Find the ratio of 450 mL to 1.8 L in simplest form. Are the ratios 6 : 8 and 15 : 20 in proportion? Justify your answer.

To find the ratio, both quantities must be in the same unit.

1.8 L = 1.8 × 1000 = 1800 mL.

$\Rightarrow$ 450 mL: 1.8 L = 450 : 1800

= $\dfrac{450}{1800}$

HCF of 450 and 1800 :

$\begin{array}{l|r} 2 & 450 \ \hline 3 & 225 \ \hline 3 & 75 \ \hline 5 & 25 \ \hline 5 & 5 \ \hline & 1 \end{array} \quad \text{ and } \quad \begin{array}{l|r} 2 & 1800 \ \hline 2 & 900 \ \hline 2 & 450 \ \hline 3 & 225 \ \hline 3 & 75 \ \hline 5 & 25 \ \hline 5 & 5 \ \hline & 1 \end{array}$

⇒ 450 = 2 × 3² × 5²

⇒ 1800 = 2³ × 3² × 5²

⇒ HCF = 2 × 3² × 5² = 2 × 9 × 25 = 450

The HCF of 450 and 1800 is 450.

$\Rightarrow \dfrac{450 \div 450}{1800 \div 450}$

= $\dfrac{1}{4}$

= 1 : 4.

To check whether 6 : 8 and 15 : 20 are in proportion, we use cross multiplication :

Product of extremes = 6 × 20 = 120.

Product of means = 8 × 15 = 120.

Since the two products are equal (120 = 120), the ratios are in proportion.

Hence, the ratio is 1 : 4, and the ratios 6 : 8 and 15 : 20 are in proportion.