Number · Ratios and Proportion · Lesson 5
Percent as a Rate per 100
"40% off!", "6% interest", "35% commission": percentages are on shop windows in Osu, on bank posters and on every salesman's payslip. A percent is simply a rate per 100. In this lesson you will find a percent of any quantity and use it to work out commissions, discounts, sale prices, interest and even the original price before a discount.
By the end you will be able to
- ✔Explain that p% means p per 100, that is .
- ✔Find a percent of a quantity by multiplying the quantity by .
- ✔Solve problems on commission, discount, sale price and simple interest for one year.
- ✔Find the original price when the sale price and the percentage discount are known.
Key words
- Percent (%)
- Per hundred. 30% means 30 out of every 100, or .
- Commission
- Money a salesperson earns as a percent of the value of what they sell.
- Discount
- An amount taken off the usual (marked) price, often given as a percent of that price.
- Sale price
- The price you pay after the discount: marked price − discount.
- Interest
- Money a bank pays you for saving, given as a percent of the amount saved each year.
Curriculum reference
Content standard: B7.1.4.1 Demonstrate an understanding of the concept of ratios and its relationship to fractions and use it to solve problems that involve rates, ratios, and proportional reasoning.
Indicator: Find a percent of a quantity as a rate per 100 (e.g. 30% of a quantity means 30/100 times the quantity).
Starter
The 40% off phone
Naa walks past a phone shop at Circle and sees a big red poster: "Phone GH₵450 — NOW 40% OFF!" Her friend says, "That means you pay GH₵410." Naa is not so sure. Is 40% off the same as GH₵40 off? It is not! By the end of this lesson you will know exactly how much Naa will pay, and you will never be fooled by a percentage again.
1. A percent is a rate per 100
In the last lessons you met rates like "GH₵9 per litre". A percent is a special rate: it is always per hundred. The word comes from the Latin per centum, meaning "for every hundred". So 35% means 35 for every 100, which we can write as the fraction or the decimal 0.35.
▶ The big square is 100%. Count the shaded squares: 35 out of 100 is 35%.
| Percent | Per 100 | Fraction (simplest) | Decimal |
|---|---|---|---|
| 50% | 50 per 100 | 0.5 | |
| 25% | 25 per 100 | 0.25 | |
| 10% | 10 per 100 | 0.1 | |
| 20% | 20 per 100 | 0.2 | |
| 6% | 6 per 100 | 0.06 |
2. Finding a percent of a quantity
Because p% means , finding p% of a quantity means **multiplying the quantity by **. For example, 30% of GH₵250 = × 250 = = GH₵75.
- 1Write the percent as a fraction over 100.
- 2Multiply by the quantity.
- 3Simplify and write the unit (GH₵, kg, pupils …).
Unitary method. You can also think "rate per 100": if 100% is GH₵700, then 1% is 700 ÷ 100 = GH₵7, and 35% is 35 × 7 = GH₵245. A third way uses 10%: 10% of 700 is 70, so 30% is 210, 5% is 35, and 35% = 210 + 35 = 245. All three methods agree.
Percent is really a proportion with 100 as the whole. To write a score of 18 out of 40 as a percent, find the equivalent fraction with 100 at the bottom: , so the score is 45%. In a table of equivalent ratios, the 100 column always gives the percent.
3. Commission, discount and interest
These three ideas all use "a percent of a quantity". What changes is which quantity is 100%.
| Situation | 100% is… | Working | Answer |
|---|---|---|---|
| 35% commission on sales of GH₵700 | the sales | × 700 | GH₵245 commission |
| 40% off a phone of GH₵450 | the marked price | × 450 = 180; 450 − 180 | pay GH₵270 |
| 6% interest on GH₵520 for one year | the savings | × 520 | GH₵31.20 interest |
So Naa pays GH₵270 for the phone, not GH₵410! A quicker way: 40% off means you pay 100% − 40% = 60% of the price, and 60% of 450 = 0.6 × 450 = 270.
4. Working backwards: the original price
Yaw paid GH₵80 for a shirt that was on sale at a 20% discount. What was the original price? The discount was 20% of the original price, so Yaw paid 100% − 20% = 80% of the original price. Now use the rate per 100 idea: 80% is GH₵80, so 1% is GH₵1 and 100% is GH₵100.
▶ Each fifth of the bar is 20% of the original price. Yaw's 4 fifths are worth GH₵80, so each fifth is GH₵20.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
A percent of a quantity
Show the solution
- Step 130% = .
- Step 2 × 250 = = 75.
Answer: GH₵75
Example 2
Commission
Show the solution
- Step 1Commission = 35% of 700 = × 700.
- Step 2700 ÷ 100 = 7, then 35 × 7 = 245.
Answer: GH₵245
Example 3
Sale price after a discount
Show the solution
- Step 1Discount = 40% of 450 = × 450 = 180.
- Step 2Sale price = 450 − 180 = GH₵270.
- Step 3Check: 60% of 450 = 0.6 × 450 = 270 ✓
Answer: GH₵270
Example 4
Finding the original price
Show the solution
- Step 1Yaw paid 100% − 20% = 80% of the original price.
- Step 280% = GH₵80, so 1% = 80 ÷ 80 = GH₵1.
- Step 3100% = 100 × 1 = GH₵100.
- Step 4Check: 20% of 100 = 20; 100 − 20 = 80 ✓
Answer: GH₵100
Example 5
Interest on savings
Show the solution
- Step 1Interest = 6% of 520 = × 520.
- Step 2520 ÷ 100 = 5.20, then 6 × 5.20 = 31.20.
Answer: GH₵31.20
Summary
- p% means p per 100, that is ; 100% is the whole quantity.
- p% of a quantity = × quantity (or find 1% first, then multiply by p).
- Commission, discount and interest are all a percent of a quantity; decide which amount is 100%.
- Sale price = marked price − discount; with x% off you pay (100 − x)% of the price.
- To find an original price, the amount paid is (100 − x)% of it: find 1%, then 100%.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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