Average - Study Mode
[#421] The average of nine number is 50. The average of first five numbers is 54 and that of the last three numbers is 52. Then the sixth number is :
Correct Answer
(C) 24
Explanation
Solution: Average of 9 numbers = 50 Hence sum of 9 numbers = 450 Average of last three numbers = 52 Hence sum of them will be = 156 Average of first five numbers is = 54 Hence sum of them will be = 54 × 5 = 270 So 6 th number will be = sum of 9 numbers - (sum of first five numbers + sum of last three numbers) = 450 - (156 + 270) = 24
[#422] From a class of 42 boys, a boy aged 10 years goes away and in his place, a new boy is admitted. If on account of this change, the average age of the boys in that class increases by 2 months, the age of the newcomer is :
Correct Answer
(B) 17 years
Explanation
Solution: According to the question, Due to newcomer average age is increased by = 2 months Total age increment in 42 boys = 42 × 2 = 84 months or 7 years Note : If the age of newcomer is same as the boy which was replaced then there is no effect on 2 months. ∴ Age of new boy = 10 + 7 = 17 years
[#423] The average of five different positive numbers is 25. x is the decrease in the average when the smallest number among them is replaced by 0. What can be said about x?
Correct Answer
(A) x is less than 5
Explanation
Solution: Let a, b, c, d, and e be the five positive numbers in the decreasing order of size such that e is the smallest number. We are given that the average of the five numbers is 25. Hence, we have the equation
$$frac{{{ ext{a}} + { ext{b}} + { ext{c}} + { ext{d}} + { ext{e}}}}{5} = 25$$ a + b + c + d + e = 125 ----------- (1) by multiplying by 5.
The smallest number in a set is at least less than the average of the numbers in the set if at least one number is different. For example, the average of 1, 2, and 3 is 2, and the smallest number in the set 1 is less than the
average 2. Hence, we have the inequality 0 < e < 25 0 > -e > -25 by multiplying both sides of the inequality by -1 and flipping the directions of the inequalities.
Adding this inequality to equation (1) yields 0 + 125 > (a + b + c + d + e) + (-e) > 125 - 25 125 > (a + b + c + d) > 100 125 > (a + b + c + d + 0) > 100 by adding by 0 25 > $$frac{{{ ext{a}} + { ext{b}} + { ext{c}} + { ext{d}} + 0}}{5}$$ xa0 xa0 ⇒ 20 by dividing the inequality by 5 25 > The average of numbers a, b, c, d and 0 > 20 Hence, x equals (Average of the numbers a, b, c, d and e) – (Average of the numbers a, b, c, and d) = 25 − (A number between 20 and 25) ⇒ A number less than 5 Hence, x is less than 5
[#424] In 2011, the arithmetic mean of the annual incomes of Ramesh and Suresh was Rs. 3800. The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800, and the arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800. What is the arithmetic mean of the incomes of the three?
Correct Answer
(D) Rs. 4800
Explanation
Solution: Let a, b, and c be the annual incomes of Ramesh, Suresh, and Pratap, respectively.
Now, we are given that
The arithmetic mean of the annual incomes of Ramesh and Suresh was Rs. 3800.
Hence,
$$frac{{{ ext{a}} + { ext{b}}}}{2}$$xa0 = 3800 ⇒ a + b = 2 × 3800 = 7600
The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800.
Hence,
$$frac{{{ ext{b}} + { ext{c}}}}{2}$$xa0 = 4800 ⇒ b + c = 2 × 4800 = 9600 The arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800. Hence, $$frac{{{ ext{c}} + { ext{a}}}}{2}$$xa0 = 5800 ⇒ c + a = 2 × 5800 = 11,600
Adding these three equations yields: (a + b) + (b + c) + (c + a) = 7600 + 9600 + 11,600 2a + 2b + 2c = 28,800 a + b + c = 14,400 The average of the incomes of the three equals the sum of the incomes divided by 3, $$eqalign{
& frac{{{ ext{a}} + { ext{b}} + { ext{c}}}}{3} cr
& = frac{{14,400}}{3} cr
& = { ext{Rs}}{ ext{.}},4800 cr} $$
[#425] In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun?
Correct Answer
(A) 67 kg.
Explanation
Solution: Let Arun's weight by X kg. According to Arun, 65 < X < 72 According to Arun's brother, 60 < X < 70 According to Arun's mother, X $$ leqslant $$ 68 The values satisfying all the above conditions are 66, 67 and 68. Required average, $$eqalign{
& = frac{{66 + 67 + 68}}{3} cr
& = frac{{201}}{3} cr
& = 67,{ ext{kg}}{ ext{.}} cr} $$