The question is about Ratio of Amounts. An application based question from progressions. With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. CAT Exam tests the idea of progression often in the CAT Quantitative Aptitude section and this could also be tested in DI LR section of the CAT Exam as a part of a puzzle.

Question 15: The salaries earned by two friends Anil and Jeetu in different years are in A.P. If the ratio of the amount earned by them in ‘p’ number of years are (4p+1) : (2p+17). Then find the ratio of amount earned by them in the 7^{th} year.

- (2p+1) : (4p+6)
- 53 : 43
- 4 : 7
- 15p : 36p

53 : 43

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Let a_{1}, a_{2} be the salaries of their first year and d_{1} and d_{2} be the difference between salaries of two consecutive years.

Now, the sum of p terms of their salaries are

S_{p} = \\frac{p}{2}\\) * [2a_{1} + (p-1)d_{1}]

S'_{p} = \\frac{p}{2}\\) * [2a_{2} + (p-1)d_{2}]

\\frac{\frac{p}{2} * [2a_1 + (p-1)d_1]}{\frac{p}{2} * [2a_2 + (p-1)d_2]}\\) = \\frac{4p+1}{2p + 17}\\)

\\frac{[2a_1 + (p-1)d_1]}{[2a_2 + (p-1)d_2]}\\) = \\frac{4p+1}{2p + 17}\\) -------------------- (1)

Now,

Ratio of Salaries earned by them in the 7^{th} year

= \\frac{a_1 + 6d_1}{a_2 + 6d_2}\\)

= \\frac{2a_1 + 12d_1}{2a_2 + 12d_2}\\) (Multiplying and Dividing by 2)

= \\frac{2a_1 + (13 - 1)d_1}{[2a_2 + (13 - 1)d_2}\\)

= \\frac{4 * 13 + 1}{2 * 13 + 17} \\)

= 53 : 43

The question is **"find the ratio of amount earned by them in the 7 ^{th} year."**

Choice B is the correct answer.

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