now find 7%, then the total, and check it against 400 × 1.07.
Check the answer
7% is 7 × 4 = 28, so the total is 400 + 28 = 428 rupees. Check: 400 × 1.07 = 428.
Check before you go on
1. A price of 240 is increased by 15%. Which multiplier gives the new price?
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. A bus fare of 50 rupees rises by 20%. What is the new fare, in rupees?
3. A bag costing 640 naira is reduced by 25%. What is the sale price, in naira?
Level 2 · Building
4. Increase 120 by 15%.
5. A town of 8000 people grows by 6% in a year. How many people live there now?
6. Which multiplier takes a price down by 12%?
Level 3 · Stretch
7. A shirt costs 250 rupees. The price rises by 20%, then the new price falls by 20%. What does the shirt cost at the end?
8. A coat costs 500 rupees. It is reduced by 20%, and then the sale price is reduced by a further 10%. What is the final price, in rupees?
Something to think about
A price is multiplied by 1.2 and then by 0.8. Is the answer the original price? What single multiplier is the same, and what percentage change is that?
One way to think about it
1.2 × 0.8 = 0.96, so the price ends at 0.96 of the original: a decrease of 4%. It is not the original, because the fall of 20% is taken from the larger amount.
Exit ticket
9. A price rises by 40% and then falls by 40%. Dilnoza says it is back to the starting price. Which reply is right?