Will, Mike, and Sue went to dinner. Will paid 1/3 of the dinner bill. Micah and Sue paid in the ratio 2:3. If Sue paid $6 more than Will, how much did the dinner cost?

Let's call their parts w, m and s

if will paid 1/3, then m+s=2w (they paid 2/3, which is twice as much as will did)

Now, we know that:

Micah and Sue paid in the ratio 2:3.

this means the 3m=2s

and m=2/3s

Again:

m+s=2w

and we substitute m:

2/3s+s=2w

5/3s=2w / / multiply both sides by 3

5s=6w

we also know that s=w+6 (from the last sentence) so we substitute:

5 (w+6) = 6w

5w+30=6w

30=w

so, Will paid 30, Sue paid 36 (six more than him), Mike paid 24 (24:36 is the same ratio as 2:3, you can check this by dividing both 24 and 36 by 12: you have 2 and 3)

and the total was 30+36+24=90.