It costs 216 dollars to refinish the wood floor of a room. A larger room has the same shape, with corresponding side lengths in the ratio 3 to 4. How much would it cost to refinish the wood floor of the larger room?

Step-by-step solution

1. 1

   Decide what the cost is actually proportional to. Refinishing is priced per square unit of floor, so the cost scales with area, not with side length. Getting this right is the whole problem: multiplying $216$ by $\tfrac43$ would be the classic mistake.

2. 2

   Convert the side ratio into an area ratio. For similar figures, corresponding lengths in the ratio $k$ give areas in the ratio $k^2$. Here

   $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$

   so the areas are in the ratio $9:16$.

3. 3

   Set the small room equal to 9 parts. The \216$covers the smaller floor, which is the$9$-part side of the ratio:

   $9 \text{ parts} = \$216 \;\Longrightarrow\; 1 \text{ part} = \frac{216}{9} = \$24$

   Working with a single part is safer than juggling the fraction, because it makes the arithmetic exact.

4. 4

   Scale up to 16 parts for the larger room.

   $16 \times \$24 = \$384$

5. 5

   Sanity check the size of the answer. The larger floor has $\tfrac{16}{9} \approx 1.78$ times the area, and $216 \times 1.78 \approx 384$. It should be noticeably more than $\tfrac43 \times 216 = 288$, which is exactly the wrong answer you would get by scaling by length instead of area.