Directions: Study the following questions carefully and answer the questions given below.
1
Consider the sequence of 5 consecutive integers. If the average of first three is t. The average of all five integers is:
» Explain it
B
Let the 3 integers be

a – 1, a, a + 1. then average is-
 
⇒  a – 1 + a + a + 1  =  3a  = t  ⇒  a = t.
3 3

Similarly, the average of 5 numbers is
 
⇒   a–1 + a + a+1 + a+2 + a+3  =  5a + 5  = (a + 1) = t + 1.
5 5

Hence, option B is correct.

2
The arithmetic mean of the following numbers

1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 and 7, 7, 7, 7, 7, 7, 7, is
» Explain it
B
Required mean =  1 × 1 + 2 × 2 + 3 × 3 + 4 × 4 + 5 × 5 + 6 × 6 + 7 × 7
1 + 2 + 3 + 4 + 5 + 6 + 7


⇒  1+ 22 + 3+ ...... + 72
1 + 2 + 3 + ........+ 7

⇒      
n(n + 1)(2n + 1)
6
                             [ sum of squares of n natural numbers]
n(n + 1)
2
                             [ sum of n natural numbers]

⇒  (2n + 1)  ⇒  2 × 7 + 1  =  15  = 5.
3 3 3

Hence, option B is correct.

3
If average weight of a family is ‘y’ kg. If a guest weighing 30 kg arrives then the average weight is increases by 1 kg. If the weight of this guest had been 18 kg then the average weight of the family would have decreased by 1 kg. Find ‘y’.
» Explain it
B
Let the number of family members be x.

Therefore, the total weight = xy

Equation in the 1st scenario:

(x + 1) (y + 1) = xy + 30

xy + x + y + 1 = xy + 30

x + y = 29                 .....(i)

Equation in the 2nd scenario:

(x + 1) (y - 1) = xy + 18

xy + y - x - 1 = xy + 18

y - x = 19           ....(ii)

Solving equtions (i) and (ii), we get
 
2y = 48, Therefore, y = 24.

Hence, option B is correct.

4
The average of 11 results is 50. If the average of the first six results is 49 and that of the last six is 52. The sixth no. is?
» Explain it
D
Sixth results = total no. of first six reuslth + total no. of last six results – total no. of 11 results 

= (6 × 49) + (6 × 52) – (11 × 50)

= 294 + 312 – 550 = 56.

Hence, option D is correct.

5
A number is such that when it is multiplied by 8, it gives another number which is as much above from 270 as the original number (itself) is below 270. The average of the original number and the resultant number is:
» Explain it
D
Let the number be x then.

270 – x = 8x – 270

⇒ x = 60 and 8x = 480

Therefore the average of 60 and 480 is 270.

Hence, option D is correct.