CAT 2023Slot 1QAQuestion & Solution
Question
The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7, respectively. In the first year, they get salary hikes of 20%, 25% and 20%, respectively. In the second year, Sita and Mita get salary hikes of 40% and 25%, respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is
Options
25%
28%
26%
30%
Solution
1. Concept Used
- Topic: Ratio and Proportion, Percentage Change, Arithmetic Mean
- Formula: $$\text{Percentage Hike} = \frac{\text{New Salary} - \text{Old Salary}}{\text{Old Salary}} \times 100$$
2. Calculation
Let the initial salaries of Sita, Gita, and Mita be $5p$, $6p$, and $7p$ respectively.
After Year 1 hikes (20%, 25%, 20%):
$$\text{Sita} = 5p \times 1.20 = 6p$$
$$\text{Gita} = 6p \times 1.25 = 7.5p$$
$$\text{Mita} = 7p \times 1.20 = 8.4p$$
After Year 2 hikes for Sita (40%) and Mita (25%):
$$\text{Sita} = 6p \times 1.40 = 8.4p$$
$$\text{Mita} = 8.4p \times 1.25 = 10.5p$$
Let Gita's salary after Year 2 hike be $g$. The condition states that Gita's salary equals the mean of all three salaries after Year 2:
$$g = \frac{8.4p + g + 10.5p}{3}$$
$$3g = 18.9p + g$$
$$2g = 18.9p$$
$$g = 9.45p$$
Now, Gita's salary before Year 2 hike was $7.5p$, and after the hike it is $9.45p$. So the percentage hike is:
$$\text{Hike} = \frac{9.45p - 7.5p}{7.5p} \times 100 = \frac{1.95p}{7.5p} \times 100 = \frac{1.95}{7.5} \times 100 = 26%$$
3. Solution
Answer = Option 3 ✅
The salary hike of Gita in the second year is 26%.