Question

# Five years later, the father’s age will be three times the age of his son. Five years ago, a father was seven times as old as his son. Find their present ages.

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Hint: Start by taking the present age of father and son to be x and y respectively. Then according to the conditions given in the question, form the equation and solve the equation by performing arithmetic operations.

Let x be the present age of father and y be the present age of son.
According to the question, after 5 years father’s age will be (x+5) and son’s age will be (y+5), then
x+5 = 3(y+5)
x-3y-10 = 0 …..(1)
According to the question, another condition is given such that,
Five years ago, the age of the father was (x-5) and son’s age was (y-5), so according to the condition,
(x-5) = 7(y-5)
x-7y+30 = 0 …..(2)
Now we will solve (1) and (2), by subtracting (1) from (2),
$- 3y + 7y - 10 - 30 = 0$
$\Rightarrow y = 10$
We have found the age of son, now to find the age of father we are going to put the value of y in one of the equations,
Let us put y = 10 in (1),
$x - 3\left( {10} \right) - 10 = 0$
$\Rightarrow x = 40$
Therefore, father’s present age is 40 and son’s present age is 10.

Note: In age problems, if we have one person we can easily solve it by taking one variable and forming an equation using the condition given. In case there are 2 people, we assign one variable to one of the people and the second variable to the other one and then form equations using the conditions given in the question.