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 of the examples of these problems and also learn about the various shortcuts that we can use to solve them. Let us start with some easier examples and concepts below.
Equation Problems of Age
Equations are a convenient way to represent conditions or relations between two or more quantities. An equation could have one, two or more unknowns. The basic rule is that if the number of unknowns is equal to the number of conditions, then these equations are solvable, otherwise not. We will see some important examples here but first, let us see the following tricks.
If the age of a person is ‘x’, then ‘n’ years after today, the age = x + n. Similarly, n years in the past, the age of this would have been x – n years.
Example 1: A father and his son decide to sum their age. The sum is equal to sixty years. Six years ago the age of the father was five times the age of the son. Six years from now the son’s age will be:
A) 23 years B) 19 years C) 20 years D) 22 years
Answer: Suppose that the present age of the son is = x years. Then the father’s age is (60 -x) years. Notice that we are trying to reduce the problem into as few variables as possible. As per the second condition of the question, we have:
The age of the father six years ago = (60 – x) – 6 years = 54 – x years.
Similarly the age of the son six years ago will be x – 6 years. Therefore as per the second condition, we have;
54 – x = 5(x – 6) or 54 – x = 5x – 30 and we can write 6x = 84
Hence, we have x = 14 years. Thus the son’s age after 6 years = (x+ 6) = (14 + 6) = 20 years. Hence the correct option is C) 20 years.
More Solved Examples
Example 2: The difference in the age of two people is 20 years. If 5 years ago, the elder one of the two was 5 times as old as the younger one, then their present ages are equal to:
A) 20 and 30 years respectively
B) 30 and 10 years respectively
C) 15 and 35 years respectively
D) 32 and 22 years respectively
Answer: The first step is to find the equation. Let the age of the younger person be x. Then the age of the second person will be (x + 20) years. Five years ago their ages would have been x – 5 years and x + 20 years. Therefore as per the question, we have: 5 (x – 5) = (x + 20 – 5) or 4x = 40 or x = 10.
Therefore the ages are 10 years for the younger one and (10 + 20) years = 30 years for the elder one.
Example 3: Yasir is fifteen years elder than Mujtaba. Five years ago, Yasir was three times as old as Mujtaba. Then Yasir’s present age will be:
A) 29 years B) 30 years C) 31 years D) 32 years
Answer: Let the age of Yasir be = x years. Then the age of Mujtaba will be equal to x – 15 years. Now let us move on to the second condition. Five years ago the age of Yasir will be equal to x – 5 years. Also, the age of Mujtaba five years ago will be x – 15 – 6 years = x – 21 years. As per the question, we have:
3(x – 21) = x – 5 or 3x – 63 = x – 5. Therefore we have: 2x = 58 and hence x = 29 years. Therefore Yasir’s present age is 29 years and the correct option is A) 29 years.
Example 4: Ten years ago, the age of a person’s mother was three times the age of her son. Ten years hence, the mother’s age will be two times the age of her son. The ratio of their present ages will be:
A) 10:19 B) 9: 5 C) 7: 4 D) 7: 3
Answer: Let the age of the son ten years ago be equal to x years. Therefore the age of the mother ten years ago will be equal to 3x. Following this logic, we see that the present age of the son will be equal to x + 10 years and that of the mother will be equal to 3x + 10 years.
The second condition says that ten years from the present, the mother’s age will be twice that of her son. After ten year’s the mother’s age will be 3x + 10 + 10 years and that of the son will be x + 10 + 10 years. As per the question we have:
(3x + 10) +10 = 2 [(x + 10) + 10] or (3x + 20) = 2[x + 20]. In other words, we can write:
x = 20 years. Thus the present age of the mother = 3(20) + 10 = 70 years. Also the present age of the son = 20 + 10 = 30 years. Thus the ratio is 7:3 and the correct option is D) 7:3.
Q 1: The age of a man is 24 years more than his son. In two years, the father’s age will be twice that of his son. Then the present age of his son is:
A) 18 years B) 21 years C) 22 years D) 24 years
Ans: C) 22 years.
Q 2: After 15 years Ramesh’s age will be 5 times his age 5 years ago. What is the present age of Ramesh?
A) 5 years B) 10 years C) 15 years D) 20 years
Ans: B) 10 years