Discussion

Question:

The average of x consecutive integers is n and the average of the next y consecutive integers is n + 6. Find the relation between x and y.

Explanation:

Let the 'x' consecutive numbers be a + 1, a + 2, ……… a + x.

Average = $\frac{(a+1+a+2+a+3+\cdots +a+x)}{x}$ = $\frac{\mathrm{a}+1+\mathrm{a}+\mathrm{x}}{2}$ = a + $\frac{x+1}{2}$ = n

Next y consecutive numbers be a + x + 1, a + x + 2, …….. a + x + y.

Average = a + x + $\frac{1+y}{2}$ = n + 6 

∴ a + x + $\frac{1+y}{2}$ = a + $\frac{x+1}{2}$ + 6

⇒ 2x + 1 + y = x + 1 + 12

⇒ x + y = 12

Alternately,
Let 3 consecutive numbers be 1, 2 and 3.

Average n = $\frac{1+2+3}{3}$ = 2

Next numbers will be 4, 5, 6 …. 

The average these numbers should be (n + 6) = 8

∴ Next numbers must be 4, 5, 6, 7, 8, 9, 10, 11 and 12.

Hence, y must be 9.

⇒ x + y = 3 + 9 = 12

Hence, option (a).

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