1.3 Operations

1. Integer Operations

Integer arithmetic follows strict sign rules. Mastering these first makes later decimal and algebra work much easier.

Operation	Rule	Example
Add (Same Signs)	Add values, keep the sign.	$(- 3) + (- 5) = - 8$
Add (Diff Signs)	Subtract smaller from larger, keep sign of larger.	$(- 7) + 3 = - 4$
Subtract	Change to adding the opposite.	$5 - (- 2) = 5 + 2 = 7$
Multiply/Divide	Same signs positive, different signs negative.	$(- 3) \times 4 = - 12$

AddAddition Rule
Same sign: add and keep the sign. Different sign: subtract and keep the sign of the larger absolute value.
SubSubtraction Rule
Convert subtraction to addition of the opposite: $a - b = a + (- b)$ .
MulMultiplication Rule
Same signs give positive, different signs give negative.
DivDivision Rule
Same sign quotient is positive; different sign quotient is negative.

Sign Guide

(+) \times (+) = (+)

(-) \times (-) = (+)

(+) \times (-) = (-)

(-) \times (+) = (-)

The same sign rules apply to division as well.

2. Decimal Operations

Action	Key Rule	Important Note
Add / Subtract	Align decimal points vertically	Fill empty places with zeros (e.g., 3.5 → 3.50)
Multiply	Ignore decimals, multiply, count total places	If $0.2 \times 0.03$ , total 3 places → 0.006
Divide	Shift divisor decimal to make it whole	Shift dividend decimal by same amount

Add & Subtract

Always align decimal points vertically before calculating.

12.50 + 3.75 = 16.25

Multiply

Multiply as integers first, then place decimal using total decimal places.

0.5 × 0.3 = 0.15

Divide

Shift decimal in divisor to make it whole; shift dividend equally.

3.6 ÷ 0.2 = 36 ÷ 2 = 18

Worked Examples

Integers: Addition

Calculate

15 + 27

Solution:

15 + 27 = 42

Evaluate

- 8 + 12

Solution:

Start at -8 on the number line and move 12 steps right.

- 8 + 12 = 4

Find the value of

- 15 + (- 7)

Solution:

Adding a negative is the same as subtracting.

- 15 - 7 = - 22

Calculate

123 + 456

Solution:

Column addition:

123

+ 456

-----

579

The temperature is -5°C. It rises by 8°C. What is the new temperature?

Solution:

- 5 + 8 = 3

°C.

A diver is at -20m. He ascends 15m. What is his new depth?

Solution:

- 20 + 15 = - 5

m (5m below surface).

Integers: Subtraction

Calculate

50 - 23

Solution:

50 - 23 = 27

Evaluate

12 - 20

Solution:

12 - 20 = - 8

Find the value of

- 5 - 12

Solution:

Start at -5 and move 12 steps left.

- 5 - 12 = - 17

Calculate

15 - (- 8)

Solution:

Subtracting a negative becomes addition.

15 + 8 = 23

The temperature drops from 5°C to -3°C. What is the difference?

Solution:

Difference = Final - Initial or Large - Small.

5 - (- 3) = 5 + 3 = 8

°C drop.

Calculate

- 12 - (- 20)

Solution:

- 12 + 20 = 8

Integers: Multiplication

Calculate

12 \times 11

Solution:

12 \times 10 = 120

12 \times 1 = 12

. Total = 132.

Evaluate

- 5 \times 9

Solution:

Negative × Positive = Negative.

- 5 \times 9 = - 45

Find the value of

- 8 \times - 7

Solution:

Negative × Negative = Positive.

8 \times 7 = 56

Calculate

15 \times - 4

Solution:

15 \times 4 = 60

. Result is negative: -60.

Calculate

(- 3)^{2}

Solution:

(- 3) \times (- 3) = 9

Evaluate

(- 2)^{3}

Solution:

(- 2) \times (- 2) \times (- 2) = 4 \times (- 2) = - 8

Integers: Division

Calculate

48 \div 6

Solution:

48 \div 6 = 8

Evaluate

- 56 \div 8

Solution:

Negative ÷ Positive = Negative.

- 56 \div 8 = - 7

Find the value of

- 36 \div - 4

Solution:

Negative ÷ Negative = Positive.

