# In need of Algebra assistance

For some reason, my Algebra is getting rusty and I need help solving for the Purchase of the following:

(Sale - Const - Purchase) / (Const + Purchase) = ROI

• Not sure I follow why that's useful to you. A simpler way to state that equation is: gain / cost = ROI. Why is it meaningful to separate purchase from construction cost?
– NL7
Commented Apr 28, 2014 at 20:15
• one reason is in the US you might have some special rules around taxes. For example, the materials (not labor) put into a house can be added to the cost basis, which therefore would affect what people would think of as ROI, ie "money in my pocket at the end of the day" Commented Apr 28, 2014 at 20:24
• Construction and other costs go into ROI, per the formula cited. I could see wanting to know the target purchase price so you know what price range to look at. But the purchase price is affected by anticipated construction, fees, and any other costs. So the purchase price is contingent on anticipated costs, and it's more meaningful to find the cost number rather than the purchase price.
– NL7
Commented Apr 28, 2014 at 20:38

I will also skip the steps, but show the result:

Purchase = (Sale/(ROI+1)) - Const

• That's exactly what I was looking for! Thanks @soakley Commented May 8, 2014 at 17:06

A simpler way to state that equation is: gain / cost = ROI. That makes the math easier.

gain / cost = ROI

multiply each side by cost

gain = ROI * cost

divide each side by ROI

gain / ROI = cost

split up the constituent parts of cost, construction and purchase

gain / ROI = constr + purch

subtract construction from each side

(gain / ROI) - constr = purch

Or, formulated verbally: divide your anticipated gain by your desired ROI to find your total cost, then subtract non-purchase costs (construction, fees, etc.) to isolate the purchase cost.

Here it is framed from your posted equation:

(Sale - Const - Purchase) / (Const + Purchase) = ROI

(Sale - Const - Purchase) = ROI * (Const + Purchase)

(Sale - Const - Purchase) / ROI = (Const + Purchase)

( (Sale - Const - Purchase) / ROI ) - Const = Purchase

Purchase = ( (Sale - Const - Purchase) / ROI ) - Const

• Purchase = ( (Sale - Const - Purchase) / ROI ) - Cons : This is what I'm looking for but Purchase needs to be on the equals side only. What I'm trying to accomplish is figuring out a max purchase price I would be willing to pay given the other variables. Commented Apr 28, 2014 at 20:56
• It's cost that matters for roi. If the property is in better or worse shape, then cost changes because of construction.
– NL7
Commented Apr 29, 2014 at 21:15

It may be easier to just go ahead and use goal seek in excel but solving for your required ROI by changing the Purchase. But the Algebra is as follows (building off where NL7 Left off)

Purchase = ((1/ROI)*(Sale-Const)-Const)/(1+1/ROI)

Forgive me for not showing the rest of the steps but the formula works when plugged into Excel.

• Yes, goal seek does do the appropriate calculation but that misses the point of this exercise. Commented May 8, 2014 at 17:00