Question

A man is four times as old as his son. After 16 years, he will be only twice as old as his son. Find their present ages.

Answer 1

Hint: Here, we need to find the present ages of the man and his son. We will use the given information to form two equations. Then, we will solve these equations to get the values of the present ages of the man and his son.

Complete step by step solution:
Let the present age of the man and his son be \[x\] and \[y\] respectively.
The ages of the man and his son 16 years later will be 16 more than their present ages.
Thus, we get the ages of the man and his son 16 years later as \[x + 16\] and \[y + 16\] respectively.
Now, it is given that the present age of the man is 4 times his son’s present age.
Thus, we get the equation
\[x = 4y\]
The age of the mean 16 years later will be 2 times the age of his son at that time.
Thus, we get the equation
\[x + 16 = 2\left( {y + 16} \right)\]
Now, we will solve the two equations to get the values of \[x\] and \[y\], and hence, the present ages of the man and his son.
Substituting \[x = 4y\] in the equation \[x + 16 = 2\left( {y + 16} \right)\], we get
\[ \Rightarrow 4y + 16 = 2\left( {y + 16} \right)\]
Multiplying the terms 2 and \[y + 16\] using the distributive law of multiplication, we get
\[ \Rightarrow 4y + 16 = 2y + 32\]
Subtracting \[2y\] from both sides of the equation, we get
\[ \Rightarrow 4y + 16 - 2y = 2y + 32 - 2y\\ \Rightarrow 2y + 16 = 32\]
Subtracting 16 from both sides of the equation, we get
\[\Rightarrow 2y + 16 - 16 = 32 - 16\\ \Rightarrow 2y = 16\]
Dividing both sides by 2, we get the value of \[y\] as
\[ \Rightarrow y = \dfrac{{16}}{2} = 8\]
Therefore, the present age of the son of the man is 8 years.
Substitute 8 for \[x\] in the expression \[x = 4y\], we get
Present age of the man \[ = 4y = 4 \times 8 = 32\] years
Therefore, we get the present ages of the man and his son as 32 years and 8 years respectively.

Note: We have used the distributive law of multiplication to find the product of 2 and \[y + 16\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We can verify the answer by using the given information.
The man’s present age is 32 years, and his son’s present age is 8 years.
We can observe that \[8 \times 4 = 32\].
Therefore, we have verified that the man’s present age is four times the age of his son.
The man’s age 16 years later will be \[32 + 16 = 48\] years.
The man’s son’s age 16 years later will be \[8 + 16 = 24\] years.
Now, we can observe that \[24 \times 2 = 48\].