Video Transcript
A farmer has 53 and one-third feet
of fencing and wants to build a square pen for his chickens? What is the area of the largest pen
he can build?
We’re told that the farmer wants to
build a square pen using 53 and one-third feet of fencing. Since the fencing encloses the
square, we can say that 53 and one-third feet is the perimeter of our square. Now, we also know that squares have
four sides of equal length. Let’s call these 𝑥 feet.
So let’s begin by finding the value
of 𝑥. To do this, we divide 53 and
one-third by four. But we can’t really do that until
we’ve converted our mixed number, 53 and one-third, into an improper or top heavy
fraction. To do that, we multiply the integer
part by the denominator of the fraction. That’s 53 times three, which is
159. We then add this value to the
numerator of our fraction. 159 plus one is 160. This part forms the numerator of
our top heavy or improper fraction. The denominator stays the same. It’s three.
And so the value of 𝑥 can be
calculated by dividing 160 over three by four. Let’s write four as four ones or
four over one and remind ourselves that to divide by a fraction we multiply by the
reciprocal of that fraction. This process is sometimes called
“keep, change, flip.”
We’re now going to cross-cancel by
noticing that both 160 and four are divisible by four. When we divide 160 by four, we get
40. 40 times one is 40, and three times
one is three. So we find that 𝑥 is equal to 40
over three or forty thirds.
Now, at this stage, we might be
tempted to convert this back into a mixed number. But let’s check we’ve actually
finished working with our fractions. We now know that the side length of
our square is forty thirds feet. And the question asked us to find
the area of the largest pen he can build. So we recall that the area of a
square is found by squaring its side lengths. So the area of the pen is forty
thirds squared.
Notice that had we converted this
back into a mixed number, we wouldn’t be able to easily calculate this. But to square a single fraction, we
square both the numerator and denominator. 40 squared is 1600, and three
squared is nine. So the area is 1600 over nine
square feet.
We’re now ready to convert this
back into a mixed number. This time, we divide the numerator
by the denominator. 1600 divided by nine is 177 with a
remainder of seven. 177 forms the integer part of our
mixed number, whereas the remainder forms the numerator of the fractional part. The denominator remains
unchanged. This means that 1600 over nine is
the same as 177 and seven-ninths. And since we’re working in feet, we
can say that the area of the pen will be 177 and seven-ninths square feet.