36 \div 4 = 9

Calculate

100 \div - 5

Solution:

100 \div 5 = 20

. Result is negative: -20.

Calculate

\frac{- 144}{12}

Solution:

144 \div 12 = 12

. Result is -12.

Evaluate

\frac{- 20}{- 5}

Solution:

20 \div 5 = 4

Decimals: Addition & Subtraction

Calculate

3.5 + 4.2

Solution:

3.5 + 4.2 = 7.7

Evaluate

12.5 + 3.75

Solution:

Align decimal points:

12.50

+ 3.75

-------

16.25

Calculate

15 - 3.2

Solution:

15.0

- 3.2

-------

11.8

Find the value of

0.05 + 0.12

Solution:

0.17

Evaluate

1.2 - 0.08

Solution:

1.20

- 0.08

------

1.12

Calculate

100 - 0.01

Solution:

99.99

Decimals: Multiplication

Calculate

0.5 \times 0.3

Solution:

5 \times 3 = 15

. Two decimal places total. 0.15.

Evaluate

1.2 \times 4

Solution:

12 \times 4 = 48

. One decimal place. 4.8.

Find the value of

0.02 \times 0.04

Solution:

2 \times 4 = 8

. Four decimal places total. 0.0008.

Calculate

1.5 \times 1.5

Solution:

15 \times 15 = 225

. Two decimal places. 2.25.

Calculate

300 \times 0.02

Solution:

3 \times 2 = 6

. Zeros offset decimals.

300 \times 0.02 = 3 \times 2 = 6

Evaluate

0.1 \times 0.1 \times 0.1

Solution:

0.001.

Decimals: Division

Calculate

3.6 \div 0.2

Solution:

Multiply both by 10 to clear decimal.

36 \div 2 = 18

Evaluate

0.15 \div 0.03

Solution:

Multiply by 100.

15 \div 3 = 5

Calculate

12 \div 0.5

Solution:

Dividing by 0.5 is same as multiplying by 2.

12 \times 2 = 24

Find value of

0.008 \div 0.02

Solution:

Multiply by 1000/1000 or simplify.

0.8 \div 2 = 0.4

Calculate

5 \div 0.001

Solution:

5 \times 1000 = 5000

Evaluate

0.4 \div 0.08

Solution:

Multiply by 100.

40 \div 8 = 5

3. Fraction Operations

Operation	Method	Formula
Addition / Subtraction	Find common denominator (LCD).	$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$
Multiplication	Multiply tops together, multiply bottoms together.	$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$
Division	KFC: Keep first, Flip second, Change to multiply.	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Add & Subtract

Find a common denominator first. Only add/subtract the numerators.

$\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

Multiply

Multiply numerators together and denominators together. Simplify at the end.

$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

Divide

KFC: Keep the first fraction, Flip the second, Change divide to multiply.

$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$

Worked Examples: Fractions

Fractions: Addition & Subtraction

Calculate

\frac{1}{5} + \frac{2}{5}

Solution:

\frac{1 + 2}{5} = \frac{3}{5}

Evaluate

\frac{1}{3} + \frac{1}{4}

Solution:

Common denominator (LCM of 3 and 4) is 12.

\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

Find the value of

1 \frac{1}{2} + 2 \frac{1}{3}

Solution:

Add integers:

1 + 2 = 3

Add fractions:

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Total:

3 \frac{5}{6}

Calculate

\frac{5}{6} - \frac{1}{2}

Solution:

Common denominator is 6.

\frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}

Evaluate

3 \frac{1}{4} - 1 \frac{3}{4}

Solution:

Convert to improper fractions:

\frac{13}{4} - \frac{7}{4} = \frac{6}{4} = \frac{3}{2} = 1 \frac{1}{2}

Calculate

2 - \frac{3}{5}

Solution:

2 = \frac{10}{5}

\frac{10}{5} - \frac{3}{5} = \frac{7}{5} = 1 \frac{2}{5}

Fractions: Multiplication

Calculate

\frac{2}{3} \times \frac{4}{5}

Solution:

Multiply tops:

2 \times 4 = 8

. Multiply bottoms:

3 \times 5 = 15

. Result:

\frac{8}{15}

Evaluate

5 \times \frac{2}{7}

Solution:

\frac{5}{1} \times \frac{2}{7} = \frac{10}{7} = 1 \frac{3}{7}

Find the value of

1 \frac{1}{2} \times 2 \frac{1}{3}

Solution:

Convert to improper:

\frac{3}{2} \times \frac{7}{3}

= \frac{21}{6} = \frac{7}{2} = 3 \frac{1}{2}

Calculate

(\frac{2}{5})^{2}

Solution:

\frac{2}{5} \times \frac{2}{5} = \frac{4}{25}

Evaluate

\frac{3}{4} of 20

Solution:

"Of" means multiply.

