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If the total cost of a housing project with f units is. . .

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If the total cost of a housing project with f units is. . .

by Vincen » Tue Nov 21, 2017 8:41 am
If the total cost of a housing project with f units is $1,750,000, what was the cost per unit for the housing project?

(1) If half as many units were built for the same total price, the cost per unit would be $250,000.
(2) If six more units had been built for the same total price, the cost per unit would be $3,750 less.

The OA is D .

Experts how can I use each statement to get an answer? I don't know how to solve it.
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Source: — Data Sufficiency |

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by ErikaPrepScholar » Tue Nov 21, 2017 11:33 am
Let x equal the cost per unit for the housing project. Then the question stem gives:

$$f\cdot x=1,750,000$$

Statement 1

This tells us that we can build f/2 units for 1,750,000, where x = 250,000. In other words:

$$\frac{f}{2}\cdot250,000\ =\ 1,750,000$$

Solving for f gives 14 units. We can then plug into our initial equation:

$$14\cdot x=1,750,000$$

Solving for x gives $125,000 per unit. Sufficient.

Statement 2

This tells us that we can build f + 6 units for 1,750,000 at x - 3,750 per unit. In other words:

$$\left(f+6\right)\cdot\left(x-3,750\right)\ =\ 1,750,000$$

We can rearrange our original equation to get

$$f\ =\ \frac{1,750,000}{x}$$

Then we substitute that in for f
$$\left(\frac{1,750,000}{x}+\ 6\right)\left(x-3750\right)=1,750,000$$

At this point, we could expand, giving a quadratic equation that we can then factor, but we should recognize at this point that we CAN do that, hence we CAN solve for x, and statement 2 is sufficient. These numbers aren't pretty, so continuing to solve the quadratic is a waste of time.
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by ErikaPrepScholar » Tue Nov 21, 2017 11:35 am
Another trick we can use on statement 2: if we have n variables, we need n distinct equations to solve for them.

In statement 2, we have 2 variables (f and x) and 2 distinct equations (the one from the question stem and the one from statement 2). So we should know even without substituting in for f that we can solve for x using these two equations together.
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