### 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.