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Video: Using Proportions to Solve Problems in a Real-World Context

Tim Burnham

The amount of stain needed to cover a wooden surface is directly proportional to the area of the surface. If 4 pints are required to cover a square deck of side 3 feet, how many pints of stain are needed to paint a square deck of side 8 feet 3 inches?

03:24

Video Transcript

The amount of stain needed to cover a wooden surface is directly proportional to the area of its surface. If four pints are required to cover a square deck of side three feet, how many pints of stain are needed to paint a square deck of side eight feet three inches?

Well first up, let’s define a couple of variables. Let’s let 𝑆 be the amount of stain in pints, and we’ll let 𝐴 be the area of the surface in square feet. Now the question tells us that the amount of stain needed is directly proportional to the area of the surface. So we can write that as 𝑆 is directly proportional to 𝐴. And this just means that 𝑆 is equal to some constant times 𝐴. That’s what direct proportionality means.

Now we know that four pints are required to cover a square deck of side three feet. So if we’ve got a square deck of side three feet, that means each of its sides are three feet. So the area of that deck is gonna be three times three is nine square feet. And we can put that into our equation. We know that the amount of stain is four pints. We don’t know what the constant is yet. That’s what we’re trying to work out here. And that goes with an area of nine square feet. So four is equal to 𝑘 times nine. If I divide both sides of that by nine, the nines cancel from the right-hand side, and it tells me that 𝑘 is equal to four-ninths. So now I’ve got a grand formula that tells me the amount of stain that I need to cover a wooden deck of a certain area. The amount of stain in pints is equal to four-ninths times the area of the deck in square feet.

Now let’s use that formula to answer this last part of the question. We’ve got a square deck of side eight feet three inches. So we need to work out the area of that deck with sides of eight feet three inches. So the calculation is gonna be eight feet three by eight feet three. Well, a foot has got twelve inches in it, so eight feet and three inches means eight and three-twelfths of a foot. So our calculation becomes eight and three-twelfths times eight and three-twelfths. Well the more observant among you will notice that three-twelfths is the same as a quarter. So an even simpler form of that calculation would be eight and a quarter times eight and a quarter.

Turning those into top heavy fractions or improper fractions, that’s 33 over four times 33 over four, which is 1089 over 16 square feet. So let’s just write that into our diagram here and make a little bit of space for some more calculation. Then when the area is 1089 over 16, the volume of stain that we need is four-ninths times 1089 over 16. Now I can cancel top and bottom a little bit. Fours go into 16 four times and nines go into 1089 121 times. So the volume is 121 over four pints. Now we can change that into a mixed number, which would be a really nice way to present our final answer of 30 and a quarter pints.