Percentage, Discount, Profit and Loss (Senior 1 Mathematics): free lessons | Ganzaa

Percentage, Discount, Profit and Loss

Lessons

Senior 1 · Mathematics

Percentage, Discount, Profit and Loss

5 lessons

Lesson 1 of 5

Percentages

22 min

Let's begin

Keza scored 42 out of 50 in a maths test and 34 out of 40 in English. In which subject did she do better? The totals are different, so the raw scores cannot be compared directly. But if both are written "out of 100", the answer is clear: 84 in maths and 85 in English. Writing a number "out of 100" is what a percentage does, and it is why percentages appear on report cards, price tags, phone bills and tax receipts.

Definition

Percentage: a number of parts per hundred. The symbol $%$ means "out of 100", so $x\% = \frac{x}{100}.$

Example: $18\% = \frac{18}{100} = 0.18$ . The word comes from the Latin per centum, "for each hundred".

1. Converting between fractions, decimals and percentages

Rule. To write a fraction or a decimal as a percentage, multiply it by $100\%$ . To write a percentage as a decimal, divide by 100; as a fraction, write it over 100 and simplify.

Why. $100\% = \frac{100}{100} = 1$ , and multiplying by 1 does not change a number's value, only the way it is written.

Fraction	Decimal	Percentage
$\frac{1}{2}$	$0.5$	$50 %$
$\frac{1}{4}$	$0.25$	$25 %$
$\frac{3}{4}$	$0.75$	$75 %$
$\frac{1}{5}$	$0.2$	$20 %$
$\frac{1}{8}$	$0.125$	$12.5\%$
$\frac{1}{3}$	$0.333\ldots$	$33\tfrac{1}{3}\%$
$\frac{9}{50}$	$0.18$	$18 %$
$\frac{3}{2}$	$1.5$	$150\%$

A percentage can be more than 100%: $150\%$ of something is one and a half times it.

2. A percentage of a quantity

Important

$x\% \text{ of } Q = \frac{x}{100} \times Q$

"Of" means multiply. Change the percentage to a fraction or a decimal, then multiply by the quantity.

Worked example

Find (a) $18 %$ of 25,000 Frw; (b) $12.5\%$ of 64 kg; (c) $35 %$ of 2,400 students

(a) $\frac{18}{100} \times 25{,}000 = 0.18 \times 25{,}000 = 4{,}500$ Frw.

(b) $12.5\% = \frac{1}{8}$ , so $\frac{1}{8} \times 64 = 8$ kg.

3. One quantity as a percentage of another

Important

$\text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\%$

Both quantities must be in the same unit first.

Worked example

(a) Compare Keza's two test scores. (b) Express 45 minutes as a percentage of 2 hours

(a) Maths: $\frac{42}{50} \times 100\% = 84\%$ . English: $\frac{34}{40} \times 100\% = 85\%$ . She did slightly better in English.

(b) Same unit: 2 hours = 120 minutes. $\frac{45}{120} \times 100\% = 37.5\%$ .

4. Finding the whole from a part

If you know a percentage of a quantity, you can find the whole quantity: find $1 %$ first, then multiply by 100.

Worked example

Ishimwe has used 1.2 GB of his monthly data bundle. His phone says this is $40 %$ of the bundle. How big is the bundle, and how much is left?

$40 %$ is 1.2 GB, so $1 %$ is $1.2 \div 40 = 0.03$ GB and $100\%$ is $0.03 \times 100 = 3$ GB. The bundle is 3 GB. What is left is $60 %$ : $3 - 1.2 = 1.8$ GB.

As one equation: $0.4 \times B = 1.2$ , so $B = \frac{1.2}{0.4} = 3$ .

Warning

A percentage of what? A percentage always refers to a whole. If 30 of 40 students passed, the passes are $\frac{30}{40} = 75\%$ of the class. But 10 students failed, so the same 30 passes are $\frac{30}{10} = 300\%$ of the failures. The same number gives two different percentages, depending on the whole. Before you divide, say aloud what the whole is, and put it underneath.

Warning

$0.5 %$ is not $50 %$ . $0.5\% = \frac{0.5}{100} = 0.005$ , half of one per cent. Moving the decimal point by the wrong number of places is the most common slip. Estimate first: $0.5 %$ of 20,000 Frw must be small, and it is 100 Frw.

Tip

Mental percentages. $10 %$ : divide by 10. $5 %$ : half of $10 %$ . $1 %$ : divide by 100. Build the rest: $15\% = 10\% + 5\%$ , and $18\% = 10\% + 5\% + 3 \times 1\%$ . For 25,000 Frw: $2,500 + 1,250 + 750 = 4,500$ Frw.

Math Antics (mathantics): "What Are Percentages?". Watch how a percentage is shown as part of a hundred-square, and how it links to fractions and decimals.

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Practice

Try the first four without a calculator.

Write $35 %$ as a decimal and as a fraction in its simplest form.
Write $\frac{3}{8}$ and $0.06$ as percentages.
Find $15 %$ of 6,000 Frw.
Find $2.5 %$ of 80 kg.
In a school of 1,250 students, 450 are boarders. What percentage are boarders?
Express 750 m as a percentage of 3 km.
A farmer has harvested 6 tonnes of maize, which is $75 %$ of her expected harvest. What harvest does she expect?
Uwase says that 15 out of 60 is a better result than 12 out of 40 "because 15 is bigger". Is she right?

Show answersHide answers

$0.35 = \frac{35}{100} = \frac{7}{20}$
$37.5\%$ and $6 %$
$600 + 300 = 900$ Frw
$0.025 \times 80 = 2$ kg
$\frac{450}{1{,}250} \times 100\% = 36\%$
$\frac{750}{3{,}000} \times 100\% = 25\%$
$1\% = 6 \div 75 = 0.08$ t, so $100\% = 8$ tonnes.
No. $\frac{15}{60} = 25\%$ but $\frac{12}{40} = 30\%$ , so 12 out of 40 is better.

Sources and credits

6.1: Understand Percent, from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
6.2: Solve General Applications of Percent, from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Math Antics - What Are Percentages?, mathantics, embedded from YouTube.

The test-score and data-bundle examples, the hundred square, the percentage bar, the practice items and the quiz questions were written by Ganzaa.

11. Converting between fractions, decimals and percentages

22. A percentage of a quantity

33. One quantity as a percentage of another

44. Finding the whole from a part

1. Converting between fractions, decimals and percentages

2. A percentage of a quantity

3. One quantity as a percentage of another

4. Finding the whole from a part