Average - Study Mode
[#321] Find the weighted arithmetic mean of the first 'n' natural numbers, the weights being the corresponding numbers.
Correct Answer
(C) $$frac{{left{ {left( {2n + 1}
ight)}
ight}}}{3}$$
Explanation
Solution: $$eqalign{
& { ext{Weighted arithmetic mean}} cr
& = frac{{1 imes 1 + 2 imes 2 + 3 imes 3 + ,.....,n imes n}}{{1 + 2 + 3 + ,....., + n}} cr
& = frac{{nleft( {n + 1}
ight)left( {2n + 1}
ight) imes 2}}{{6 imes nleft( {n + 1}
ight)}} cr
& = frac{{2n + 1}}{3} cr} $$
[#322] The average of three numbers a, b and c is 2 more than c. The average of a and b is 48. If d is 10 less than c, then the average of c and d is:
Correct Answer
(D) 40
Explanation
Solution: $$eqalign{
& frac{{a + b + c}}{3} = c + 2 cr
& a + b + c = 3c + 6 cr
& a + b = 2c + 6 o left( { ext{i}}
ight) cr
& frac{{a + b}}{2} = 48 cr
& a + b = 96 cr
& { ext{From equation }}left( { ext{i}}
ight) cr
& 96 = 2c + 6 cr
& c = 45 cr
& d = 35 cr
& { ext{The average of }}left( {c + d}
ight) cr
& = frac{{45 + 35}}{2} cr
& = 40 cr} $$
[#323] What is the difference between the average of first 148 even positive numbers and the average of first 129 odd positive numbers?
Correct Answer
(A) 20
Explanation
Solution: Average of first 148 positive number $$eqalign{
& = frac{{Nleft( {N + 1}
ight)}}{N} cr
& = N + 1 cr
& = 148 + 1 cr
& = 149 cr} $$ Average of first 129 odd number $$ = frac{{{N^2}}}{N} = N = 129$$ Difference = 149 - 129 = 20
[#324] The average of all odd numbers from 113 to 159 is . . . . . . . .
Correct Answer
(D) 136
Explanation
Solution: Average of all odd numbers from 113 to 159, by using Arithmetic progression formula, $$eqalign{
& { ext{Average}} = frac{{frac{n}{2}left[ {a + l}
ight]}}{n} cr
& = frac{1}{2}left[ {a + l}
ight] cr
& = frac{1}{2} imes left[ {113 + 159}
ight] cr
& = frac{{272}}{2} cr
& = 136 cr} $$
[#325] A school has only four classes that contain 10, 20, 30 and 40 students respectively. The pass percentage of these classes are 20%, 30%, 60% and 100% respectively. Find the pass % of the entire school.
Correct Answer
(D) 66%
Explanation
Solution: The number of pass candidates are 2 + 6 + 18 + 40 = 66 out of total 100. Hence, Pass percentage = 66%