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Hint: First of all, take Aftab’s age as A years and his daughter’s age as D. Then find their ages seven years ago and 3 years later. Use the conditions given in the question to find the values of A and D.

Complete step-by-step answer:

We are given that, seven years ago, Aftab was 7 times as old as his daughter was then. Also, three years from now, he shall be three times as old as his daughter would be. We have to represent this situation algebraically.

Let us assume that the present age of Aftab is A years. Also, let us assume that the present age of his daughter is D years.

So, seven years ago, the age of Aftab = (A – 7) years

Seven years ago, the age of his daughter = (D – 7) years

Now we are given that 7 years ago, Aftab was 7 times as old as his daughter, so we get,

(Aftab’s age seven years ago) = 7 (His daughter’s age seven years ago)

By substituting the value of LHS and RHS, we get,

\[\left( A7 \right)=7\left( D7 \right)\]

By simplifying the above equation, we get,

\[A-7=7D-49\]

Or, \[7D-A=49-7\]

\[\Rightarrow 7D-A=42....\left( i \right)\]

Also, three years later, the age of Aftab would be = (A + 3) years.

Three years later, the age of his daughter would be = (D + 3) years.

We are given that, three years from now, Aftab will be 3 times as old as his daughter will be. So, we will get,

(Aftab’s age after 3 years) = 3 (His daughter’s age after 3 years)

By substituting the value of LHS and RHS, we get,

\[\left( A+3 \right)=3\left( D+3 \right)\]

By simplifying the above equation, we get,

\[\left( A+3 \right)=3D+9\]

Or, \[A-3D=6....\left( ii \right)\]

So, we get two algebraic equations involving their ages as,

\[\begin{align}

& 7D-A=42...\left( i \right) \\

& A-3D=6....\left( ii \right) \\

\end{align}\]

Now by adding equation (i) and (ii), we get,

\[\left( 7D-A \right)+\left( A-3D \right)=42+6\]

By simplifying the above equation and canceling the terms, we get,

\[\Rightarrow 4D=48\]

By dividing 4 on both sides of the above equation, we get,

\[\Rightarrow D=12\text{ years}\]

By substituting D = 12 in equation (i), we get,

\[\begin{align}

& 7\left( 12 \right)-A=42 \\

& 84-A=42 \\

& A=84-42 \\

\end{align}\]

So, we get, A = 42 years.

Therefore, we get Aftab’s and his daughter’s age as 42 years and 12 years respectively.

Note: Many students make the mistake while writing the equation for the given conditions because they often write the ages of the people some years ago and some years later incorrectly. Also, students often make the mistake of writing the age of one person properly while the other person incorrectly. So this must be taken care of. Whenever there is a condition given for ‘m’ years ago, then subtract ‘n’’ from the age of each person while if it is for ‘n’ years later, then add ‘n’ to the age of each person

Complete step-by-step answer:

We are given that, seven years ago, Aftab was 7 times as old as his daughter was then. Also, three years from now, he shall be three times as old as his daughter would be. We have to represent this situation algebraically.

Let us assume that the present age of Aftab is A years. Also, let us assume that the present age of his daughter is D years.

So, seven years ago, the age of Aftab = (A – 7) years

Seven years ago, the age of his daughter = (D – 7) years

Now we are given that 7 years ago, Aftab was 7 times as old as his daughter, so we get,

(Aftab’s age seven years ago) = 7 (His daughter’s age seven years ago)

By substituting the value of LHS and RHS, we get,

\[\left( A7 \right)=7\left( D7 \right)\]

By simplifying the above equation, we get,

\[A-7=7D-49\]

Or, \[7D-A=49-7\]

\[\Rightarrow 7D-A=42....\left( i \right)\]

Also, three years later, the age of Aftab would be = (A + 3) years.

Three years later, the age of his daughter would be = (D + 3) years.

We are given that, three years from now, Aftab will be 3 times as old as his daughter will be. So, we will get,

(Aftab’s age after 3 years) = 3 (His daughter’s age after 3 years)

By substituting the value of LHS and RHS, we get,

\[\left( A+3 \right)=3\left( D+3 \right)\]

By simplifying the above equation, we get,

\[\left( A+3 \right)=3D+9\]

Or, \[A-3D=6....\left( ii \right)\]

So, we get two algebraic equations involving their ages as,

\[\begin{align}

& 7D-A=42...\left( i \right) \\

& A-3D=6....\left( ii \right) \\

\end{align}\]

Now by adding equation (i) and (ii), we get,

\[\left( 7D-A \right)+\left( A-3D \right)=42+6\]

By simplifying the above equation and canceling the terms, we get,

\[\Rightarrow 4D=48\]

By dividing 4 on both sides of the above equation, we get,

\[\Rightarrow D=12\text{ years}\]

By substituting D = 12 in equation (i), we get,

\[\begin{align}

& 7\left( 12 \right)-A=42 \\

& 84-A=42 \\

& A=84-42 \\

\end{align}\]

So, we get, A = 42 years.

Therefore, we get Aftab’s and his daughter’s age as 42 years and 12 years respectively.

Note: Many students make the mistake while writing the equation for the given conditions because they often write the ages of the people some years ago and some years later incorrectly. Also, students often make the mistake of writing the age of one person properly while the other person incorrectly. So this must be taken care of. Whenever there is a condition given for ‘m’ years ago, then subtract ‘n’’ from the age of each person while if it is for ‘n’ years later, then add ‘n’ to the age of each person

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