Question

The ratio of the ages of Mandy and Sandy is 2:5. After 8 years, their ages will be in the ratio 1:2. What is the difference between their present ages?
Important!!

Answer 1

Let $x$ represent Mandy's age and $y$ represent Sandy's.

From the first statement we have:
$5x=2y$

From the second statement we have:
$2*(x + 8) = 1*(y + 8)$

So we need to solve this system of equations:
$\left \{ {{5x=2y} \atop {2(x+8)=y+8}} \right.$

Express $y$ from first equation equation:
$2y=5x$
$y=\frac{5}{2}x$

Replace found value in second equation and solve the resulting equation:
$2(x+8)=\frac{5}{2}x+8$
$2x+16=\frac{5}{2}x+8$
$2x-\frac{5}{2}x=8-16$
$-\frac{1}{2}x=-8$
$\frac{1}{2}x=8$
$x=8*2=16$

Replace found value in first equation and solve the resulting equation:
$5*16=2y$
$2y=80$
$y=40$

So, Mandy is 16 years old and Sandy is 40 years old. The difference between their ages is 40 - 16 = 24 years.