- Convert between fractions, decimals and percentages
- Recognise and solve the three basic types of percentage problems
- Calculate increases, decreases and successive changes with a multiplier
- Solve problems on bank interest and the concentration of mixtures, and tell a percent change from a change in percentage points
One shop takes 30% off a 100-manat jacket in one go. Another first takes 20% off and then another 10% off the new price. Is that the same? At first sight 20% + 10% = 30%, but in the second shop the jacket costs 72 manat, not 70. Prices, discounts, bank deposits and test results are all given in percentages, and in this lesson you will see that the key question is always the same: “percent of what?” The lesson builds on “Decimal fractions”: every percentage is really a decimal.
What is a percent?
One hundredth of a number: 1% = 1/100 = 0.01. The whole amount is 100%. The sign “%” simply means “divided by 100”.
- pthe number of percent
To turn a percentage into a decimal, divide by 100 (the point moves two places to the left); to write a number as a percentage, multiply it by 100% (the point moves two places to the right).
1) Write as a decimal: 7%; 45%; 120%; 0.5%. 2) Write as a percentage: 0.3; 1.25; 0.045. 3) Write as a percentage: 3/5; 7/8; 2/3.
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2) 0.3 = 30%; 1.25 = 125%; 0.045 = 4.5%.
3) 3/5 = 60/100 = 60%; 7/8 = 7 ÷ 8 = 0.875 = 87.5%; 2/3 = 0.666… ≈ 66.7%.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/20 | 0.05 | 5% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.3̅ | ≈ 33.3% |
| 3/2 | 1.5 | 150% |
Three basic percentage problems
Every percentage problem links three quantities: the whole a (it is 100%), the part b and the number of percent p. Two of them are given and the third is unknown — so there are three basic problems, and all of them come from one formula.
- athe whole (the amount taken as 100%)
- bthe part — p% of a
- pthe number of percent
Finding a percentage of a number: write the percentage as a decimal and multiply.
1) Trainers costing 40 manat are 25% off. How big is the discount, and what is the new price? 2) 12% of the 250 pupils of a school go to the chess club. How many pupils is that?
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2) 250 · 0.12 = 30 pupils.
- athe whole we are looking for (100%)
- bthe known part
- pthe percentage that the part makes up
Finding a number from its percentage: divide the part by the percentage written as a decimal.
1) Aysel spent 15% of her money, that is 30 manat, on books. How much money did she have? 2) 8 pupils, that is 32% of the class, got top marks. How many pupils are in the class?
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2) a = 8 ÷ 0.32 = 800 ÷ 32 = 25 pupils. Check: 25 · 0.32 = 8.
- pwhat percentage b is of a
- athe whole
- bthe part
What percentage one number is of another: divide the part by the whole and multiply by 100%.
1) Elvin answered 34 of 40 test questions correctly. What percentage is that? 2) 9 of the 25 pupils in a class go swimming. What percentage of the class is that? 3) A 60-manat bag was sold for 45 manat. By what percentage was the price reduced?
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2) 9/25 = 0.36 = 36%.
3) The discount is 60 − 45 = 15 manat, and we compare it with the old price: 15/60 = 0.25 = 25%.
- 1Find the whole
Ask yourself: “percent of what?” That amount is 100%: the price before the change, the whole class, the whole solution.
- 2Name the unknown
Is the part, the whole or the number of percent unknown? That tells you which of the three formulas you need.
- 3Calculate and check
Write the percentage as a decimal, calculate, and check backwards: the part must be p% of the whole.
The same problems can also be solved with a proportion: a is 100% and b is p%, so a : b = 100 : p (see the lesson “Ratio and proportion”).
Increase and decrease: the multiplier
When a price goes up by 15%, the new price is 100% + 15% = 115% of the old one; when it goes down by 15%, it is 100% − 15% = 85%. So a single multiplication does the whole job:
- athe starting amount (100%)
- 1 + p/1001 + p/100the increase multiplier: +15% → · 1.15
- 1 − p/1001 − p/100the decrease multiplier: −15% → · 0.85
Increasing by p% means multiplying by (1 + p/100); decreasing by p% means multiplying by (1 − p/100).
1) Shoes cost 80 manat, and the price falls by 15%. 2) A salary of 600 manat rises by 12%. 3) After a 20% price rise a book costs 54 manat. What did it cost before?
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2) 600 · 1.12 = 672 manat.
3) old price · 1.2 = 54, so old price = 54 ÷ 1.2 = 540 ÷ 12 = 45 manat. Not 54 · 0.8 = 43.2: the 20% was taken from the old price, not from 54.
- differencethe difference between the new and the old value (smaller taken from larger)
- old valuethe value before the change — it is 100%
A percent change is always measured from the old (starting) value.
By what percentage did the price change: 1) from 40 to 50 manat; 2) from 50 back to 40 manat; 3) from 20 to 50 manat?
