The problem asks us to find the values of the variables y, t, w, and m in the given proportions.

1. Problem Description

The problem asks us to find the values of the variables y, t, w, and m in the given proportions.

2. Solution Steps

a. We are given the proportion $10 : y = 2 : 3$.

This can be written as $\frac{10}{y} = \frac{2}{3}$.

Cross-multiplying, we get $2y = 10 \times 3 = 30$.

Dividing by 2, we get $y = \frac{30}{2} = 15$.

b. We are given the proportion $20 : 25 = t : 5$.

This can be written as $\frac{20}{25} = \frac{t}{5}$.

Cross-multiplying, we get $25t = 20 \times 5 = 100$.

Dividing by 25, we get $t = \frac{100}{25} = 4$.

c. We are given the proportion $y : 15 = 9 : 45$.

This can be written as $\frac{y}{15} = \frac{9}{45}$.

Cross-multiplying, we get $45y = 15 \times 9 = 135$.

Dividing by 45, we get $y = \frac{135}{45} = 3$.

d. We are given the proportion $2 : w = 10 : 5$.

This can be written as $\frac{2}{w} = \frac{10}{5}$.

Cross-multiplying, we get $10w = 2 \times 5 = 10$.

Dividing by 10, we get $w = \frac{10}{10} = 1$.

e. We are given the proportion $72 : 27 = 8 : m$.

This can be written as $\frac{72}{27} = \frac{8}{m}$.

Cross-multiplying, we get $72m = 27 \times 8 = 216$.

Dividing by 72, we get $m = \frac{216}{72} = 3$.

3. Final Answer

a. $y = 15$

b. $t = 4$

c. $y = 3$

d. $w = 1$

e. $m = 3$