\frac{3}{4} \times 20 = \frac{60}{4} = 15

Find area of rectangle with sides

\frac{2}{3}

m and

\frac{3}{4}

Solution:

Area = \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}

m².

Fractions: Division

Calculate

\frac{2}{3} \div \frac{4}{5}

Solution:

Keep, Change, Flip (KCF).

\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

Evaluate

4 \div \frac{1}{2}

Solution:

4 \times \frac{2}{1} = 8

Calculate

\frac{3}{4} \div 2

Solution:

\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}

Find value of

2 \frac{1}{2} \div 1 \frac{1}{4}

Solution:

Convert to improper:

\frac{5}{2} \div \frac{5}{4}

Flip second fraction:

\frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2

Evaluate

\frac{1 / 2}{3 / 4}

Solution:

Same as

\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}

A rope is

3 \frac{3}{4}

m long. Cut into pieces of

\frac{3}{4}

m. How many pieces?

Solution:

3 \frac{3}{4} = \frac{15}{4}

\frac{15}{4} \div \frac{3}{4} = \frac{15}{4} \times \frac{4}{3} = \frac{60}{12} = 5

pieces.

4. Order of Operations

When multiple operations appear in one expression, follow the order of BIDMAS (or BODMAS/PEMDAS).

Order	Letter	Operation	Example
1	B	Brackets	$2 \times (3 + 4) = 14$
2	I/O	Indices / Orders	$5 + 2^{2} = 9$
3	D/M	Division & Multiplication	Left to Right: $10 \div 2 \times 5 = 25$
4	A/S	Addition & Subtraction	Left to Right: $10 - 4 + 2 = 8$

Common Mistake: Thinking Addition comes before Subtraction. They are equal! Always work from Left to Right for D/M and A/S.

Worked Examples: Order of Operations

Structured Questions (Past-paper style) — Questions 1–10

a) Calculate:

8 + 12 \div 3

b) Calculate:

(8 + 12) \div 3

c) Calculate:

20 - 4 \times 3 + 2

d) Explain why the answers in parts (a) and (b) are different.

Solution:

12 \div 3 = 4

, so

8 + 4 = 12

8 + 12 = 20

, so

20 \div 3 = 6 \frac{2}{3}

4 \times 3 = 12

, so

20 - 12 + 2 = 8 + 2 = 10

d) Brackets change the order by forcing addition first.

a) Calculate:

5 + 2^{3} \times 4

b) Calculate:

(5 + 2^{3}) \times 4

c) Calculate:

18 - 3^{2} + 6 \div 3

d) State which operation is carried out first in part (c) and why.

Solution:

2^{3} = 8

8 \times 4 = 32

, so

5 + 32 = 37

5 + 8 = 13

, so

13 \times 4 = 52

3^{2} = 9

6 \div 3 = 2

, so

18 - 9 + 2 = 11

d) Powers (

3^{2}

) come before multiplication/division/addition/subtraction.

a) Calculate:

6.4 - 1.2 \times 2

b) Calculate:

(6.4 - 1.2) \times 2

c) Calculate:

3.6 \div 0.3 + 1.4

d) Explain the role of brackets in part (b).

Solution:

1.2 \times 2 = 2.4

, so

6.4 - 2.4 = 4.0

6.4 - 1.2 = 5.2

, so

5.2 \times 2 = 10.4

3.6 \div 0.3 = 12

, so

12 + 1.4 = 13.4

d) Brackets make subtraction happen before multiplication.

a) Calculate:

\frac{1}{2} + \frac{3}{4} \times 2

b) Calculate:

(\frac{1}{2} + \frac{3}{4}) \times 2

c) Calculate:

\frac{5}{6} - \frac{1}{3} \times \frac{3}{5}

d) State which operation is performed first in part (c).

