Average of 3 numbers is 23. First number is 5 less than the average of the other two. Third number is 13 more than the average of other two. What is the difference between first and third number?

Average of 3 numbers = 23

First number is 5 less than the average of the other two.

Third number is 13 more than the average of the other two. 

Formula Used: 

Average of numbers = $$\frac{\text{Sum of numbers}}{\text{total number}}$$ 

Solution: 

Let the first number be x, Second number be y and third number be z.

Total sum of 3 numbers (x + y + z) = $$23\times 3 = 69$$ 

Now,  According to question;

From first condition; 

  $$x =\frac{y + z}{2} -5 \\ 2x = y+z -10 \\ z = 2x -y+10 ..........(i)$$ 

From second condition;

​$$z= \frac{x+y}{2}+13 ................(ii)$$ 

equating z from (i) and (ii) ;

​$$2x-y+10 = \frac{x+y}2+13 \\ \ \\ 4x-2y+20 = x+y+26 \\ \ \\ 3x -3y = 6 \\ \ \\ x-y = 2$$

Substituting x - y in eq(i)

$$z = 2x -y +10 \\ z = x+x-y+10 \\ z = x+2+10 \\ z -x = 12$$​  

thus, the difference between third and first is 12.