Shortcut Trick
Let a, b and c be the first number, second number and third number respectively.
a + b + c = 23 × 3 = 69 ---(1)
Now,
a + 5 = (b + c)/2
⇒ 2a + 10 = b + c
⇒ 2a + 10 = 69 - a (from equation (1))
⇒ 2a + a = 69 - 10
⇒ a = 59/3
Again,
c - 13 = (a + b)/2
⇒ 2c - 26 = 69 - c (from equation (1))
⇒ 2c + c = 69 + 26
⇒ c = 95/3
Now, the difference between first and third numbers
⇒ 95/3 - 59/3 = (95 - 59)/3 = 36/3 = 12
The difference between the first and third numbers is 12.
Alternate Method
Given:
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 = Sum of numbers / Number of numbers
Calculation:
Let the three numbers be A, B, and C.
Given, (A + B + C) / 3 = 23
⇒ A + B + C = 69
First number (A) is 5 less than the average of the other two (B and C).
⇒ A = (B + C) / 2 - 5
Third number (C) is 13 more than the average of the other two (A and B).
⇒ C = (A + B) / 2 + 13
We have two equations:
-
A + B + C = 69
-
A = (B + C) / 2 - 5
-
C = (A + B) / 2 + 13
From equation 2:
⇒ 2A = B + C - 10
⇒ B + C = 2A + 10
From equation 3:
⇒ 2C = A + B + 26
⇒ A + B = 2C - 26
Substitute B + C from the second equation into the first equation:
⇒ A + (2A + 10) = 69
⇒ 3A + 10 = 69
⇒ 3A = 59
⇒ A = 19.67
Substitute A into B + C = 2A + 10:
⇒ B + C = 2(19.67) + 10
⇒ B + C = 39.34 + 10
⇒ B + C = 49.34
Substitute A into A + B = 2C - 26:
⇒ 19.67 + B = 2C - 26
⇒ B = 2C - 45.67
We know that B + C = 49.34:
⇒ (2C - 45.67) + C = 49.34
⇒ 3C - 45.67 = 49.34
⇒ 3C = 95.01
⇒ C = 31.67
Substitute C back into B + C = 49.34:
⇒ B + 31.67 = 49.34
⇒ B = 17.67
The first number (A) = 19.67
The third number (C) = 31.67
Difference between the first and third numbers:
⇒ 31.67 - 19.67 = 12
The difference between the first and third number is 12.
The correct answer is option 4.