Question

Of the three numbers, the average of the first and the second is greater than the average of the second and third number by 15. What is the difference between the first and the third of these numbers?
A. $15$
B. $30$
C. $45$
D. Data inadequate
E. None of these

Answer 1

Hint: According to the question given in the question we have to determine the difference between the first and the third of these numbers if the three numbers, the average of the first and the second is greater than the average of the second and third number by 15. So, first of all we have to let all the three numbers.
Now, we have to determine the average between the first and the second number with the help of the formula to determine the average as mentioned below:

Formula used: $ \Rightarrow $Average of two numbers$ = \dfrac{{x + y}}{2}$……………………………(A)
Where, x and y are the two numbers.
Now, we have to determine the average between the second and the third number with the help of the formula (A) as mentioned above.
Now, as mentioned in the question we have to determine the average of the first and the second is greater than the average of the second and third number by 15 by substituting all the values.

Complete step-by-step solution:
Step 1: First of all we have to let all the three numbers so, let all the three numbers are A, B, and C.
Step 2: Now, we have to determine the average between the first and the second number with the help of the formula (A) to determine the average as mentioned in the solution hint. Hence,
$ = \dfrac{{A + B}}{2}..................(1)$
Step 2: Same as the step 2 we have to determine the average between the second and the third number with the help of the formula (A) as mentioned in the solution hint. Hence,
$ = \dfrac{{B + C}}{2}.................(2)$
Step 3: Now, as mentioned in the question that the average of the first and the second is greater than the average of the second and third number by 15 hence, with the help of the equations (1) and (2).
$ \Rightarrow \dfrac{{A + B}}{2} - \dfrac{{B + C}}{2} = 15$……………………..(3)
Step 4: Now, on solving the expression (3) as obtained in the solution step 3 we can easily determine the difference between the first and the third terms. Hence,
$
   \Rightarrow \dfrac{{A + B - B - C}}{2} = 15 \\
   \Rightarrow \dfrac{{A - C}}{2} = 15 \\
   \Rightarrow A - C = 30
 $
Hence, with the help of formula (A) we have determined the difference between the first and the third of these numbers which is $A - C = 30$.

Therefore option (B) is correct.

Note: To determine the difference between first and the third term it is necessary that first of all we have to determine the average between first and the second term and same as the difference between the second and the third term.
As the average of the first and the second is greater than the average of the second and third number by 15 so, we just have to substitute the averages we obtained between the three numbers.