average of A and B is 71; of B and C is 76; of A and C is 79. How to find the average of A,B,C. I know the way of solving this problem using equations.But it takes time.So wanted to know if any other logic can be applied to it to solve faster.
Fastest Method to solve this average problem
-
0An even faster method is: \frac{2}{3} \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{bmatrix}^{-1} \begin{bmatrix} 71 \\ 76 \\ 79 \end{bmatrix}. – 2012-05-10
4 Answers
A trick you might use (if you're doing this in your head) is to "center" the numbers on a guess near the average, and then compute the average offset and add back the guess.
For example, for your three numbers $71, 76, 79,$ I would guess the average is around $75$. So now I take the offsets from my guess: $-4, 1,$ and $4$. Next, take the average of these 3 numbers, which is clearly $1/3$. And last, add back $75$, so the answer becomes $75 \frac{1}{3}$.
Just add the averages and divide by $3$: $\frac{A+B}2+\frac{A+C}2+\frac{B+C}2=A+B+C\;\;.$
Add the three averages, you get $A+B+C$ (because first average is $\frac{A}{2}+\frac{B}{2}$, etc); then divide by $3$. So $\frac{1}{3}(71+76+79) = \frac{226}{3}.$
What could be faster than:
$\frac{A+B+C}{3} = \frac{\frac{A+B}{2}+\frac{B+C}{2}+\frac{A+C}{2}}{3} = \frac{71+76+79}{3}$