# 2007 iTest Problems/Problem 10

## Problem

My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages? $\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007$

## Solution

Let the youngest grandparent's age be $x$. The youngest grandparent must be a woman because each grandfather is two years older than his wife. That means the oldest grandparent's age (a grandfather) is $x+4$, the older grandmother's age is $x+2$, and the younger grandfather's age is $x+2$. Because Bertha is younger than Dolores, Bertha is the youngest grandparent of them all. Since the average age of the grandparents is $\frac{x + (x+2) + (x+2) + (x+4)}{4} = x+2$, the difference between Bertha's age and the mean of the grandparents' ages is $\boxed{\textbf{(C) }2}$ years.

## Solution 2

Let the youngest grandparent's age by $x,$ just like in the previous solution. We can make a table and fill it out:

  | x+4 | x+3 | x+2 | x+1 | x
-----------------------------
A |     |     |     |     |
B |     |     |     |     |
C |     |     |     |     |
D |     |     |     |     |


Since each grandfather is $2$ years older than his wife, then we can cross out some things already in the table:

  | x+4 | x+3 | x+2 | x+1 | x
-----------------------------
A |     |     |     |  X  | X
B |     |     |     |     |
C |     |     |     |  X  | X
D |     |     |     |     |


Then, since Dolores is older than Bertha, Bertha is the youngest. Thus, her age is $x.$ So, since the average is $x+2,$ the answer is $\boxed{\textbf{(C) }2}.$