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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

25. The arithmetic mean of the scores of a group of students in a test was 52. The brightest 20% of them secured a mean score of 80 and the dullest 25% of a mean score of 31. The mean score of remaining 55% is
 
  • A. 40.8 approx
  • B. 45.3 approx
  • C. 51.4 approx
  • D. 53.8 approx
Ans: C.
Sol.
Let the required mean score be x. Then,
20 × 80 + 25 × 31 + 55 × x = 52 × 100
⇔ 1600 + 775 + 55x = 5200
⇔ 55x = 2825
⇔ x = 565 / 11 = 51.4.
 
26. Find the average of all prime numbers between 30 and 50.
 
  • A. 30
  • B. 33.6
  • C. 39.8
  • D. 44.9
Ans: C.
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.
 
 
27. A library has an average of 510 visitors on Sunday and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is
 
  • A. 250
  • B. 276
  • C. 280
  • D. 285
Ans: D.
Sol.
Since the month begins with a Sunday, so there will be five Sundays in the month.
∴ Required average = [510 x 5 + 240 x 25 / 30] = 8550 / 30 = 285.