06 December, 2015

Average Formulas, Shortcuts and Problems

Arithmetic Average Formulas:

  • An average or an arithmetic or mean of a no. of quantities of the same type is equal to their sum divided by the no. of those quantities.
               Average = (Sum of Observations)/(No. of Observations)

  • Average Speed
              A Person covers certain distance at 'X' kmph and the equal distance at 'Y' kmph. Then average speed of whole journey is given by the following formula.

             Average Speed = (2XY)/(X + Y)
  • If  'M' no. of items with 'X' average and their sum is 'MX', 'N' no. of items with 'Y' average and their sum is 'NY'. Then the combined average is given by following formula.
             Average = (MX + NY)/(M + N)

Problems on Average:
 1. Find the average of 11, 14, 16, 24 and 30.
A) 17                     B) 19                   C) 20                   D) 21


Ans.B
Solution:
Average = (11 + 14 + 16 + 24 + 30)/5
Average = 95/5
Average = 19.

2. If the average score of 15 boys is 85 and the average score of 10 girls in the same class is 95. What is the average of total class?
A) 85                   B) 87                     C) 88                  D) 89


Ans.D
Solution:
15 boys avg. score is 85 then sum = 15 * 85
10 girls avg. score is 95 then sum = 10 * 95
Average = [(15 * 85) + (10 * 95)]/(15 + 10)
Average = 2225/25
Average = 89.

3. The average temperature on Monday, Tuesday, Wednesday is 24.40C. If average temperature on Monday, Tuesday was 240C. Find the temperature on Wednesday?
A) 25.20C           B) 25.40C              C) 25.50C               D) 25.60C


Ans.A
Solution:
Let Temperature on Wednesday = X;
X = [(3 * 24.4) - (2 * 24)]/(3 - 2)
X = 73.2 - 48
X = 25.20C.

4. A batsman made 160 runs in 4 innings but was out of duck in the 5th innings. What is the average of five innings ?
A) 30                  B) 32                     C) 34                      D) 36


Ans.B
Solution:
Total score made by batsman:
In four innings = 160;
In five innings = Total score in four innings + 5th innings score ;
In five innings = 160 + 0 = 160 runs ;
Total innings = 5
Average = 160/5 = 32.

5. If two kinds of dates costing Rs.25 per Kg, Rs. 20 per Kg are mixed in the ratio of 2 : 3. Find the average cost of mixture per Kg ?
A) Rs. 17          B) Rs. 18                 C) Rs. 20                 D) Rs. 22


Ans.D
Solution:
Average cost/Kg = [(25 * 2) + (20 * 3)]/5
Average cost/Kg = 110/5 = Rs. 22/Kg.

6. The average of Seven numbers is 18. The average of first three numbers is 14 and the average of last 3 numbers is 19. What is the middle number?
A) 24               B) 25                        C) 26                       D) 27


Ans.D
Solution:
Total of seven numbers = 7 * 18 = 126
Total of first three numbers = 3 * 14 = 42
Total of last three numbers = 3 * 19 = 57
Middle number = 126 - 42 - 57 = 27.

7. The average weight of a group of 13 persons is 62.875. When anew person joining the group the average weight increases to 62.985. What is the weight of the new person joined in the group?
A) 64.2           B) 64.4                     C) 64.5                     D) 64.6


Ans.B
Solution:
Sum of weights of 13 persons = 13 * 62.875 = 817.375
Sum of weights of 14 persons = 14 * 62.985 = 881.790
Weight of the new person joined = 881.790 - 817.375 = 64.41.

                [OR]                
Short Cut Method:
Age of New Person = 62.875 + 14(62.985 - 62.875)
Age of New Person = 62.875 + 1.54
Age of New Person = 64.415.

8. The average weight of 8 persons is increased by 1.5 Kg when a person weighting 65 Kg is replaced by a new person.Then what could be the weight of the new person?
A) 77 Kg         B) 78.85 Kg            C) 79 Kg                 D) 79.65 Kg


Ans.A
Solution:
Let the average weight of 8 persons = X Kg
The average weight of new person = Y Kg
(8X - 65 + Y)/8 = X + 1.5
8X - 65 + Y = 8X + 12
Y = 77 Kg.

                [OR]                
Short Cut Method:
Age of New Person = 65 + (8*1.5)
Age of New Person = 65 + 12
Age of New Person = 77.

9. The arithmetic mean of 75 numbers is calculated to be 35. If each number  is increased by 5 then what is the average of new numbers ?
A) 30              B) 35                       C) 40                        D) 45


Ans.C
Solution:
Average = [(75 * 35) + (75 * 5)]/75
Average = 35 + 5 = 40

10. The average monthly income of A & B is Rs. 5050, B & C is Rs. 6250 and C & A is Rs. 5200. Find the monthly income of B ?
A) 5200         B) 5550                    C) 6050                  D) 6100


Ans.D
Solution:
A and B's Monthly income = 5050 * 2 = Rs. 10,100 ----- (1)
B and C's Monthly income = 6250 * 2 = Rs. 12,500 ----- (2)
C and A's Monthly income = 5200 * 2 = Rs. 10,400 ----- (3)
(1) + (2) + (3)==>
2(A + B + C) = 33,000
A + B + C = 16,500
B = 16,500 - 10,400
B = 6,100.

Important Points:

  1. If every number is increased by 'k' in the series then their average also increased by 'k'.
  2. If every number is decreased by 'k' in the series then their average also decreased by 'k'.
  3. If every number is multiplied by 'k' in the series then their average also multiplied by 'k'.
  4. If every number is divided by 'k' in the series then their average also divided by 'k'.

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