John's earnings increased by 12% to Sh. 5600. The ratio of John's initial earnings to Musa's initial earnings was 5:3. The sum of their new earnings is Sh. 9600. What is the percentage change in Musa'

Let John's initial earnings be $J_0$. His earnings increased by $12\%$, and his new earnings ($J_{new}$) are Sh. 5600.

$$J_{new} = J_0 \times (1 + 0.12)$$ $$5600 = J_0 \times 1.12$$ $$J_0 = \frac{5600}{1.12}$$ $$J_0 = 5000 \text{ Sh.}$$

The ratio of John's initial earnings to Musa's initial earnings ($M_0$) is $5:3$.

$$\frac{J_0}{M_0} = \frac{5}{3}$$

Substitute $J_0 = 5000$:

$$\frac{5000}{M_0} = \frac{5}{3}$$ $$5 M_0 = 5000 \times 3$$ $$5 M_0 = 15000$$ $$M_0 = \frac{15000}{5}$$ $$M_0 = 3000 \text{ Sh.}$$

The sum of their new earnings is Sh. 9600. We know John's new earnings ($J_{new}$) are Sh. 5600.
Let Musa's new earnings be $M_{new}$.

$$J_{new} + M_{new} = 9600$$ $$5600 + M_{new} = 9600$$ $$M_{new} = 9600 - 5600$$ $$M_{new} = 4000 \text{ Sh.}$$

The percentage change is calculated as $\frac{\text{New earnings} - \text{Initial earnings}}{\text{Initial earnings}} \times 100\%$.

$$\text{Percentage change} = \frac{M_{new} - M_0}{M_0} \times 100\%$$ $$\text{Percentage change} = \frac{4000 - 3000}{3000} \times 100\%$$ $$\text{Percentage change} = \frac{1000}{3000} \times 100\%$$ $$\text{Percentage change} = \frac{1}{3} \times 100\%$$ $$\text{Percentage change} = 33\frac{1}{3}\%$$ $$\text{Percentage change} \approx 33.33\%$$

The percentage change in Musa's earnings is:

$$\mathbf{33\frac{1}{3}\% \text{ or } 33.33\%}$$