Question

Shona’s mother is four times as old as Shona. After five years, her mother will be three times as old as Shona (at that time). What are their present ages?

Answer 1

Hint:
We will assume that Shona is \[x\] years old. We will formulate a linear equation using the information given in the question. We will solve the equation for \[x\] and find the ages of Shona and her mother.

Complete step by step solution:
Let Shona is \[x\] years old.
We know that Shona’s mother is four times as old as Shona. Therefore, her mother’s age will be \[4x\].
After 5 years, Shona’s age will be \[x + 5\] and her mother’s age will be \[4x + 5\].
We know that after five years, Shona’s mother will be three times as old as Shona at that time. We will formulate a linear equation for this:
\[\Rightarrow 4x + 5 = 3\left( {x + 5} \right)\]
We know that according to the distributive property multiplying the sum of two or more numbers by a number will give the same result as multiplying each number individually by the number and then adding the products together:
\[\Rightarrow a\left( {b + c} \right) = ab + ac\]
We will solve the bracket on the right-hand side using the distributive property. We will substitute 3 for \[a\], \[x\] for \[b\] and 5 for \[c\]in the above formula:
\[\begin{array}{l}\Rightarrow 4x + 5 = 3 \cdot x + 3 \cdot 5\\ \Rightarrow 4x + 5 = 3x + 15\end{array}\]
We will subtract \[3x\] from both the sides and collect the like terms:
\[\begin{array}{l} \Rightarrow 4x + 5 - 3x = 3x + 15 - 3x\\ \Rightarrow \left( {4x - 3x} \right) + 5 = \left( {3x - 3x} \right) + 15\\ \Rightarrow x + 5 = 15\end{array}\]
We will subtract 5 from the sides:
\[\begin{array}{l} \Rightarrow x + 5 - 5 = 15 - 5\\ \Rightarrow x = 10\end{array}\]
We have found that \[x\] is 10. We will find the value of \[4x\]:
\[\Rightarrow 4x = 4 \cdot 10 = 40\]

So, we can conclude that Shona is 10 years old and her mother is 40 years old.

Note:
We can assume any unknown quantity as \[x\] and solve the equations formed accordingly to obtain the ages. For example, we can assume that Shona’s mother’s age is \[x\]. So, Shona’s age will be \[\dfrac{x}{4}\] because her mother’s age is 4 times that of Shona. After 5 years, Shona’s age will be \[\dfrac{x}{4} + 5\] and her mother’s age will be \[x + 5\]. We know that after five years, Shona’s mother will be three times as old as Shona at that time. We will formulate a linear equation for this:
\[\Rightarrow x + 5 = 3\left( {\dfrac{x}{4} + 5} \right)\]
We will solve the bracket on the right-hand side using the distributive property. We will substitute 3 for \[a\], \[\dfrac{x}{4}\] for \[b\] and 5 for \[c\] in the above formula:
\[x + 5 = \dfrac{{3x}}{4} + 15\]
We will subtract \[\dfrac{{3x}}{4}\] from both the sides and collect the like terms:
\[\begin{array}{l} \Rightarrow x + 5 - \dfrac{{3x}}{4} = \dfrac{{3x}}{4} + 15 - \dfrac{{3x}}{4}\\\left( {x - \dfrac{{3x}}{4}} \right) + 5 = \left( {\dfrac{{3x - 3x}}{4}} \right) + 15\\\dfrac{x}{4} + 5 = 15\end{array}\]
We will subtract 5 from the sides:
\[\begin{array}{l}\Rightarrow\dfrac{x}{4} + 5 - 5 = 15 - 5\\ \Rightarrow \dfrac{x}{4} = 10\end{array}\]
We have found that \[\dfrac{x}{4}\] is 10. We will find the value of \[x\]:
\[4 \cdot \dfrac{x}{4} = 4 \cdot 10 = 40\]
 So, we can conclude that Shona is 10 years old and her mother is 40 years old.