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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

7. The average runs of a cricket player of 10 innings was 32. How many runs must he makes ih his next innings so as to increase his average of runs by 4?
 
  • A. 2
  • B. 4
  • C. 70
  • D. 76
Ans: D.
Sol.
Average after 11 innings = 36.
Therefore required numbert of runs
= (36 × 11) – (32 × 10)
= 396 - 320
= 76.
 
8. The mean of 50 observations was 36. It was found later that an observation 48 was wrongly taken as 23. The corrected new mean is
 
  • A. 35.2
  • B. 36.1
  • C. 36.5
  • D. 39.1
Ans: C.
Sol.
Therefore correct sum
= ( 36 × 50 + 48 – 23)
= 1825.
Therefore correct mean
= 1825 / 50
= 36.5.
 
 
9. The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?
 
  • A. 0
  • B. 1
  • C. 10
  • D. 19
Ans: D.
Sol.
Average of 20 numbers = 0.
Therefore sum of 20 numbers = (0 × 20 ) = 0.
It is quite possible that 19 of these numbers may be positive and if their sum is a, then 20th number is (- a).