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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 -> Examples

1. In an examination, a pupil’s average marks were 63 per paper. If he had obtained 20 more marks for his Geography paper and 2 more marks for his History paper, his average per paper would have been 65. How many papers were there in the examination?
 
  • A. 8
  • B. 10
  • C. 11
  • D. 12
Ans: C.
Sol.
Let the numbers of papers be x.
Then, 63x + 20 + 2 = 65x (or)
2x = 22 (or)
x = 11.
 
2. The average age of the mother and her six children is 12 years which is reduced by 5 years if the age of the mother is excluded. How old is the mother?
 
  • A. 40 years
  • B. 42 years
  • C. 48 years
  • D. 50 years
Ans: B.
Sol.
Therefore age of the mother
= ( 12 × 7 – 7 × 6)
= 42 years.
 
 
3. The average of five numbers is 27. If one number is excluded, the average becomes 25. The excluded number is
 
  • A. 25
  • B. 27
  • C. 30
  • D. 35
Ans: D.
Sol.
Therefore excluded number
= (27 × 5) - ( 25 × 4)
= 135 – 100
= 35.