Question

Six years ago a father was seven times as old as his son. After six years he will be three times as old as his son. Determine their present ages.

Answer 1

Hint: To find the son and father ‘s age first find the age of the son by assuming son’s age as an unknown variable and then place that variable according to the age instruction given in the question. To find the son’s age we need to equate the present age of the son to that of the father and then find the value of the common unknown variable.

Complete step-by-step answer:
Let us take the present age of the son in son and father combination as \[x\] .
Hence, six years ago the age of the son was around \[x-6\] years.
Meanwhile when compared to the father his age was around \[7\left( x-6 \right)\] years.
Thereby, the present age of the father is calculated as \[7\left( x-6 \right)+6\] years.
And similarly, the age after six years for the son is \[x+6\] years.
And also similarly, the father‘s age after six years will change to \[7\left( x-6 \right)+6+6\] years.
Now as given in the question, the father’s current age will be that of three times that of the son’s age. Hence, placing the values in equation we get the age of the father equal to that of the son as:
 \[\Rightarrow 7\left( x-6 \right)+6+6=3\left( x+6 \right)\]
Shifting the values of x and the variables in LHS and RHS respectively, we get the value of the x variable as:
 \[\Rightarrow 7x-42+12=3x+18\]
 \[\Rightarrow 4x-30=18\]
 \[\Rightarrow 4x=48\]
 \[\Rightarrow x=\dfrac{48}{4}\]
 \[\Rightarrow x=12\]
Therefore, the value of the unknown/common variable \[x\] is \[12\] .
Now putting the value of \[x\] in the age equation of father and son, we get the age of father and son as:
x and \[7\left( x-6 \right)\] years respectively.
Hence, the ages of son and father are calculated as \[12\] and \[7\left( 12-6 \right)\] .
 \[12,48\] years respectively.
So, the correct answer is “ \[12,48\] years”.

Note: Equating age doesn’t have to be present age, it can be any age period future, past or present, the only thing one must remember that the age period during equating should be same as that of the father and son, if it is present then present for both of them, if past then past for both of them and so on.