A trim costs two-thirds as much as a haircut, but twice as much as shaving. If a man gets a trim 4 times and a shave once every month, the frequency of his haircuts for the cost of the haircut to be equal to that for shaving and trimming combined is,

Correct option is D

Given:
Cost of shave = ₹S

​$$\text{Let the cost of a haircut} = H \\\text{Cost of a trim} = \frac{2}{3}H \\\text{Trim} = 2 \times \text{Shave} \Rightarrow \frac{2}{3}H = 2S \Rightarrow S = \frac{1}{3}H$$

Concept:

$$\text{Total haircut cost in } y \text{ months} = x \cdot H \\\text{Total trim + shave cost in } y \text{ months} = y \cdot \left(4 \cdot \frac{2}{3}H + \frac{1}{3}H \right)$$

Solution:

$$\text{Trim cost per month} = 4 \cdot \frac{2}{3}H = \frac{8}{3}H \\\text{Shave cost per month} = \frac{1}{3}H \\\text{Total monthly trim + shave cost} = \frac{8}{3}H + \frac{1}{3}H = 3H \\\text{So over } y \text{ months, trim + shave cost} = y \cdot 3H\\\ \\\text{Equating with haircut cost:} \\xH = 3Hy \\\Rightarrow x = 3y$$

$$x = 3, \quad y = 1 \Rightarrow x = 3y \quad \text{(True)}$$

Final Answer: D