Question

Ten years ago a father was six times as old as his daughter. After \[10\] years, he will be twice as old as his daughter. Determine the sum of their present ages.

Answer 1

Hint: Here, we will assume the present age of the father and the daughter to be some variable. Then using the given information we will form two equations and simplify them. Then we will equate the simplified equation and solve it further to get the value of one variable. Then we will substitute this value in any one equation to find the value of the second variable and hence the present age of both father and daughter. We will add their ages to get the required sum.

Complete step-by-step answer:
Let us consider that the present age of the father is \[x\] and the present age of the daughter is \[y\]
Now, it is given that ten years ago father’s age was six times of its daughter’s age.
So, we can mathematically write it as:
\[(x - 10) = 6(y - 10)\]
Now multiplying the terms using distributive property, we get
\[ \Rightarrow x - 10 = 6y - 60\]
Rewriting the equation, we get
\[x = 6y + 10 - 60\]
Subtracting the like terms, we get
\[ \Rightarrow x = 6y - 50\]………………………\[\left( 1 \right)\]
Next, it is given that after 10 years father age will be twice the daughter's age.
Therefore, we will form the equation using the given data.
\[(x + 10) = 2(y + 10)\]
Now multiplying the terms using distributive property, we get
\[ \Rightarrow x + 10 = 2y + 20\]
Rewriting the equation, we get
\[x = 2y + 20 - 10\]
Subtracting the like terms, we get
\[ \Rightarrow x = 2y + 10\]………………….\[\left( 2 \right)\]
Now equating both the equation \[\left( 1 \right)\] and \[\left( 2 \right)\], we get
 \[6y - 50 = 2y + 10\]
Now, taking variable \[y\] on one side, we get
 \[ \Rightarrow 6y - 2y = 50 + 10\]
Adding and subtracting the like terms, we get
\[ \Rightarrow 4y = 60\]
Dividing both sides by 4, we get
\[ \Rightarrow y = 15\]
Now, substituting value of \[y\] in equation \[\left( 1 \right)\] we get
\[x = 6 \times 15 - 50\]
Simplifying the expression, we get
\[ \Rightarrow x = 40\]
Therefore, the age of the father is \[40\] and the age of daughter is \[15\].
Now we will find the sum of their age. Therefore, we get
\[{\rm{Sum}} = 40 + 15 = 55\]
Hence, the sum of their present age is 55 years.

Note: Here we have formed a linear equation in two variables. A linear equation is an equation which has the highest degree of one and has only one solution. A linear equation can be divided on the basis of the number of variables present such as linear equation in one variable, linear equation in two variables, etc. In order to find two distinct variables, we need to have two equations. This means that to find the values of the variable, the number of equations should be equal to the number of variables.