Arithmetic

Lessons

Senior 4 · Basic Mathematics

Arithmetic

5 lessons

Lesson 1 of 5

Negative numbers, decimals and fractions

25 min

A tailor's half-year

Ingabire runs a small tailoring shop. She records each month's result: a profit as a positive number, a loss as a negative one. For the first half of the year her records read, in Frw (illustrative):

Month	Jan	Feb	Mar	Apr	May	Jun
Result (Frw)	$+45\,000$	$-12\,000$	$+30\,500$	$-8\,250$	$+17\,750$	$-3\,000$

Did the shop make money over the six months, and how much on average per month? To answer, she must add negative numbers, divide, and round a decimal. In the same shop she cuts cloth into pieces of $\frac{3}{4}$ of a metre, which is a question about fractions. This lesson makes sure these three kinds of number hold no surprises.

Definition

Integers are the whole numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ . The negative numbers are those less than 0, written with a minus sign: $- 5$ is read "negative five". On a number line they lie to the left of 0, so further left means smaller: $-7 \lt -2$ .

Example: a loss of 12 000 Frw is $-12\,000$ ; a temperature 4 degrees below zero is $-4^\circ\mathrm{C}$ .

Calculating with negative numbers

Adding a positive number moves right on the number line; subtracting it moves left. Subtracting a negative number is the same as adding its opposite: $5 - (- 3) = 5 + 3 = 8$ (removing a debt of 3 makes you 3 richer).

For multiplication and division, only the signs need a rule:

Signs	Result	Examples
same signs	positive	$(-4) \times (-5) = 20$ , $\ (-18) \div (-3) = 6$
different signs	negative	$(-4) \times 5 = -20$ , $\ 18 \div (-3) = -6$

Worked example 1: the tailor's six months

Find Ingabire's total result and her mean monthly result, to the nearest franc.

Solution

Add the profits and the losses separately, then combine:

45\,000 + 30\,500 + 17\,750 = 93\,250, \qquad -12\,000 - 8\,250 - 3\,000 = -23\,250,

\text{total} = 93\,250 + (-23\,250) = 70\,000 \text{ Frw}.

Mean per month: $\frac{70\,000}{6} = 11\,666.66\ldots \approx 11\,667$ Frw. The shop made a profit of 70 000 Frw, an average of about 11 667 Frw a month.

Warning

Order of operations and powers of negatives. Work in the order brackets, powers, then $\times$ and $\div$ (left to right), then $+$ and $-$ (left to right). Watch the brackets on powers: $(- 3)^{2} = (- 3) (- 3) = 9$ , but $- 3^{2} = - (3^{2}) = - 9$ . And $2 + 3 \times 4 = 14$ , not 20.

Decimals

A decimal writes tenths, hundredths, thousandths, … after a decimal point: $4.375 = 4 + \frac{3}{10} + \frac{7}{100} + \frac{5}{1000}$ . Multiply decimals as whole numbers, then place the point so the answer has as many decimal places as the two factors together: $1.2 \times 0.35 = 0.420$ .

Rounding. To round to a number of decimal places, count places after the point; to round to a number of significant figures (s.f.), count from the first non-zero digit. Look at the next digit: 5 or more rounds up. So $0.0045678 \approx 0.0046$ (2 s.f.) $\approx 0.005$ (3 decimal places).

Fractions to decimals. Divide: $\frac{3}{8} = 0.375$ (a terminating decimal), $\frac{2}{3} = 0.666\ldots$ (a recurring decimal, written $0.\overline{6}$ ).

Worked example 2: decimals at the market, and a recurring decimal

(a) A trader buys 12.5 kg of beans at 850 Frw per kg and 3.75 kg of rice at 1 240 Frw per kg. What does she pay? (b) Write $0.\overline{27} = 0.272727\ldots$ as a fraction in its simplest form.

Solution

$12.5 \times 850 = 10\,625$ and $3.75 \times 1\,240 = 4\,650$ , so she pays $10\,625 + 4\,650 = 15\,275$ Frw.
Let $x = 0.2727\ldots$ . Two digits repeat, so multiply by 100: $100x = 27.2727\ldots$ . Subtract: $99 x = 27$ , so $x = \frac{27}{99} = \frac{3}{11}$ .

Fractions

A fraction $\frac{a}{b}$ ( $b \ne 0$ ) means $a$ parts out of $b$ equal parts. Dividing top and bottom by a common factor gives an equal fraction in simplest form: $\frac{18}{24} = \frac{3}{4}$ .

