Question

Veronika has 3 daughters. Today the eldest is 7 years older the the second who is 2 years older than the youngest. The sum of their ages 12 years from now will be 56. How old are they today?

Answer 1

We define the following variables:

• x = age (in years) of the eldest,

,

• y = age (in years) of the second,

,

• z = age (in years) of the youngest.

From the statement, we know that:

0. the eldest is 7 years older than the second → , x = 7 + y,,

,

1. the second is 2 older than the youngest → ,y = 2 + z,,

,

2. the sum of the ages 12 years from now will be 56 → x + y + z + 12 = 56 → ,x + y + z = 56 - 12 = 44,.

We have the following system of equations:

$\begin{gathered} x=7+y, \\ y=2+z, \\ x+y+z=44. \end{gathered}$

i) Replacing the second equation in the first one, we have:

$x=7+y=7+(2+z)=9+z\text{.}$

ii) Summing the ages of the three daughters, we have:

$x+y+z=(9+z)+(2+z)+z=11+3z\text{.}$

iii) Equalling the last equation with the third one, we have:

$\begin{gathered} 11+3z=44, \\ 3z=44-11=33, \\ z=\frac{33}{3}=11. \end{gathered}$

Replacing the value of z in the equation of x and y, we get:

$\begin{gathered} x=9+11=20, \\ y=2+11=13. \end{gathered}$

Answer

The ages of the daughters are 20, 13 and 11.