# 9 years ago Jane was twice as old as Millie. The sum of their ages now is 35. How old is Millie now?

Jun 4, 2017

Millie is $\frac{26}{3}$ and Jane is $\frac{79}{3}$

#### Explanation:

Let's translate the word problem into a system of equations. Let $J$ be Jane and $M$ be Millie.

Jane was twice as old as Millie $9$ years ago. That means we will have to double $m$ and subtract $9$ from J.

$J - 9 = 2 M$

Adding their ages together will give $35$

$M + J = 35$

Our problem is in an easy situation for substitution. However, you could use elimination if you wanted to. I will stick with substitution. Solve for $J$ in the first equation:

$J = 2 M + 9$

Now plug it into the other equation:

$M + 2 M + 9 = 35$

Consolidate $M$s:

$3 M + 9 = 35$

Subtract $9$ on both sides:

$3 M + 9 - 9 = 35 - 9$

This becomes:

$3 M = 26$

Divide both sides by $3$:

$\frac{3 M}{3} = \frac{26}{3}$

This becomes:

$M = \frac{26}{3}$

Now we can plug this back into one of the original equations. I will plug it into the second equation:

$\frac{26}{3} + J = 35$

Subtract both sides by $\frac{26}{3}$:

$\frac{26}{3} + J - \frac{26}{3} = 35 - \frac{26}{3}$

This becomes:

$J = \frac{79}{3}$

So Millie is not $13$ and Jane is not $22$

Jul 5, 2017

Millie is 14 years and 8 months old and
Jane is 20 years and 4 months old.

$14 \frac{2}{3} + 20 \frac{1}{3} = 35$

#### Explanation:

We are working with two people and 2 periods of time.
Jane and Millie were both $9$ years younger than their present age.

Millie is the younger of the two, let her present age be $x$ years.

color(white)(wwwwww9 years ago$\textcolor{w h i t e}{w w w w}$present age

Millie:$\text{ "x-9" "larr color(white)(wwwww)color(red)(x)}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots} \downarrow$
Jane:$\text{ "color(blue)(2(x-9))" "rarr" } \textcolor{red}{2 \left(x - 9\right) + 9}$

$9$ years ago Jane was twice as old as Millie. ($\textcolor{b l u e}{2 \left(x - 9\right)}$)

Jane's present age is $9$ years older than $9$ years ago.

The $\textcolor{red}{\text{sum of their present ages is } 35}$

$\textcolor{red}{x + 2 \left(x - 9\right) + 9 = 35}$

$x + 2 x - 18 + 9 = 35$

$3 x - 9 = 35$

$3 x = 44$

$x = \frac{44}{3} = 14 \frac{2}{3} \text{ } \leftarrow$ Millie's present age

$35 - 14 \frac{2}{3} = 20 \frac{1}{3} \text{ } \leftarrow$ Jane's present age

Millie is 14 years and 8 months old and
Jane is 20 years and 4 months old.