Operation	Rule	Example
Add or subtract	Use a common denominator (the LCM of the denominators)	$\frac{2}{5} + \frac{1}{4} = \frac{8}{20} + \frac{5}{20} = \frac{13}{20}$
Multiply	Multiply tops and multiply bottoms	$\frac{2}{3} \times \frac{9}{10} = \frac{18}{30} = \frac{3}{5}$
Divide	Multiply by the reciprocal of the divisor	$\frac{3}{4} \div \frac{5}{8} = \frac{3}{4} \times \frac{8}{5} = \frac{6}{5}$
Mixed numbers	Change to improper fractions first	$2\tfrac{1}{2} = \frac{5}{2}$

Worked example 3: sharing a profit

A cooperative shares its yearly profit of 1 200 000 Frw: $\frac{2}{5}$ goes to members as dividends, $\frac{1}{4}$ to savings, and the rest buys equipment. What fraction buys equipment, and how much money is that?

Solution

1 - \frac{2}{5} - \frac{1}{4} = \frac{20}{20} - \frac{8}{20} - \frac{5}{20} = \frac{7}{20}, \qquad \frac{7}{20} \times 1\,200\,000 = 420\,000 \text{ Frw}.

Worked example 4: cutting cloth

Ingabire has $6\tfrac{1}{2}$ m of cloth and cuts pieces of $\frac{3}{4}$ m. How many whole pieces does she get, and how much cloth is left?

Solution

6\tfrac{1}{2} \div \frac{3}{4} = \frac{13}{2} \times \frac{4}{3} = \frac{52}{6} = \frac{26}{3} = 8\tfrac{2}{3} .

She gets 8 whole pieces. They use $8 \times \frac{3}{4} = 6$ m, so $\frac{1}{2}$ m is left (which is the $\frac{2}{3}$ of a piece: $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$ ).

Tip

Before any long calculation, estimate. $12.5 \times 850$ is about $12 \times 900 = 10\,800$ , so an answer of 1 062.5 or 106 250 must be a slip with the decimal point. In fraction problems, "of" means multiply: $\frac{2}{5}$ of 1 200 000 is $\frac{2}{5} \times 1\,200\,000$ .

Math with Mr. J: Fraction Review, how to add, subtract, multiply and divide fractions. Watch for the common denominator in adding and the reciprocal in dividing.

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Practice

Try each one before opening the answers. They get harder as you go.

Calculate: (a) $- 7 + 12 - (- 5)$ (b) $(-4) \times (-6) \div (-3)$ (c) $- 2^{3} + (- 2)^{2}$ .
At dawn on a mountain the temperature was $-5^\circ\mathrm{C}$ . It rose by $11^\circ\mathrm{C}$ by noon and fell by $8^\circ\mathrm{C}$ by night. What was the night temperature?
Round $0.0076849$ to (a) 3 decimal places (b) 2 significant figures.
Write $0.\overline{36} = 0.3636\ldots$ as a fraction in simplest form.
Calculate $2\tfrac{1}{3} + \frac{3}{4} - 1\tfrac{1}{6}$ .
A farmer plants beans on $\frac{3}{8}$ of her land and maize on $\frac{2}{5}$ of the rest. What fraction of the land is maize, and what fraction is left?
Strips of $\frac{2}{3}$ m are cut from $7\tfrac{1}{2}$ m of cloth. How many whole strips are there, and how much cloth is left?

Show answersHide answers

(a) 10 (b) $24 \div (-3) = -8$ (c) $- 8 + 4 = - 4$ .
$-5 + 11 - 8 = -2^\circ\mathrm{C}$ .
(a) 0.008 (b) 0.0077.
$100 x - x = 36$ , so $x = \frac{36}{99} = \frac{4}{11}$ .
$\frac{28}{12} + \frac{9}{12} - \frac{14}{12} = \frac{23}{12} = 1\tfrac{11}{12}$ .
The rest is $\frac{5}{8}$ ; maize $= \frac{2}{5} \times \frac{5}{8} = \frac{1}{4}$ . Left: $1 - \frac{3}{8} - \frac{1}{4} = \frac{3}{8}$ .
$\frac{15}{2} \div \frac{2}{3} = \frac{45}{4} = 11\tfrac{1}{4}$ : 11 strips, using $\frac{22}{3} = 7\tfrac{1}{3}$ m, so $\frac{1}{6}$ m is left.

Quick check

What is (−6) × (−3) ÷ (−2)?

Key points

Subtracting a negative adds; same signs multiply or divide to a positive, different signs to a negative.
Round by decimal places or significant figures; recurring decimals become fractions by multiplying by 10, 100, ….
Add fractions over a common denominator; divide by multiplying by the reciprocal; "of" means multiply.
Estimate first to catch slips with the decimal point.

Sources and credits

Written by Ganzaa: the explanations, the diagram, the worked examples, the practice items and the questions. The unit follows the topic list of REB's S4 Basic Mathematics (2026–2027); no REB text is reproduced. The shop, market and cooperative figures are illustrative.
Photo: Tailor Lady, IGANZE, Wikimedia Commons, CC BY-SA 4.0.
Video: Fraction Review: How to Add, Subtract, Multiply, and Divide Fractions, Math with Mr. J, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1A tailor's half-year

2Calculating with negative numbers

3Decimals

4Fractions

A tailor's half-year

Calculating with negative numbers

Decimals

Fractions