Question 1032723: Joan is 10 years older than Fiona. They decide to share $400 in the ratio of their ages.
How much do they each get?
[Hint: if Joan = x, then Fiona = x –10.]
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ratio will change depending on the age of joan and fiona.
for example:
joan is 10 years older than fiona.
when fiona is 5, joan is 15, and the ratio of fiona's age to joan's age is 1/3.
when fiona is 15, joan is 25, and the ratio of fiona's age to joan's age is 3/5.
let x = fiona's age.
then x + 10 = joan's age.
the ratio of fiona's age to joan's age is 
if we let y = fiona's share of the pot, then joan's share of the pot will be (400 - y).
we get the ratio:
this means that the ratio of fiona's age to joan's age is equal to the ratio of fiona's share of the pot to joan's share of the pot.
cross multiply to get x * (400-y) = (x+10) * y
simplify to get 400x - xy = xy + 10y
add xy to both sides of this equation to get:
400x = 2xy + 10y
factor out y from the right side of the equation to get:
400x = y * (2x+10)
divide both sides of the equation by (2x+10) to get:
400x / (2x+10) = y
simplify to get:
200x / (x+5) = y
this is the same as:
y = 200x / (x+5)
since 
the ratio of becomes:
when fiona is 5, x is equal to 5.
the ratio of becomes  which becomes (5/15) = 100/300 which becomes 1/3 = 1/3.
the ratios are the same.
fiona's share is 100 when fiona is 5.
joan's share is 300 when fiona is 5.
when fiona is 15, the formula of becomes  which becomes 15/25 = 150/250 which becomes 3/5 = 3/5.
the ratios are the same.
fiona's share is 150 when fiona is 15.
joan's share is 250 when joan is 25.
as the ratio of their ages gets closer, so does each's share of the 400.
when fiona is 70 and joan is 80, you would expect fiona's share of the pot to be 186 and 2/3 while joan's share of the pot would be 213.3333
the ratio of their ages would be 7/8.
the ratio of their share of the pot should be the same.
186 and 2/3 divided by 213 and 1/3 is equal to 7/8 once you simplify it.
the ratio of 7/8 is closer to 1 than the ratio of 3/5 and the ratio of 3/5 is closer to 1 than the ratio of 1/3.
if they could live forever, then at some point in time the ratios would become very very close to 1.
this is, of course, impossible, since their life span is realistically limited to somewhere around 80 to 150 years, depending on the progress of modern science.
the general idea is that, as they get older, the ratio between them becomes closer and their share of the pot becomes closer as well.
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