Mathematics (Syllabus D) 4024/11 — May/June 2019

Approach

To add two fractions with different denominators, we must first find a common denominator. The least common multiple (LCM) of $9$ and $5$ is used to convert each fraction into an equivalent form.

Working

The denominators are $9$ and $5$. Since they have no common factors other than $1$, their LCM is:

$$\text{LCM}(9, 5) = 45$$

Convert each fraction to have a denominator of $45$:

$$\frac{4}{9} = \frac{4 \times 5}{9 \times 5} = \frac{20}{45}$$ $$\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}$$

Add the two fractions:

$$\frac{20}{45} + \frac{18}{45} = \frac{20 + 18}{45} = \frac{38}{45}$$

Check if the result can be simplified. The factors of $38$ are $1, 2, 19, 38$. None of these divide $45$ evenly. Therefore, the fraction is in its simplest form.

Answer

$$\frac{38}{45}$$

Approach

According to the order of operations (often remembered as BODMAS or PEMDAS), division must be performed before addition. We first evaluate $0.6 \div 0.02$, then add $1$ to the result.

Working

Evaluate the division part first: $0.6 \div 0.02$.

To make this easier to calculate by hand, we can eliminate the decimals by multiplying both the dividend and the divisor by $100$:

$$0.6 \div 0.02 = \frac{0.6}{0.02}$$

Multiply numerator and denominator by $100$:

$$\frac{0.6 \times 100}{0.02 \times 100} = \frac{60}{2}$$

Perform the division:

$$\frac{60}{2} = 30$$

Now substitute this back into the original expression and perform the addition:

$$1 + 30 = 31$$

Answer

$$31$$