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CAT 2023 Slot 1 — QA Question 16

Percentage ChangeEasy

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

Answer & solution

  • 26%

  • B

    25%

  • C

    30%

  • D

    28%

Solution

Easy

Track each salary as a product of growth factors. Sita and Mita are fully specified by their two hikes, so compute them directly. Gita's final salary is pinned by the rule "Gita = mean of the three," and since the mean of three values equals the average of the other two when one of them is the mean, Gita finally equals the average of Sita and Mita. Back out Gita's year-2 hike from there.

1

Set up the starting salaries using the 5:6:75:6:7 ratio:

Sita=5x,Gita=6x,Mita=7x\text{Sita}=5x,\qquad \text{Gita}=6x,\qquad \text{Mita}=7x
2

Apply Sita's and Mita's two hikes as multiplying factors (1+rate)(1+\text{rate}):

Sita=5x×1.20×1.40=8.4xMita=7x×1.20×1.25=10.5x\begin{aligned} &\text{Sita}=5x\times 1.20\times 1.40=8.4x\\ &\text{Mita}=7x\times 1.20\times 1.25=10.5x \end{aligned}
3

Use the mean condition. Gita's final salary equals the mean of all three. If one number equals the average of the set, it must equal the average of the other two:

Gitafinal=8.4x+10.5x2=9.45x\text{Gita}_{\text{final}}=\frac{8.4x+10.5x}{2}=9.45x
4

Back out Gita's year-2 hike. After year 1 (+25%+25\%) Gita is 6x×1.25=7.5x6x\times1.25=7.5x. Let the year-2 rate be P%P\%:

7.5x(1+P100)=9.45x1+P100=9.457.5=1.26⇒ P=26\begin{aligned} &7.5x\left(1+\tfrac{P}{100}\right)=9.45x\\ &1+\tfrac{P}{100}=\frac{9.45}{7.5}=1.26\\ &\Rightarrow\ P=26 \end{aligned}

Gita's second-year hike is 26%\mathbf{26\%}.

Skip xx entirely: compare against Gita's post-year-1 value 7.57.5. Final 9.459.45 over 7.57.5 is 1.261.26, i.e. a 26%26\% rise — one division does it.

CAT 2023 Slot 1 QA Q16: The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7, respectively. In the f — Solution | TheCATExam