Problems based on age Solved with shortcut methods

Equation Problems of Age are part of the quantitative aptitude section. In the equation problems of age, the questions are such that they result in equations. These equations could become either linear or non-linear and they will have solutions that will represent the age of the people in the question. In the following sections, we will some examples of these problems and also learn about the various shortcuts that we can use to solve them.

In such Problems there may be three situations:
i. Age some years ago
ii. Present age
iii. Age some year hence

Now we are going to solve some questions:

Age some years ago:

Ex.1 Ram is a 40-year-old Mohan who is 60 years old. How many years ago was the ratio of their ages of 3:5.
Sol. (40-x)/ 60-x = 3/5
X=10

Present age:

Ex.2 The ratio of the present age of Rahul and Vikram is 2:1. The ratio of their age after 30 years will be 7:6. What is the present age of Rahul?
Sol. Conventional Method.
Let the present age of Rahul and Ram be 2x and x.
After 30 year
(2x+30)/(x+30) = 7/6
12x +180 = 7x + 210
12x-7x = 210-180
5x = 30
X= 6
Rahul’s present age is 2 *6 = 12

Short trick

R0 : V0 = 2:1
R30: V30 = 7:6
Difference between R0 and R30 = 7- 2= 5
Difference between V0 and V30 = 6-1 = 5
This means, 5 Ratio = 30 years
1 ratio = 6
The present age of Rahul and Vikram is = 6*2: 6*1 = 12 and 6 years

Age some year hence:

Ex .3 Neeraj is as younger than Gopal as he is older than Deepak. If the sum of the ages of Gopal and Deepak is 58 yrs. What is Neeraj's age?
Sol. G-D = N-d
G+D = 2N
58 =2N
N= 29 years Ans.

Type 1.

If t1 yrs earlier the father's age was x times that of his son. At present, the father's age is y times that of his son. What are the present ages of the son and the father?
then,
son's age = t1(x-1)/(x-y)

 
Ex. 4 The age of the father 3 years ago was 7 times his son. At present fathers, age is five times that of his son. What are the present ages of the father and the son?
Sol 1 Let the present age of son = x yrs
3 year ago,
7(x-3) = 5x-3
7x-21 = 5x-3
2x= 18
x= 9 yrs
Therefore son's age = 9 yrs
Father's age = 45

Short trick:
Method 1.
 Son's age = 3(7-1)/ (7-5 ) = 9 yrs
 Fathers's age = 9*5 = 45 yrs.

Method 2.
              S: F 
3yrs        1:7 , 1-7 = 6 Difference between ratio
Present  1:5  ,1-5 = 4 Difference between ratio
then,

3yrs        (1:7) *4 =  4: 28
present  (1:5) *6 =  6: 30

2 ratio = 3 yerars
1 = 3/2 years
Present age of son and father = (6 : 30) * 3/2=  9 : 45
The present age of son = 9
The present age of father = 45

Type 2.

The present age of the father is y times the age of his son. T2 years hence, the father's age becomes z times the age of his son. What are the present ages of the father and his son?

Son's age = (z-1)T2/(y-z)
Ex.5. 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 ages of the Father and the son.

Sol. Let the present age of son= x
Then, the present age of father = 5x
3 years hence,
4(x+3) = 5x+3
x = 9 yrs
son's age =9 years
Father's age =45

Shortcut 

Method 1.

Son's age = (4-1)*3/(5-4) = 9 years
Father's age = 9*5 = 45 years

Method 2.

                        S: F

Present  age   (1:5) *3 = 3:15
3 year hence  (1:4) *4 = 4:16

1 ratio = 3

The present age of father and son = (3:15)*3 = 9 years and 45 years.

Type 3.

t1 years earlier, the age of the father was x times the age of his son. t2 years hence, the age of the father becomes z times the age of his son. What are the present ages of the son and the father?

Son's age =t2(z-1) +t1(x-1)/ (x-z)

Ex.6 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 his son?
Sol. Let the present age of son = x years
The present age of father = years
3 years earlier, 7(x-3)= y-3
3year hence, 4(x+3)= y+3
4x+12= y+3
 son solving equation we got
son's age = 9 years
Father's age =  45 years

Shortcut

Method 1.
Son's age = 3(4-1)+ 3(7-1)/ (7-4) =9 years
Fathers' age = 45 years

Method 2.

S: F
Age 3 year ago      (1:7)*3= 3:21
Present age      
Age 3 year hence  (1:4)*6= 6:24

3 ratio = 6 years
1 ratio = 2 years
.5 ratio = 1 year

Age 3 year ago      (1:7)*3= 3   :21
Present age                           4.5:22.5 
Age 3 year hence  (1:4)*6= 6   :24
1 ratio = 2 years

 Present age of son and father = (4.5 : 22.5)*2 = 9 years and 45 years