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# [Smart Math] Averages Problem 9

Here’s and example of a SMART MATH problem for AVERAGES.

### Problem

The average age of Allen and Beth is 40 years. If Chris were to replace Allen, the average would be 38 years and if he were to replace Beth, the average would be 42 years. What are the ages of Allen, Beth and Chris respectively?

1. 44, 44, 36
2. 36, 44, 40
3. 44, 40, 36
4. 36, 40, 44
5. 44, 36, 40

### The Usual Method

Let the ages of Allen, Beth and Chris be ‘1’, ‘2’ and ‘3’ years respectively.

Hence,

$\frac{a+b}{2}=40$                 $\therefore a+b=80$                  …… Eq. 1

$\frac{b+c}{2}=38$                 $\therefore b+c=76$                  …… Eq. 2 and

$\frac{a+c}{2}=42$                 $\therefore a+c=84$                  …… Eq. 3

$2a+2b+2c=80+76+84=240$

$\therefore a+b+c=120$

Now substituting Eq. 1 in this we get:

$80+c=120$

$\therefore c=40$

Similarly by substituting Eq. 2 in $a+b+c=120$, we get:

$a+76=120$

$\therefore a=44$

Hence,

$44+b+40=120$

$\therefore b=36$

Hence the ages of Allen, Beth and Chris are 44, 36 and 40 years respectively.

(Ans: 5)

Estimated Time to arrive at the answer = 60 seconds.

### Using Technique

The average age of Allen and Beth = 40 years. Note from options, that only options ‘2’ and ‘5’ satisfies this condition ( $\frac{36+44}{2}=40$ from option ‘2’ and $\frac{44+36}{2}=40$ from option ‘5’). This eliminates all other options.

We also know from the question that the average age of Beth and Chris is 38 years. Now between the options ‘2’ and ‘5’, only option ‘5’ satisfies this condition ( $\frac{36+40}{2}=38$).