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2) The difference is again 10 manat, but the old price is now 50: 10/50 = 0.2 — down 20%. The difference is the same, but the bases differ, so the percentages differ.
3) The difference is 30 manat: 30/20 = 1.5 — up 150%. A percentage can be more than 100.
When changes follow one another, each new percentage is taken from the new amount. That is why the percentages are not added — the multipliers are multiplied.
- p₁the percentage of the first change
- p₂the percentage of the second change (of the new amount)
- ±“+” for an increase, “−” for a decrease
For successive changes multiply the multipliers; the total change is read from their product.
1) The jacket from the start of the lesson: first 20% off, then 10% off. What is the total discount? 2) A price of 80 manat first went up by 10% and then down by 10%. 3) Trainers costing 40 manat sell for 30 manat after a 25% discount; then this price is raised by 25%.
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2) 80 · 1.1 = 88; 88 · 0.9 = 79.2 manat, because 1.1 · 0.9 = 0.99 — a 1% drop in total.
3) 30 · 1.25 = 37.5 manat, not 40: the second 25% is taken from 30, the first one from 40.
Bank interest: simple and compound
A bank pays interest on a deposit: the annual rate says what percentage of the money is added each year. With simple interest the interest is always calculated from the initial deposit, so the same amount is added every year. With compound interest the interest is added to the deposit, and the next year interest is paid on that interest too — the amount is multiplied by the same multiplier every year.
- Sthe amount after n years
- Pthe initial deposit
- pthe annual rate, in percent
- nthe number of years
Simple interest: P · p/100 is added every year.
- (1 + p/100)ⁿ(1 + p/100)ⁿthe increase multiplier applied n times
- S, P, p, nas in the simple interest formula
Compound interest: n successive increases of p% each.
1) 1000 manat is deposited at 10% a year for 3 years. How much will be in the account with simple and with compound interest? 2) 2000 manat is deposited for 2 years at 8% a year compound interest. How much will there be at the end? 3) A phone bought for 800 manat loses 20% of its value every year. What is it worth after 2 years?
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2) 2000 · 1.08² = 2000 · 1.1664 = 2332.8 manat (with simple interest it would be 2320 manat).
3) The multiplier is 0.8 every year: 800 · 0.8² = 800 · 0.64 = 512 manat.
Mixtures and percentage points
The percentage of a substance (salt, sugar) in the total mass of a mixture or solution.
- cthe concentration, in percent
- mthe mass of the substance
- Mthe mass of the whole mixture (substance + water)
Adding water changes M but not m; when two solutions are mixed, both the m’s and the M’s add up.
1) 50 g of salt is dissolved in 200 g of water. What is the concentration of the solution? 2) How much sugar is there in 300 g of a 15% syrup? 3) 200 g of a 10% salt solution is mixed with 300 g of a 20% salt solution. What is the concentration of the mixture? 4) How much water must be added to 400 g of a 30% salt solution to get a 20% solution?
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2) m = 300 · 0.15 = 45 g.
3) Salt: 200 · 0.1 + 300 · 0.2 = 20 + 60 = 80 g; mixture: 200 + 300 = 500 g; c = 80/500 = 0.16 = 16%. Not 15%, halfway between 10% and 20%, because there is more of the stronger solution.
4) Salt: 400 · 0.3 = 120 g, and it does not change. 120 g must be 20% of the new mass: 120 ÷ 0.2 = 600 g. Water to add: 600 − 400 = 200 g.
Be careful when a percentage itself changes. If a bank raises its rate from 8% to 10%, the rate goes up by 2 percentage points (10 − 8), but by 25% compared with the old rate (2/8 = 0.25). The two are often mixed up in the news.
- p₁the old percentage
- p₂the new percentage
Percentage points measure the difference between two percentages; the relative change compares this difference with the old percentage (for a decrease, take p₁ − p₂).
Give each change in percentage points and in percent: 1) a deposit rate rose from 8% to 10%; 2) the share of pupils who passed an exam rose from 60% to 75%; 3) a shop’s share of online sales fell from 20% to 16%.
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2) 75 − 60 = 15 percentage points up; 15/60 = 0.25, again 25% up.
3) 20 − 16 = 4 percentage points down; 4/20 = 0.2, that is 20% down.
Key points
- 1% = 1/100 = 0.01; to turn a percentage into a decimal divide by 100, to write a number as a percentage multiply by 100%.
- b = a · p/100: find the part by multiplying, the whole by dividing (a = b ÷ p/100), and the percentage as b/a · 100%.
- An increase by p% is multiplication by 1 + p/100, a decrease by 1 − p/100; for successive changes the multipliers are multiplied.
- Always ask “percent of what?”: a change is measured from the starting value.
- Simple interest: S = P · (1 + n · p/100); compound interest: S = P · (1 + p/100)ⁿ.
- Concentration = substance / whole mixture · 100%; the difference of two percentages is measured in percentage points.
Check yourself
12 questions. Every correct answer earns XP.