Question

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

Answer 1

Millie is `26/3` and Jane is `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=2M`

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=2M+9`

Now plug it into the other equation:

`M+2M+9=35`

Consolidate `M`s:

`3M+9=35`

Subtract `9` on both sides:

`3M+9-9=35-9`

This becomes:

`3M=26`

Divide both sides by `3`:

`(3M)/3=26/3`

This becomes:

`M=26/3`

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

`26/3+J=35`

Subtract both sides by `26/3`:

`26/3+J-26/3=35-26/3`

This becomes:

`J=79/3`

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

Answer 2

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

`14 2/3 +20 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)(wwwwww`9 years ago`color(white)(wwww)`present age

Millie:`" "x-9" "larr color(white)(wwwww)color(red)(x)"`
`color(white)(..................)darr`
Jane:`" "color(blue)(2(x-9))" "rarr" "color(red)(2(x-9)+9)`

`9` years ago Jane was twice as old as Millie. (`color(blue)(2(x-9))`)

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

The `color(red)("sum of their present ages is "35)`

`color(red)(x+2(x-9)+9 =35)`

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

`3x-9 =35`

`3x = 44`

`x = 44/3 = 14 2/3" "larr` Millie's present age

`35- 14 2/3 = 20 1/3" "larr` Jane's present age

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