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Average Problems

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AVERAGE -> IMPORTANT FACTS AND FORMULAE

I. Average = [Sum of observations / Number of observations]
II. Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is [2xy / x + y] kmph.

AVERAGE -> SOLVED EXAMPLES

1. Find the average of all prime numbers between 30 and 50.
  Sol. There are five prime numbers between 30 and 50.
They are 31, 37, 41, 43 and 47.
∴ Required average = [(31 + 37 + 41 + 43 + 47) / 5] = 199/5 = 39.8.
2. Find the average of first 40 natural numbers.
  Sol. Sum of first n natural numbers = 40 * 41 / 2 = 820.
∴ Required average = 820 / 40 = 20.5.
3. Find the average of first 20 multiples of 7.
  Sol. Required average = 7(1+2+3 ... + 20) / 20 = [7 * 20 * 21 / 20 * 2] = [147 / 2] = 73.5.
4. The average weight of 10 oarsmen in a boat is increased by 1.8 kg when one of the crew, who weighs 53kg is replaced by a new man. find the weight of the new man.
  Sol. Total weight increased = (1.8 * 10)kg = 18kg.
∴ Weight of the new man = (53 + 18)kg = 71kg.
5. A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. find his average after 17th inning.
  Sol. Let the average after 17th inning = x.
Then, average after 16th inning = (x - 3)
∴ 16(x - 3) + 87 = 17x or x = (87 - 48) = 39.

Average -> EXERCISE

31. 3 years ago, the average age of a family of 5 members was 17 years. A baby having been born, the average age of the family is the same today. The present age of the baby is
 
  • A. 1 years
  • B. 2 years
  • C. 3 years
  • D. 4 years
Ans: B.
Sol.
Total age of 5 members, 3 years ago = (17 x 5)years = 85 years.
Total age of 5 members now = (85 + 3 x 5)years = 100 years.
Total age of 6 members now = (17 x 6)years = 102 years.
∴ Age of the baby = (102 - 100)years = 2 years
 
32. A grocer has a sale of Rs. 6345, Rs. 6927, Rs.6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500?
 
  • A. Rs. 3000
  • B. Rs. 4991
  • C. Rs. 5991
  • D. Rs. 6001
Ans: B.
Sol.
Total sale for 5 months = Rs. (6435 + 6927 + 6855 + 7230 + 6562)
= Rs. 34009.
∴ Required sale = Rs.[(6500 x 6) - 34009]
= Rs. (39000 - 34009)
= Rs. 4991.
 
 
33. A cricketer has a certain average for 10 innings. In the eleventh inning, he scored 108 runs, thereby increasing his average by 6 runs. His new average is
 
  • A. 42 runs
  • B. 45 runs
  • C. 48 runs
  • D. 52 runs
Ans: C.
Sol.
Let the average for 10 innings be x. Then,
10x + 108 / 11 = x+6 ⇔ 11x + 66 = 10x + 108 ⇔ x = 42.
∴ New average = (x+6) = 48 runs.