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?

Answer 1

Have you seen systems of equations at all? Like solving by elimination method or substitution method?

Answer 2

Yes I have

Answer 3

but this section which is asking this questions based on rational expressions

Answer 4

Alright, we'll deal with it that way then. I just thought the work might be a little bit uglier this way was all :)
Alright, so we'll let Mandy's age be x and Sandy's age be y. A ratio of 2/5 between their ages simply means:
\[\frac{x}{y}=\frac{2}{5}\]we increase their ages by 8, then their ratio is 1/2, meaning we would have:
\[\frac{x+8}{y+8} = \frac{1}{2}\]So what we can do with this is solve for x or y in one of the equations and do a substitution. This seems like it would be the easiest way of going about this kind of problem. So is this okay so far so I can continue?

Answer 5

Yes understanding so far

Answer 6

Alright, cool. Well, in order to do this, we need some sort of relationship between x and y in order to do this. The easiest thing is to use the first ratio and solve for x in that ratio. Given:
\[\frac{x}{y} = \frac{2}{5}\], we can multiply both sides of the equation by y, which would accomplish getting x by itself. Doing this we have:
\[x = \frac{2y}{5}\]. Because we have this relationship now, we can substitute this value of x into the other equation and we'll be able to solve for y. So with this substitution, ill have:
\[\frac{x+8}{y+8} = \frac{1}{2} \implies \frac{(\frac{2y}{5})+8}{y+8}= \frac{1}{2}\]Not the neatest thing in the world, but do you understand the substitution?

Answer 7

yes i do.

Answer 8

Alright, so all we need to do is properly solve for y then. There are a couple methods for this, so let's see if you're okay with this method. In the complex fraction on the left, I can simply this somewhat by multiplying every term in both the numerator and denominator by the lowest common denominator. Among 2y/5, 8, y, 8, the common denominator is a 5. Multiplying every term by 5 allows us to make that complex fraction a little neater before we proceed. So doing that gives me:
\[\frac{ 5*(\frac{ 2y }{ 5 })+5*8 }{ 5*y+5*8 }\implies \frac{ 2y+40 }{ 5y+40 }\]. The 5's on the top left cancelled eliminating the fraction and the other numbers were simply multiplied. From here I would cross multiply:


Thats probably the most complicated part of the algebra. You can then solve that equation for y. Once you find your y value, plug in into the equation x = 2y/5 and see what you get for x.