7. Average age of 18 men is decreased by 1 year when one man whose age is 49 years is replaced by a new man. Find age of the new man.
a. 29
b. 31
c. 33
d. 35

Answer: B

Explanation:
Let the initial average = $x$ years
Sum of the ages of 18 men = $18 \times x$ = $18x$
Let the age of the new man = $k$ years
New sum of the ages = $18x$ – $49$ $+k$
New average = $\dfrac{{18x - 49 + k}}{{18}} = x - 1$
$ \Rightarrow $ ${18x - 49 + k}$ $= 18\left( {x - 1} \right)$
$ \Rightarrow $ ${18x - 49 + k}$ $=18x - 18$
$ \Rightarrow $ $k = 49 - 18 = 31$

Shortcut:
In cases of replacement, $\Delta A = \dfrac{{\Delta S}}{n}$
$\therefore 1 = \dfrac{{49 - k}}{{18}}$
Here average has reduced so new member age $k$ is less than existing person's age. So $49 - k$ has taken.
$ \Rightarrow 49 - k = 18$
$ \Rightarrow k = 31$

Alternate Method:
If the replaced persons age is same with the existing average, there is no change in the average. But by replacement overall decrease in the age is 18 × 1 = 18 years. This is the change bought by the new man.
Age of new man = Age of man replaced - Total decrease in age = 49 – (1 × 18) = 31 years

8. The average weight of 12 persons is 50 kg. On replacing a man whose weight is 53 kg. with a new man, new average becomes 49 kg. The weight of the new man is:
a. 35
b. 37
c. 39
d. 41

Answer: D

Explanation:
Sum of the weights of all 12 persons $= 12 × 50 = 600$
Sum of the weights of all 12 persons after replacement $= 12 × 49 = 588$
Difference between the ages $= 600 - 588 = 12$
The difference is because of one person with $53$ kg is replaced by another one.
So his weight must $12$ less than $53$.
$\therefore $ New persons weight $ = 53 – 12 = 41$

Shortcut:
In cases of replacement, $\Delta A = \dfrac{{\Delta S}}{n}$
$\therefore 1 = \dfrac{{53 - k}}{{12}}$
Here average has reduced so new member age $k$ is less than existing person's age. So $53 - k$ has taken.
$ \Rightarrow 53 - k = 12$
$ \Rightarrow k = 41$

Alternative method:
If the replaced person weight is the same as the existing average, there is no change in the average. But by replacement overall decrease in the age is $12 × 1 = 12$ kg. This is the change bought by the new man.
Weight of new man = Weight of man replaced - Total decrease in weight
$= 53 – (1 × 12) = 41$ kg.

9. The average age of 12 men in a group is increased by 2 years when two men whose ages are 20 years and 22 years, are replaced by new members. What is the average age of the new men included?
a. 35
b. 33
c. 31
d. 29

Answer: B

Explanation:
Total age of two men replaced $= 20 + 22 = 42$ years
Total increase in age on replacement $= 2 × 12 = 24$ years
Total age two new persons included $= 42 + 24 = 66$ years
Therefore, Average age of new persons = $\displaystyle\frac{{{\rm{66}}}}{{\rm{2}}}$ $= 33$ years

Shortcut:
In cases of replacement, $\Delta A = \dfrac{{\Delta S}}{n}$
$\therefore 2 = \dfrac{{(k_1 + k_2) - (20 + 22)}}{{12}}$
Here average has increased so new member age $(k_1 + k_2)$ is more than sum of existing person's ages. So $(k_1 + k_2) - (20 + 22)$ has taken.
$ \Rightarrow (k_1 + k_2) - 42 = 24$
$ \Rightarrow (k_1 + k_2) = 66$
Average of two new persons $= \dfrac{{{k_1} + {k_2}}}{2} = \dfrac{{66}}{2} = 33$

10. Average age of 7 members of a family is 29 years. If present age of the youngest member is 5 years, find average age of the remaining members at the time of birth of the youngest member.
a. 24
b. 26
c. 28
d. 30

Answer: C

Explanation:
Sum of the ages of 7 members $= 29 \times 7 = 203$ years
Sum of the ages of 6 members $= 203 - 5 = 198$ years
Sum of the ages of 6 members 5 years ago $= 198 - (6 \times 5) = 168$
($\because $ six persons age increases by 5 years each in 5 years.)
Average age of the remaining six members $ = \dfrac{{168}}{6} = 28$

11. Average weight of 8 persons is 48 kg. If one man weighing 34 kg, is died, what is a average age of the remaining 7 persons.
a. 50
b. 52
c. 54
d. 56

Answer: A

Explanation:
Sum of the ages of 8 members $= 48 \times 8 = 384$ years
Sum of the ages of remaining 7 members $= 384 - 34 = 350$ years
Average age of the remaining 7 members $ = \dfrac{{350}}{7} = 50$ kg

Alternative Method:
Average weight of 8 persons $= 48$ kg
Therefore, Excess of average weight than the weight of man died $= 48 – 34 = 14$ kg
Therefore, Increase in weight of the remaining 7 persons $= \displaystyle\frac{{{\rm{14}}}}{{\rm{7}}} = 2$ kg
Therefore, Weight of remaining 7 persons $= 48 + 2 = 50$ kg

12. The average expenditure of a man for 10 days is Rs. 45 per day. If his average expenditure for the first 3 days is Rs. 52 per day, find his average expenditure for the remaining 7 days.
a. 35
b. 37
c. 39
d. 42

Answer: D

Explanation:
Total expenditure for 10 days $= 45 \times 10 = 450$
Sum of expenditure for first 3 days $= 52 \times 3 = 156$
Remaining expenditure for the 7 days $=450 - 156 = 294$
Average expenditure for the 7 days $ = \dfrac{{294}}{7} = \text{Rs}.42$