Quants 10 - Average

Practice Questions:

Question 1:

If the mean of 5 observations x, x+2, x+4, x+6 and x+8 is 11, then the mean of the last three observations is ___.

[ ( x + (x+2) + (x+4) + (x+6) + (x+8) ] / 5 = 11
( 5x + 20 ) / 5 = 11
5x + 20 = 55
5x = 35
x = 7
So, the numbers are, 7, 9, 11, 13, 15.
Required Mean = (11 + 13 + 15) / 3 = 39 / 3 = 13.

Question 2:

If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a² + b² + c² is ___

(a + b + c) / 3 = M
a + b + c = 3M
(a+b+c)² = 3M² = 9M²
a² + b² + c² + 2 ( ab + bc + ca ) = 9M²
a² + b² + c² + 2(0) = 9M²
Required Mean = ( a² + b² + c² ) / 3 = 9M² / 3 = 3M².

Question 3:
The average of the two - digit numbers, which remain the same when the digits interchange their positions is ___.

Average = ( 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 ) / 9
( ( 11 + 99 ) + ( 22 + 88 ) + ( 33 + 77 ) + ( 44 + 66 ) + 55 ) / 9
( 4 * 110 + 55 ) / 9
( 440 + 55 ) / 9
495 / 9 = 55
The average of the two - digit numbers, which remain the same when the digits interchange their positions is 55.

Question 4:
The average of five consecutive odd numbers is 61. What is the difference between the highest and lowest numbers?

Let the numbers be x, x+2, x+4, x+6, x+8.
Then, ( x + (x+2) + (x+4) + (x+6) + (x+8) ) / 5 = 61
(5x + 20 ) / 5 = 61
5x + 20 = 305
5x = 285
x = 57.
The difference between the highest and lower number = (x + 8) - x = 65 - 57 = 8.


Question 5:

The average of 8 numbers is 20. The average of first two numbers is 15 1/2 and that of the next three is 21 1/3. If the sixth number be less than the seventh and the eighth numbers by 4 and 7 respectively, then the eighth numbers is ___.

Let the eighth number be x.
Then, sixth number = x - 7
Seventh number = (x-7)  + 4 = x - 3
So, 2 * (15 1/2) + 3* (21 1/3 ) + ( x - 7 ) + ( x - 3 ) + x = 8 * 20
2 * (31/2) + 3 * (64/3) + 3x -10 = 160
31 + 64 - 10 + 3x = 160
85 + 3x = 160
3x = 75
x = 25
So, the eighth number is 25.


Time to Think:

If a, b, c, d and e are five odd consecutive numbers, their average is ___.