Question

Emile and Gertrude are brotherand sister. Emile has twice as many sisters as brothers and Gertrude has just as many brothers as sisters. How many children are there in this family?

Short answer

The family has 7 children.

Step-by-step solution

Given that Emile and Gertrude are brother and sister.

Emile has twice as many sisters as brothers, and Gertrude has just as many brothers as sisters. We need to find the total number of children in the family.

Making the equation:

Let be the number of brothers andbe the number of sisters.

As per the question, Emile has twice as many sisters as brothers.

This means he hassisters andbrothers.

So the equation is $\mathrm{y}=2\left(\mathrm{x}-1\right)$……. (1)

Gertrude has as many brothers as sisters. This means she has x brothers and y-1 sisters.

So the equation is $\mathrm{x}=\mathrm{y}-1$……… (2)

Substituting $\mathrm{y}=\left(2\mathrm{x}-1\right)$in this equation

$$\begin{gathered} \mathrm{x}=\mathrm{y}-1 \\ \mathrm{x}=2\left(\mathrm{x}-1\right)-1 \\ \mathrm{x}=2\mathrm{x}-2-1 \\ \mathrm{x}=2\mathrm{x}-3 \\ \mathrm{x}=3 \end{gathered}$$

Now

$$\begin{gathered} \mathrm{y}=\mathrm{x}+1 \\ \mathrm{y}=3+1 \\ \mathrm{y}=4 \end{gathered}$$

So, the family has 3brothers and 4 sisters.

Hence, the total number of children in the family is:

$\begin{array}{rcl}\mathrm{x}+\mathrm{y}& =& 3+4\\ & =& 7\end{array}$

Final answer

Thus the family has 7 children.