‘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 146:100:161, respectively. Which of the following statement(s) is/are true regarding ‘y’?
I. Sum of squares of each digit of ‘y’ is 70.
II. 12y < √900 × √900
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 = 146:100 Or, 12,000 + 24y = 14,016 Or, 24y = 2,016 Or, ‘y’ = 84 Statement I:
Required sum = 8² + 4² = 64 + 16 = 80
Statement says 70 → false. Statement II:
12y = 12 × 84 = 1,008
√900 × √900 = 30 × 30 = 900
Here, 12y > √900 × √900
So, statement II is false. Statement III:
Tens digit of ‘y’ = 8, which is an even number.
So, statement III is true. Therefore, only statement III is true.