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

Question

Accepted Answer

The problem involves calculating the percentage of total income paid as house rent in October, given the income ratios and rent percentages for Kamal, Amal, and Vimal. Here's the step-by-step solution:

**Income Ratios in September:**

- Kamal, Amal, and Vimal have incomes in the ratio $$8 : 6 : 5$$.
   - Assume total income in September is $$x$$. Their individual incomes are:
     - Kamal's income: $$\frac{8}{19}x$$
     - Amal's income: $$\frac{6}{19}x$$
     - Vimal's income: $$\frac{5}{19}x$$

**Rent Contributions in September:**

- Kamal pays 15%, Amal pays 12%, and Vimal pays 18% of their respective incomes as rent:
     - Kamal's rent: $$0.15 	imes \frac{8}{19}x$$
     - Amal's rent: $$0.12 	imes \frac{6}{19}x$$
     - Vimal's rent: $$0.18 	imes \frac{5}{19}x$$
   - Total house rent:
     $$
     	ext{Total Rent} = \left(0.15 	imes \frac{8}{19}xight) + \left(0.12 	imes \frac{6}{19}xight) + \left(0.18 	imes \frac{5}{19}xight)
     $$
     Simplify:
     $$
     	ext{Total Rent} = 0.1484x
     $$

**Income Increases in October:**

- Kamal's income increases by 10%, Amal's by 12%, and Vimal's by 15%. Their new incomes are:
     - Kamal's October income: $$\frac{8}{19}x 	imes 1.10$$
     - Amal's October income: $$\frac{6}{19}x 	imes 1.12$$
     - Vimal's October income: $$\frac{5}{19}x 	imes 1.15$$
   - Total income in October:
     $$
     	ext{Total Income (October)} = \left(\frac{8}{19}x 	imes 1.10ight) + \left(\frac{6}{19}x 	imes 1.12ight) + \left(\frac{5}{19}x 	imes 1.15ight)
     $$
     Simplify:
     $$
     	ext{Total Income (October)} = 1.1195x
     $$

**Percentage of Income Paid as Rent in October:**

- The total house rent remains unchanged from September ($$0.1484x$$).
   - Percentage of total income paid as rent:
     $$
     	ext{Percentage Rent (October)} = \left(\frac{	ext{Total Rent}}{	ext{Total Income (October)}}ight) 	imes 100
     $$
     Substituting values:
     $$
     	ext{Percentage Rent (October)} = \left(\frac{0.1484x}{1.1195x}ight) 	imes 100 = 13.26\%
     $$

### Final Answer:

The percentage of their total income paid as house rent in October is **13.26%** (nearest to **13.26**).  
Correct option: **13.26**

---

Answer

Answer

Answer