‘P’ and ‘Q’ started a business with investment of Rs. 3,000 and Rs. 4,200, respectively. After 8 months, ‘P’ increased his investment by Rs. ‘8y’. After another 8 more months, ‘Q’ decreased his investment by Rs. ‘8y’ and ‘R’ joined the business with Rs. 4,800. At the end of 32 months, the ratio of profit share of ‘P’, ‘R’ and ‘Q’ was 134:100:169, respectively. Which of the following statement(s) is/are true regarding ‘y’?
I. Sum of squares of each digit of ‘y’ is 45.
II. 12y > √400 × √441
III. The tens digit of ‘y’ is an even number.

Solution

Ratio of profit share of ‘P’, ‘Q’ and ‘R’, respectively: = [3,000 × 8 + (3,000 + 8y) × 24]:[4,200 × 16 + (4,200 − 8y) × 16]:[4,800 × 16] = (12,000 + 24y) : (16,800 − 16y) : 9,600 So, (12,000 + 24y):9,600 = 134:100 Or, 12,000 + 24y = 12,864 Or, 24y = 864 Or, ‘y’ = 36 Statement I:
Required sum = 3² + 6² = 9 + 36 = 45
So, statement I is true. Statement II:
12y = 12 × 36 = 432
√400 × √441 = 20 × 21 = 420
Here, 12y > √400 × √441
So, statement II is true. Statement III:
Tens digit of ‘y’ = 3, which is not an even number.
So, statement III is false. Therefore, statements I and II are true.