In a set of three numbers, the average of first two numbers is 2, the average of the last two numbers is 3, and the average of the first and the last numbers is 4. What is the average of three numbers?

12
22.5
33
43.5

Answer:- 1

Explanation:-

Solution:

\begin{aligned} Let the three numbers be x, y, and z . \\ We are given that \\ \frac{(x + y)}{2} = 2 \\ \frac{(y + z)}{2} = 3 \\ \frac{(x + z)}{2} = 4 \\ Adding three equations, \\ \frac{(x + y)}{2} + \frac{(y + z)}{2} + \frac{(x + z)}{2} = 2 + 3 + 4 \\ x + y + z = 9 \\ The average of the three numbers is, \\ \frac{(x + y + z)}{3} = \frac{9}{3} = 3 \end{aligned}

$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{three}}\,{\text{numbers}}\,{\text{be}}\,x,\,y,\,{\text{and}}\,z.\, \cr & {\text{We}}\,{\text{are}}\,{\text{given}}\,{\text{that}}\, \cr & \frac{{\left( {x + y} \right)}}{2} = 2 \cr & \frac{{\left( {y + z} \right)}}{2} = 3 \cr & \frac{{\left( {x + z} \right)}}{2} = 4 \cr & {\text{Adding}}\,{\text{three}}\,{\text{equations}}, \cr & \frac{{\left( {x + y} \right)}}{2} + \frac{{\left( {y + z} \right)}}{2} + \frac{{\left( {x + z} \right)}}{2} = 2 + 3 + 4 \cr & x + y + z = 9 \cr & {\text{The}}\,{\text{average}}\,{\text{of}}\,{\text{the}}\,{\text{three}}\,{\text{numbers}}\,{\text{is}}, \cr & \frac{{\left( {x + y + z} \right)}}{3} = \frac{9}{3} = 3 \cr}$