Averages

1.The arithmetic mean of the scores of a group of students in a test was 52. The brightest 20% of them secured a mean score of 80 and the dullest 25% a mean score of 31. The mean score of remaining 55% is:
A.51.4
B.52.6
C.56.1
D.55.3
E.None of these
Answer – A (51.4)
Explanation – Let the required mean score be a
Then, 20 x 80 + 25 x 31 + 55 x a = 52 x 100
1600 + 775 + 55a = 5200
55a = 2825
a = 51.4

2.The average of a non-zero number and its square is 5 times the number. The number is:
A.0 , 7
B.0 , 6
C.5 , 7
D.0 , 9
E.None of these
Answer – D (0 , 9)
Explanation – Let the number be x.
Then,
(x + x^2) / 2 = 5x
x^2 – 9x = 0
x (x – 9) = 0
x = 0 or x = 9.

3.If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a^2, b^2, c^2 is:
A.3 M x M
B.3 M
C.9 M
D.9 M x M
E.None of these
Answer – A (3 M x M)
Explanation –
We have :
(a + b + c) / 3 = M or (a + b + c) = 3M.
Now, (a + b + c)^2 = (3M)^2 = 9M^2
a^2 + b^2 + c^2 + 2 (ab + bc + ca) = 9M^2
a^2 + b^2 + c^2 = 9M^2
Required mean =
(a^2 + b^2 + c^2)/3= 9M^2/ 3 = 3 M^ 2


4.The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one them weighing 65 kg. What might be the weight of the new person?
A.65 kg
B.70 kg
C.85 kg
D.92 kg
E.None of these
Answer – C (85 kg)
Explanation – Total weight increased = (8 x 2.5) kg = 20 kg
Weight of new person = (65 + 20) kg = 85 kg

5.The average age of the boys in a class is 16 years and that of the girls is 15 years. The average age for the whole class is:
A.15
B.16
C.17
D.Data inadequate
E.None of these
Answer – D (Data inadequate)
Explanation – Clearly, to find the average, we ought to know the number of boys, girls or students in the class, neither of which has been given
So, the data provided is inadequate