Q17
Given below are two statements:
Statement I : The set of numbers (7,8,9,a,b,10,8,7) has an arithmetic mean of 9 and mode(most frequently occurring number) as 8, Then a × b = 120.
Statement II : Let a and b be two positive integers such that a + b + a x b = 84, then a + b =20.
In the light of the above statements, choose the most appropriate answer from the options given below
Answer: A
Statement I
- There are 8 numbers with mean 9, so their sum is 72.
- Known numbers: 7 + 8 + 9 + 10 + 8 + 7 = 49, so a + b = 23.
- 7 and 8 each already appear twice. For 8 to be the mode, 8 must occur more often than 7, so one of a, b must be 8.
- Then the other is 23 − 8 = 15, and a × b = 8 × 15 = 120.
Statement I is correct.
Statement II
- Add 1 to both sides: a + b + ab + 1 = 85, which factorises as (a + 1)(b + 1) = 85.
- 85 = 5 × 17 (the factorisation 1 × 85 would make a = 0, which is not positive). So a = 4 and b = 16.
- a + b = 20. Check: 4 + 16 + 64 = 84. ✓
Statement II is correct.
Answer: Both Statement I and Statement II are correct (option 1)