Solve 1 rac34 x ((4)/(9) + (1)/(3)) / (1(1)/(5) + 1(1)/(2)) using the rule of BODMAS.

$$1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$$ $$1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}$$ $$1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$$

The expression becomes:

$$\frac{7}{4} \times \left(\frac{4}{9} + \frac{1}{3}\right) \div \left(\frac{6}{5} + \frac{3}{2}\right)$$

$$\left(\frac{6}{5} + \frac{3}{2}\right)$$

The least common multiple of 5 and 2 is 10.

$$\left(\frac{6 \times 2}{5 \times 2} + \frac{3 \times 5}{2 \times 5}\right) = \left(\frac{12}{10} + \frac{15}{10}\right) = \frac{12+15}{10} = \frac{27}{10}$$

First, multiply.

$$\frac{7}{4} \times \frac{7}{9} = \frac{7 \times 7}{4 \times 9} = \frac{49}{36}$$

The expression becomes:

$$\frac{49}{36} \div \frac{27}{10}$$

$$\frac{49}{36} \times \frac{10}{27}$$

Simplify before multiplying: 10 and 36 are both divisible by 2.

$$\frac{49}{18 \times 2} \times \frac{5 \times 2}{27} = \frac{49 \times 5}{18 \times 27}$$ $$= \frac{245}{486}$$

The final answer is:

• $\boxed{\frac{245}{486}}$

.1.b: Using your knowledge on decimals, solve this: $\frac{4.2 \times 1.6 \times 3.4}{0.8 \times 3.2 \times 5.1}$

Since there are an equal number of decimal places in the numerator and denominator (one in each of the three numbers, totaling three decimal places for both), we can multiply the numerator and denominator by $10^3 = 1000$ to clear the decimals.

$$\frac{42 \times 16 \times 34}{8 \times 32 \times 51}$$

$$\frac{42 \times 16 \times 34}{8 \times 32 \times 51}$$

• Cancel 16 with 32: $\frac{16}{32} = \frac{1}{2}$. The expression becomes: $\frac{42 \times 1 \times 34}{8 \times 2 \times 51}$
• Cancel 34