In September, the incomes of Kamal, Amal and Vimal are in the ratio 8 ∶ 6 ∶ 5. They rent a house together, and Kamal pays 15%, Amal pays 12% and Vimal pays 18% of their respective incomes to cover the total house rent in that month. In October, the house rent remains unchanged while their incomes increase by 10%, 12% and 15%, respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to

The correct option is 13.26.

Let us break down the calculation step-by-step to find the percentage of their total income paid as house rent in October.

Step 1: Define the incomes in September
Let the incomes of Kamal, Amal, and Vimal in September be represented as:
$$I_K=8x$$
$$I_A=6x$$
$$I_V=5x$$
For simplicity, let we assume $x=100$. This gives the September incomes as:
Kamal's income = 800
Amal's income = 600
Vimal's income = 500

Step 2: Calculate the total house rent
Each person pays a specific percentage of their income toward the rent:
Kamal's contribution = 15% of 800 = $0.15\times 800=120$
Amal's contribution = 12% of 600 = $0.12\times 600=72$
Vimal's contribution = 18% of 500 = $0.18\times 500=90$
Therefore, the total rent in September is:
$$\text{Total Rent}=120+72+90=282$$
Since the problem states that the house rent remains unchanged in October, the total house rent in October is also 282.

Step 3: Calculate the new incomes in October
In October, their incomes increase by 10%, 12%, and 15%, respectively:
Kamal's new income = 800 + 10% of 800 = $800+80=880$
Amal's new income = 600 + 12% of 600 = $600+72=672$
Vimal's new income = 500 + 15% of 500 = $500+75=575$

Step 4: Find the total income in October
Adding up their October incomes:
$$\text{Total Income in October}=880+672+575=2127$$

Step 5: Calculate the percentage of total income paid as rent
Now, we calculate what percentage the unchanged house rent (282) is of the new total income (2127):
$$\text{Percentage}=\left(\frac{282}{2127}\right)\times 100$$
Let us perform the division:
$$\text{Percentage}\approx 0.132581\times 100\approx 13.26\%$$
Rounding to two decimal places, we get 13.26%, which matches the correct option.