Help Solving this Equation for X

[BSmith32]

I am having trouble getting this equation down to where you are solving for X. I am looking to buy investment properties and I want to obtain a certain Rate of Return. So the purpose of the equation is to find out what Purchase Price you would need in order to obtain a specific ROI with given Costs and a given Sale Price.
If ROI = R, Costs = C, S = Sale Price, and X = Purchase Price, then the equation should be as follows:

ROI = ((S - (C+X)) / (C + X)

So from here is there a way to get this down to where you are solving for X?

Thanks for any help in advance.

 

[DrPhil]

I am having trouble getting this equation down to where you are solving for X. I am looking to buy investment properties and I want to obtain a certain Rate of Return. So the purpose of the equation is to find out what Purchase Price you would need in order to obtain a specific ROI with given Costs and a given Sale Price.
If ROI = R, Costs = C, S = Sale Price, and X = Purchase Price, then the equation should be as follows:

ROI = ((S - (C+X)) / (C + X)

So from here is there a way to get this down to where you are solving for X?

Thanks for any help in advance.

First, solve for (C+X). After that, subtract C from both sides.

If you need further hints, we need to see your work.

 

[JeffM]

I am having trouble getting this equation down to where you are solving for X. I am looking to buy investment properties and I want to obtain a certain Rate of Return. So the purpose of the equation is to find out what Purchase Price you would need in order to obtain a specific ROI with given Costs and a given Sale Price.
If ROI = R, Costs = C, S = Sale Price, and X = Purchase Price, then the equation should be as follows:

ROI = ((S - (C+X)) / (C + X)

So from here is there a way to get this down to where you are solving for X?

Thanks for any help in advance.

Yes, but your equation does not measure ROI, which usually has a time dimension. It will, however, be a good approximation if the entire transaction can be completed in approximately a year.

\(\displaystyle R = \dfrac{S - (C + X)}{C + X} \implies R(C + X) = S - (C + X) \implies RC + RX = S - C - X \implies X + RX = S - C - RC \implies\)

\(\displaystyle X(1 + R) = S - C(1 + R) \implies X = \dfrac{S}{1 + R} - C.\)

Let's check. Say you want a return of 10%, expect costs of 20K and a sales price of 100K. The formula says X should be

\(\displaystyle \dfrac{100}{1 + 0.1} - 20 = \dfrac{100}{1.1} - 20 \approx 70.91K\).

So your investment would be \(\displaystyle 20K + 70.91K = 90.91K,\ and\ \dfrac{100K - 90.91K}{90.91K} = \dfrac{9.09}{90.91} \approx 10\%.\)

 

[BSmith32]

Very Helpful. Thank you

Yes, but your equation does not measure ROI, which usually has a time dimension. It will, however, be a good approximation if the entire transaction can be completed in approximately a year.

\(\displaystyle R = \dfrac{S - (C + X)}{C + X} \implies R(C + X) = S - (C + X) \implies RC + RX = S - C - X \implies X + RX = S - C - RC \implies\)

\(\displaystyle X(1 + R) = S - C(1 + R) \implies X = \dfrac{S}{1 + R} - C.\)

Let's check. Say you want a return of 10%, expect costs of 20K and a sales price of 100K. The formula says X should be

\(\displaystyle \dfrac{100}{1 + 0.1} - 20 = \dfrac{100}{1.1} - 20 \approx 70.91K\).

So your investment would be \(\displaystyle 20K + 70.91K = 90.91K,\ and\ \dfrac{100K - 90.91K}{90.91K} = \dfrac{9.09}{90.91} \approx 10\%.\)