Question

The product of the ages of two sisters is $104$. The difference between their ages is $5$ years. Find their ages.

Answer 1

Step 1: Determining the quadratic equation in $x$

Let the age of the first sister be $x$ years.

Given that the product of their ages is $104$.

So, the age of the second sister will be $\frac{104}{x}$ years.

The difference between their ages is $5$ years.

$$\begin{gathered} \Rightarrow x-\frac{104}{x}=5 \\ \Rightarrow x^2-104=5x \\ \Rightarrow x^2-5x-104=0 \end{gathered}$$

Step 2: Determining the age of the two sisters

By factorization method,

$$\begin{gathered} \Rightarrow x^2-5x-104=0 \\ \Rightarrow x^2-13x+8x-104=0 \\ \Rightarrow x(x-13)+8(x-13)=0 \\ \Rightarrow (x-13)(x+8)=0 \\ \Rightarrow x=-8,13 \end{gathered}$$

The age of a person cannot be negative.

So, the age of the first sister is $x=13$ years

and the age of the second sister is $\frac{104}{13}=8$ years

Therefore, the age of the two sisters are $13$ years and $8$ years.