Question

I am currently 5 times as old as my son. In 6 years time I will be three times as old as he was then. What are our ages now?

Answer 1

Hint:
We will first assume the age of the father and the age of the son to be any variable. Then we will equate the age of the son with five times the age of the father. This will be our first equation. Then we will add 6 years to each of their ages and then we will equate this age of the son with three times the age of the father to get the second equation. We will solve these two equations to get the value of their present age of the father and his son.

Complete step by step solution:
Let the present age of the father be \[x\] and let the present age of his son be \[y\].
It is given that the father is 5 times as old as his son.
Therefore, the equation formed will be
\[x = 5y\] …………. \[\left( 1 \right)\]
It is given that after six years, the age of the father will be 3 times as the age of his son.
After 6 years, the age of the father will become \[x + 6\] and the age of his son will become \[y + 6\].
According to question, after 6 years, the relation between their ages is given by
\[ \Rightarrow x + 6 = 3\left( {y + 6} \right)\]
Now, we will use the distributive property of multiplication to multiply the terms. Therefore, we get
\[ \Rightarrow x + 6 = y \times 3 + 6 \times 3\]
On multiplying the terms, we get
\[ \Rightarrow x + 6 = 3y + 18\]
Now, subtracting 6 from both sides, we get
\[\begin{array}{l} \Rightarrow x + 6 - 6 = 3y + 18 - 6\\ \Rightarrow x = 3y + 12\end{array}\]
On further simplification, we get
\[ \Rightarrow x - 3y = 12\] …………. \[\left( 2 \right)\]
Now, we will substitute the value of \[x\] from equation \[\left( 1 \right)\] in equation \[\left( 2 \right)\]. Therefore, we get
\[ \Rightarrow 5y - 3y = 12\]
On subtracting the like terms, we get
\[ \Rightarrow 2y = 12\]
On dividing both sides by 2, we get
\[ \Rightarrow y = \dfrac{{12}}{2} = 6\]
Now, we will substitute the value of \[x\] in equation \[\left( 1 \right)\].
\[x = 5 \times 6 = 30\]

Hence, the present age of the father is 30 years and the present age of his son is 6 years.

Note:
Here we have calculated the present age of the father and his son. We need to keep in mind that when we were obtaining the relation between their ages after 6 years, we added 6 to both of their ages, this is necessary to add because after ages will not equal to their present ages.
We have used the distributive property to solve the equation. The distributive property of multiplication which states that if \[a\] , \[b\] and \[c\] are any numbers then according to this property, \[\left( {a + b} \right)c = a \cdot c + b \cdot c\].