Question

In September, the incomes of $K$, $A$, and $V$ are in the ratio $8:6:5$. They rent a house, contributing $15\%$, $12\%$, and $18\%$ of their respective incomes. In October, the rent remains the same, but the salaries of $K$, $A$, and $V$ increase by $10\%$, $12\%$, and $15\%$, respectively. What percentage of their total income is paid as house rent in October? (Round to the nearest percentage.)

Answer 1

Step 1: Represent the incomes in September
Let the incomes of $K$, $A$, and $V$ in September be:
$8x, \, 6x, \, \text{and} \, 5x,$
where $x$ is a common factor.
Step 2: Calculate the rent paid by each
Rent paid by $K$:
$15\%$ of $8x$:
$\text{Rent}_K = 0.15 \times 8x = 1.2x.$
Rent paid by $A$:
$12\%$ of $6x$:
$\text{Rent}_A = 0.12 \times 6x = 0.72x.$
Rent paid by $V$:
$18\%$ of $5x$:
$\text{Rent}_V = 0.18 \times 5x = 0.9x.$
The total rent for the house is:
$\text{Total Rent} = \text{Rent}_K + \text{Rent}_A + \text{Rent}_V,$
$\text{Total Rent} = 1.2x + 0.72x + 0.9x = 2.82x.$
Step 3: Calculate the October incomes
Income of $K$ in October:
$10\%$ increase in $8x$:
$\text{Income}_K^\text{Oct} = 8x + 0.1 \times 8x = 8x \times 1.1 = 8.8x.$
Income of $A$ in October:
$12\%$ increase in $6x$:
$\text{Income}_A^\text{Oct} = 6x + 0.12 \times 6x = 6x \times 1.12 = 6.72x.$
Income of $V$ in October:
$15\%$ increase in $5x$:
$\text{Income}_V^\text{Oct} = 5x + 0.15 \times 5x = 5x \times 1.15 = 5.75x.$
The total income in October is:
$\text{Total Income}_\text{Oct} = \text{Income}_K^\text{Oct} + \text{Income}_A^\text{Oct} + \text{Income}_V^\text{Oct},$
$\text{Total Income}_\text{Oct} = 8.8x + 6.72x + 5.75x = 21.27x.$
Step 4: Calculate the percentage of income spent on rent
The percentage of the total income spent on rent in October is:
$\text{Percentage} = \frac{\text{Total Rent}}{\text{Total Income}_\text{Oct}} \times 100.$
Substitute the values:
$\text{Percentage} = \frac{2.82x}{21.27x} \times 100 = \frac{2.82}{21.27} \times 100.$
Simplify:
$\text{Percentage} \approx 13.26\%.$
Final Answer
The percentage of income spent on rent in October is approximately:
$\boxed{13.2\%}.$