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Problems on Ages
1.Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age?

Explanation:

Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{Ronit's}\phantom{\rule{thinmathspace}{0ex}}\text{present}\phantom{\rule{thinmathspace}{0ex}}\text{age}\phantom{\rule{thinmathspace}{0ex}}\text{be}\phantom{\rule{thinmathspace}{0ex}}x\phantom{\rule{thinmathspace}{0ex}}\text{years}.\\ & \text{Then,}\phantom{\rule{thinmathspace}{0ex}}\text{father's}\phantom{\rule{thinmathspace}{0ex}}\text{present}\phantom{\rule{thinmathspace}{0ex}}\text{age}\phantom{\rule{thinmathspace}{0ex}}\\ & =\left(x+3x\right)\phantom{\rule{thinmathspace}{0ex}}\text{years}\\ & =4x\phantom{\rule{thinmathspace}{0ex}}\text{years}\\ & \therefore \left(4x+8\right)=\frac{5}{2}\left(x+8\right)\\ & ⇒8x+16=5x+40\\ & ⇒3x=24\\ & ⇒x=8\\ & \text{Hence,}\phantom{\rule{thinmathspace}{0ex}}\text{required}\phantom{\rule{thinmathspace}{0ex}}\text{ratio}\\ & =\frac{\left(4x+16\right)}{\left(x+16\right)}=\frac{48}{24}=2\end{array}$

2.The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?

Explanation:

Solution:
Let the ages of children be x, (x + 3), (<x + 6), (<x + 9) and (x + 12) years.
Then, x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50
⇒ 5x = 20
x = 4.
∴ Age of the youngest child = x = 4 years.

3.A father said to his son, "I was as old as you are at the present at the time of your birth". If the father's age is 38 years now, the son's age five years back was:

Explanation:

Solution:
Let the son's present age be x years. Then, (38 - x) = x
⇒ 2x = 38.
x = 19.
∴ Son's age 5 years back (19 - 5) = 14 years.

4.Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand's present age in years?

Explanation:

Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{the}\phantom{\rule{thinmathspace}{0ex}}\text{present}\phantom{\rule{thinmathspace}{0ex}}\text{ages}\phantom{\rule{thinmathspace}{0ex}}\text{of}\phantom{\rule{thinmathspace}{0ex}}\text{Sameer}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\text{Anand}\phantom{\rule{thinmathspace}{0ex}}\text{be}\phantom{\rule{thinmathspace}{0ex}}\\ & 5x\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}4x\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\text{respectively}\\ & \text{Then,}\phantom{\rule{thinmathspace}{0ex}}\frac{5x+3}{4x+3}=\frac{11}{9}\\ & ⇒9\left(5x+3\right)=11\left(4x+3\right)\\ & ⇒45x+27=44x+33\\ & ⇒45x-44x=33-27\\ & ⇒x=6\\ & \therefore \text{Anand's}\phantom{\rule{thinmathspace}{0ex}}\text{present}\phantom{\rule{thinmathspace}{0ex}}\text{age}\\ & =4x\\ & =24\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\end{array}$

5.A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:

Explanation:

Solution:
Let the son's present age be x years. Then, man's present age = (x + 24) years.
∴ (x + 24) + 2 = 2(x + 2)
x + 26 = 2x + 4
x = 22.

6.Six years ago, the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present?

Explanation:

Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{the}\phantom{\rule{thinmathspace}{0ex}}\text{ages}\phantom{\rule{thinmathspace}{0ex}}\text{of}\phantom{\rule{thinmathspace}{0ex}}\text{Kunal}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\text{Sagar}\phantom{\rule{thinmathspace}{0ex}}\text{6}\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\text{ago}\phantom{\rule{thinmathspace}{0ex}}\\ & \text{be}\phantom{\rule{thinmathspace}{0ex}}6x\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}5x\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\text{respectively}\\ & \text{Then,}\phantom{\rule{thinmathspace}{0ex}}\frac{\left(6x+6\right)+4}{\left(5x+6\right)+4}=\frac{11}{10}\\ & ⇒10\left(6x+10\right)=11\left(5x+10\right)\\ & ⇒5x=10\\ & ⇒x=2\\ & \therefore \text{Sagar's}\phantom{\rule{thinmathspace}{0ex}}\text{present}\phantom{\rule{thinmathspace}{0ex}}\text{age}\\ & =\left(5x+6\right)\phantom{\rule{thinmathspace}{0ex}}\text{years}\\ & =16\phantom{\rule{thinmathspace}{0ex}}\text{years}\phantom{\rule{thinmathspace}{0ex}}\end{array}$

7.The sum of the present ages of a father and his son is 60 years. Six years ago, father's age was five times the age of the son. After 6 years, son's age will be:

Explanation:

Solution:
Let the present ages of son and father be x and (60 -x) years respectively.
Then, (60 - x) - 6 = 5(x - 6)
⇒ 54 - x = 5x - 30
⇒ 6x = 84
x = 14.
∴ Son's age after 6 years = (x+ 6) = 20 years.

8.Sachin is younger than Rahul by 7 years. If their ages are in the respective ratio of 7 : 9, how old is Sachin?

Explanation:

Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{Rahul's}\phantom{\rule{thinmathspace}{0ex}}\text{age}\phantom{\rule{thinmathspace}{0ex}}\text{be}\phantom{\rule{thinmathspace}{0ex}}x\phantom{\rule{thinmathspace}{0ex}}\text{years}.\\ & \text{Then,}\phantom{\rule{thinmathspace}{0ex}}\text{Sachin's}\phantom{\rule{thinmathspace}{0ex}}\text{age}=\left(x-7\right)\phantom{\rule{thinmathspace}{0ex}}\text{years}.\\ & \therefore \frac{x-7}{x}=\frac{7}{9}\\ & ⇒9x-63=7x\\ & ⇒2x=63\\ & ⇒x=31.5\\ & \text{Hence,}\phantom{\rule{thinmathspace}{0ex}}\text{Sachin's}\phantom{\rule{thinmathspace}{0ex}}\text{age}\\ & =\left(x-7\right)\phantom{\rule{thinmathspace}{0ex}}\text{years}\\ & =24.5\phantom{\rule{thinmathspace}{0ex}}\text{years}\end{array}$

9.The present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages (in years).

Explanation:

Solution:
Let their present ages be 4x, 7x and 9x years respectively.
Then, (4x - 8) + (7x - 8) + (9x - 8) = 56
⇒ 20x = 80
x = 4.
∴ Their present ages are 4x = 16 years, 7x = 28 years and 9x = 36 years respectively.

10.Ayesha's father was 38 years of age when she was born while her mother was 36 years old when her brother four years younger to her was born. What is the difference between the ages of her parents?