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

Correct Answer: 13.2

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:
\[\boxed{13.2\%}.\]