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Age problems are commonly asked in competitive exams. In these quantitative ability related conceptual questions, the people involved are part of the same family or friends or are related in some form or the other. One needs to know the basics of linear equations in a single variable and two variables to solve such problems. The first task is to form an equation around the unknowns from the data given in the question and then solve it to get the value of the required variable.
Problems on Ages
We will be solving some examples of problems based on age and explaining how to go about them easily and efficiently.
Question 1
The age of the father is three times the age of his son Rohan. After eight years, the father’s age will be 2.5 times the age of his son. After another six years, what will be the ratio of their ages?
SOLN:
Let us assume that the age of the son is x (We take the son’s age to be x as it would be smaller, and thus we will need to multiply three by it to get the father’s age).
Age of son (initially) = x
Age of father (initially) = 3x
Age of son (after 8 years) = x+8
Age of father (after 8 years) = 3x+8 = 2.5(x+8)
3x+8 = 2.5x+20 is our required equation.
Solving this we get: 0.5x = 12; x = 24
The age of the son is currently 24 years.
The age of the father is currently 72 years.
After 14 years (8+6), the age of the son will be 38 years.
After 14 years (8+6), the age of the father will be 86 years.
After 14 years, the ratio of their ages will be 19:43.
Question 2
The sum of ages of 5 children born at an interval of 2 years is 50 years. Find the age of the youngest child.
SOLN:
Let the age of the youngest child be y (We take the age of the youngest child as y because then we get the age of the other children in the form of addition instead of subtraction).
The age of the 2nd child will be y+2.
The age of the 3rd child will be y+4.
The age of the 4th child will be y+6.
The age of the 5th child will be y+8.
Sum= y+y+2+y+4+y+6+y+8 = 5y+20 =50
5y + 20 = 50 is our required equation
5y = 30; y =6 years
The age of the youngest child is six years.
(A more efficient way of assuming the ages would have been to take ages to be y-4, y-2, y, y+2, y+4 as in this way, the positive and negative terms cancel each other out, and we directly obtain the value of y with the minimal calculation).
NOTE: In this method, y is the age of the middle child, so to find the age of the youngest child, we will have to subtract it by 4 and report the value of y-4 as the answer.
Question 3
What is the current age of Om if, after 20 years, his age will be ten times what it was 10 years ago? What is his present age?
SOLN:
Let the current age of Om be x years.
Age of Om after 20 years = x+20
Age of Om 10 years ago= x-10
Given that 10(x-10) = x+20
10x-100 = x+20
9x = 120
x = 120/9 = 13.33 years.
The current age of Om is 13.33 years.
Question 4
The ages of Ram and Shyam are in the ratio of 4:5. Five years ago, their ages were in the ratio of 7:9. Calculate the age of Ram and Shyam.
SOLN:
This question is based on the usage of linear equations in two variables. We first make two equations using the data given in the question and then solve them simultaneously using substitution or elimination to get the values of the two unknowns.
Let the age of Ram and Shyam be x and y, respectively.
5x-4y=0
Age of Ram 5 years ago = x-5
Age of Shyam 5 years ago = y-5
9x-45=7y-35
9x-7y=10 is the first equation
5x-4y=0 is the second equation
Multiplying the 1st equation by 5 and the 2nd equation by 9, we get:
45x-35y=50
45x-36y=0
Subtracting the two equations, we get: y = 50
Solving for x, we get x =40
Conclusion
Problems based on age require a basic knowledge of linear equations in a single variable and linear equations in two variables. We are given a set of conditions and data, which we use to make equations and use them to calculate the unknowns. We first need to identify the unknown values from the given data set and assume its value to be a variable like x or y. We need to use the given data and conditions to form equations involving the variables. Depending on the number of equations, the number of variables might vary. Generally, we deal with one variable or two variables. We need to solve the formed equations by using elimination or substitution to get the value of the unknowns.