Solution:

\frac{3}{4} \times 2 = \frac{6}{4} = \frac{3}{2}

, so

\frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2

\frac{2}{4} + \frac{3}{4} = \frac{5}{4}

, so

\frac{5}{4} \times 2 = \frac{10}{4} = \frac{5}{2}

\frac{1}{3} \times \frac{3}{5} = \frac{1}{5}

, so

\frac{5}{6} - \frac{1}{5} = \frac{25 - 6}{30} = \frac{19}{30}

d) Multiplication (

\frac{1}{3} \times \frac{3}{5}

a) Calculate:

2 + \frac{3}{5} \times 10

b) Calculate:

(2 + \frac{3}{5}) \times 10

c) Calculate:

4 - \frac{1}{2} \div \frac{1}{4}

d) Explain why part (c) gives a negative answer.

Solution:

\frac{3}{5} \times 10 = 6

. Total:

2 + 6 = 8

2 + \frac{3}{5} = \frac{13}{5}

. Total:

\frac{13}{5} \times 10 = 26

\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2

. Total:

4 - 2 = 2

d) Division (

\frac{1}{2} \div \frac{1}{4} = 2

) is done first, giving a value that is subtracted from 4.

a) Calculate:

- 6 + 4 \times 5

b) Calculate:

(- 6 + 4) \times 5

c) Calculate:

- 12 \div 3^{2} + 1

d) Explain how powers affect the order of operations in part (c).

Solution:

4 \times 5 = 20

, so

- 6 + 20 = 14

- 6 + 4 = - 2

, so

- 2 \times 5 = - 10

3^{2} = 9

, then

- 12 \div 9 = - \frac{4}{3}

, so

- \frac{4}{3} + 1 = - \frac{1}{3}

d) The power is evaluated before division.

a) Calculate:

15 - 9 \div 3 \times 2

b) Calculate:

(15 - 9) \div 3 \times 2

c) Calculate:

10 + 6^{2} \div 3 - 4

d) Write down the order in which the operations are carried out in part (c).

Solution:

9 \div 3 = 3

3 \times 2 = 6

, so

15 - 6 = 9

15 - 9 = 6

6 \div 3 = 2

, so

2 \times 2 = 4

6^{2} = 36

36 \div 3 = 12

, so

10 + 12 - 4 = 18

d) Power → Division → Addition → Subtraction.

a) Calculate:

0.5 + 2^{2} \times 0.3

b) Calculate:

(0.5 + 2^{2}) \times 0.3

c) Calculate:

\frac{3}{4} + 1.6 \div 0.4

d) Explain why division is done before addition in part (c).

Solution:

2^{2} = 4

4 \times 0.3 = 1.2

, so

0.5 + 1.2 = 1.7

0.5 + 4 = 4.5

, so

4.5 \times 0.3 = 1.35

1.6 \div 0.4 = 4

, so

\frac{3}{4} + 4 = 4 \frac{3}{4} = 4.75

d) Division has higher priority than addition.

a) Evaluate:

24 \div 6 + 2^{2}

b) Evaluate:

24 \div (6 + 2)^{2}

c) Which expression, (a) or (b), gives the larger value?

d) Explain how brackets and powers change the result.

Solution:

24 \div 6 = 4

2^{2} = 4

, so

4 + 4 = 8

6 + 2 = 8

8^{2} = 64

, so

24 \div 64 = \frac{24}{64} = \frac{3}{8} = 0.375

c) Part (a) is larger (8 vs 0.375).

d) Brackets and powers greatly change the order of calculation.

10.

A taxi fare is calculated using the rule:

Fare = 4 + 1.5 \times d + 2^{2}

where

d

is the distance in km.

a) Find the fare when

d = 6

b) Find the fare when

d = 10

c) Explain why

2^{2}

is calculated before multiplication.

d) State where brackets could be added to change the fare calculation.

Solution:

4 + 1.5 (6) + 4 = 4 + 9 + 4 = 17

4 + 1.5 (10) + 4 = 4 + 15 + 4 = 23

c) Powers (

2^{2}

) are done before multiplication.

d) For example:

(4 + 1.5) \times d + 2^{2}