### Problem Based On Ages

To solve the problems based on ages, students require the knowledge of linear equations. This method needs some basic concepts as well as some more time than it deserves. Sometimes it is easier to solve the problems by taking the given choices in the account. But this hit- and- trial method proves costly sometimes when we reach our solution much later. We have tried to evaluate some easier as well as quicker methods to solve this type of question. Although , we are not able to cover each type of questions in this section, our attempt is to minimize your difficulties.

### Important Tips

1) If the current age of a person be x, then
– age after n years = x + n
– age n years ago = x – n
– n times the age = nx
– If ages in the numerical are mentioned in ratio a : b, then a : b will be ax and bx
1n of the age is xn

2)If sum of ages of x and y is A and ratio of their ages is a : b respectively, then u can determine age of y by using the formula shown below:

Age of y = Ratio of ySum of ratios × sum of ages
Age of y = a(a + b) × A

3)Father : son
Present age = x : y
After T yrs = a : b
Then, son’s age = T(a – b)difference of cross product * y
And, father’s age = T(a – b)difference of cross product * x

### Problem Based On Ages

1.) The age of the father 3 years ago was 7 times the age of his son. At present, the father’s age is five times of his son. What are the present ages of the father and the son?
Solution:
Let the present age of son = x yrs
Then, the present age of father = 5x yrs
3 years ago,
7(x – 3) = 5x – 3
Or, 7x – 21 = 5x – 3
Or, 2x = 18 thus x = 9 yrs
Therefore, the son’s age = 9 yrs
Father’s age = 45 yrs

2). At present, the age of the father is five times the age of his son. Three years hence, the father’s age would be four times that of his son. Find the present age of the father and son?
Solution:
Let the present age of son = x yrs
Then, the present age of father = 5x yrs
3 yrs hence,
4(x + 3) = 5x + 3
Or, 4x + 12 = 5x + 3
Thus x = 9 yrs.
Therefore, the son’s age = 9 yrs
And father’s age = 45 yrs.

3). Three years earlier, the father was 7 times as old as his son. Three years hence, the father’s age would be four times that of his son. What are the present ages of the father and the son?
Solution:
Let the present age of son = x yrs and the present age of father = y yrs
3 yrs earlier,
7(x – 3) y – 3 or,
7x – y = 18 …………….(1)
3 yrs hence,
4(x + 3) = y + 3
Or, 4x + 12 = y + 3 or,
4x – y = -9…………………….(2)
Solving (1) & (2) we get ,
X = 9 yrs & y = 45 yrs.

4). The sum of the ages of mother and her daughter is 50 yrs. Also 5 yrs ago, the mother’s age was 7 times the age of the daughter. What are the present ages of the mother and the daughter?
Solution:
Let the age of the daughter be x yrs.
Then, the age of the mother is (50-x) yrs.
5 yrs ago, 7(x – 5) = 50 – x = 5
Or, 8x = 50 – 5 + 35 = 80
Thus, x = 10
Therefore, the daughter’s age = 10 yrs
And mother’s age = 40 yrs.

5). The sum of the ages of a son and father is 56 yrs. After 4 yrs, the age of the father will be three times that of a son. What is the age of the son?
Solution:
Let the age of the son be x yrs.
Then, the age of the father is (56 – x) yrs.
After 4 yrs, 3(x + 4) = 56 – x + 4
Or, 4x = 56 + 4 – 12 = 48
Thus, x = 12 yrs.
Son’s age = 12 yrs.

6).The ratio of the father’s age to the son’s age is 4:1. The product of their ages is 196. What will be the ratio of their ages after 5 years?
Solution:
let the ratio of proportionality be x , then
4x*x = 196 or, 4×2 = 196 or, x = 7
Thus, father’s age = 28 yrs, son’s age = 7 yrs
After 5 yrs, father’s age = 33 yrs.
Son’s age = 12 yrs
Thus ratio = 33 : 12 = 11 : 4

7). The ratio of Rita’s age to the age of her mother is 3:11. The difference of their ages is 24 yrs. What will be the ratio of their ages after 3 yrs?
Solution:
Difference in ratios = 8
Then, 8 = 24 thus, 1 = 3
i.e., value of 1 in the ratio is equivalent to 3 yrs
thus, Rita’s age = 3 x 3 = 9 yrs.
Mother’s age = 11 x 3 = 33 yrs.
After 3 years, the ratio = 12 : 36 = 1 : 3.

8). 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. What will the age of son after 6 years?
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.

9). The ratio of the ages of the father and the son at present is 6 : 1. After 5 yrs, the ratio will become 7 : 2. What is the present age of the son?
Solution:
The ratio of father & Son on present age = 6 : 1
After 5 yrs = 7 : 2
Son’s age = 5(7 – 2)6 x 2 – 7 x 1 x 1 = 5 yrs. …(from point 3 formula)
Father’s age =5(7 – 2)6 x 2 – 7 x 1 x 6 = 30 yrs. ..(from point 3 formula)

10). The ratio of the ages of the father and the son at present is 3:1. 4 years earlier, the ratio was 4:1. What are the present ages of the son the father?
Solution:
Father : Son
Present age = 3 : 1
4 years before = 4 : 1
Son’s age = 4(4 – 1)4 x 1 – 3 x 1 x 1 = 12 yrs. –(from formula 3)
Father’s age = 4(4 – 1)4 x 1 – 3 x 1 x 3 = 36 yrs.–(from formula 3)

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