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1. Average of first 10 natural numbers =
a. 5
b. 6
c. 5.5
d. 5.2

Answer: C

Explanation:
Sum of the first 10 natrual numbers $ = \dfrac{{n(n + 1)}}{2}$$ = \dfrac{{10(11 + 1)}}{2} = 55$
Average = $\dfrac{\text{Sum of the observations}}{\text{Number of observations}}$$ = \dfrac{{55}}{{10}} = 5.5$

2. The average age of three boys is 15 years. If their ages are in the ratio 3 : 5 : 7, the age of the youngest boy is :
a. 9 years
b. 15 years
c. 18 years
d. 21 years

Answer: A

Explanation:
Let their ages be = $3x, 5x, 7x$
Average of their ages $ = \dfrac{{3x + 5x + 7x}}{3} = 5x$
Given, $5x = 15$
$ \Rightarrow x = 5$
Age of youngest $= 3x = 3 \times 5 = 9$ years

3. The average of ages of 10 persons in a club was 32. What should be the age of the new person joining in the club so as to increase the average by 4 ?
a. 68
b. 70
c. 71
d. 72

Answer: D

Total age of 10 persons $= 10 × 32 = 320$
Total age of 11 persons $= 11 × 36 = 396$ ($\because $ the new average is 4 more than present average)
So the age of the person joining is $= 396 - 320 = 76$

Shortcut:
New persons age = Existing average + Total persons in group × Change in average
$\therefore $ $32 + (10 + 1)4$ $ = 76$

Concept of Shortcut:
If the age of the new person joining the club is 32 then there is no change in the average.
If the new average has to be 36, the person who is joining must contribute 4 years to all 11 persons. That is he must have an age extra 44 years above 32. So new average is 76

4. The average weight of the teacher and six students is 12 kg which is reduced by 5 kg if the weight of the teacher is excluded. How much does the teacher weigh ?
a. 40
b. 42
c. 36
d. 38

Answer: B

Explanation:
Total age of the students and teacher together $= 7 × 12 = 84$
New average after excluding teacher $= 6 × 7 = 42$
Teachers weight $= 74 - 42 = 42$

Shortcut:
Teachers age = Existing average + Remaining students × Change in average
$\therefore $ $12 + (6)5$ $ = 42$

Concept of Shortcut:
Teacher has taken her contribution of 5 kgs from each of the students.
As she is contributing 30 kgs to all students, once she is excluded those 30 kgs remains with her along with original weight of 12 kgs. So her weight is 30 + 12 = 42 kgs

5. The average age of 40 students in a class is 15 years. If the age of teacher is also included, the average becomes 16 years, find the age of the teacher.
a. 50
b. 52
c. 54
d. 56

Answer: D

Explanation:
Sum of the ages of 40 students $= 40 × 15 = 600$
Sum of the ages of 40 students and teacher $= 41 × 16 = 656$
Teacher's age $= 656 - 600 = 56$ years

Shortcut:
Teachers age = Existing average + Remaining students × Change in average
$\therefore $ $15 + (41)1$ $ = 56$

6. Average age of 9 members of a club is 29 years. If 2 more persons with the average age of 40 years have become the members of the club, find average age of all the 11 members.
a. 29
b. 30
c. 31
d. 32

Answer: C

Explanation:
Sum of the ages of 9 members $= 9 × 29 = 261$
Sum of the ages of 11 members $= 261 + 2 × 40 = 341$
Average age of 11 members = $\dfrac{{341}}{{11}}$ = $31$ years.

Alternate Method:
Average age of 9 members $= 29$ years;
Excess age of 2 new members $= (40 - 29) × 2 = 22$ years
This excess age will be distributed among all $11$ members.
Therefore, Increase in the average age on inclusion of 2 new members = $\displaystyle\frac{{{\rm{22}}}}{{{\rm{11}}}}$ = 2 years
Therefore, Average age of 11 members $= 29 + 2 = 31$ years

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