Sarah is going to put a fence around her rectangular garden. If the garden is 14 and two-sevenths by 12 and six-sevenths yards, how many feet of fencing will she need?
To help us visualize the problem, let’s sketch Sarah’s garden. There we are, we’re told it’s rectangular, so we’ve drawn a rectangle. The next piece of information we’re given is the rectangle’s dimensions. We’re told it’s 14 and two-sevenths by 12 and six-sevenths yards. So, let’s label our rectangle’s length, 14 and two-sevenths yards, and its width, 12 and six-sevenths yards. Now, we’re told that Sarah is going to put a fence all the way around her garden. And the question that we’re asked is, how many feet of fencing is she going to need?
So, we can see that this question is all about perimeter, the distance around a shape. And because we know the length and the width of this rectangle, surely we can add the length twice and the width twice and that will give us the perimeter. Is that what we need to do? Let’s read the question very carefully. If the garden is 14 and two-sevenths by 12 and six-sevenths yards, how many feet of fencing will she need?
We need to give the perimeter using a different unit of measurement, so we’re going to have to convert these measurements into feet. Now, we can either do this at the beginning or the end of our calculation. Let’s do it at the end, and then we only have to convert once. So, to begin with, let’s calculate the perimeter in yards. We know this distance here is worth 14 and two-sevenths yards. And we also know that the side that’s parallel to it is the same length. So, we’ve got two lots of 14 and two-sevenths.
We know that to fence this side of the garden, Sarah is going to need 12 and six-sevenths yards of fencing. And once again, the side that’s parallel to that is the same, 12 and six-sevenths. Now, each of these distances is a mixed number; it’s made up of a whole number and a fraction. For example, 14 is the whole number and two-sevenths is the fraction. Let’s start by adding all the whole numbers together.
14 plus 14, we know, is 28. And two lots of 12 equals 24. To add 24 to 28, we can partition it into 20 and four. 28 plus 20 equals 48. And then, if we add four to 48, we get the answer 52. So, if we add all the whole numbers of yards that we have, we get the answer 52 yards. Now, let’s add all the fraction parts of our mixed numbers.
Two-sevenths plus two-sevenths equals four-sevenths. And six-sevenths plus six-sevenths equals twelve-sevenths. This is an improper fraction; it’s greater than one. Now, how many sevenths do we have if we add four-sevenths and twelve-sevenths together? Four plus 12 equals 16. So, we have sixteen-sevenths altogether. Now, as we’ve said already, this is an improper fraction. It’s greater than one because the numerator, which is the top number, is larger than the denominator.
We know that seven-sevenths equal one whole. This must mean that fourteen-sevenths equal two wholes. And so, if we’ve got sixteen-sevenths, we’ve got two wholes and two-sevenths left over. Now, if we add this to 52, we get the distance in yards of the perimeter all around her garden. And the answer is 54 and two-sevenths. But we haven’t solved the problem yet. Remember, the question asked us, how many feet of fencing will she need?
So, now, it’s time to convert our yards into feet. We know that one yard is equal to three feet. And so, we need to multiply 54 and two-sevenths by three. Perhaps the best way to do this is to multiply the whole number part first and then the fraction part. Four lots of three equals 12. Five threes are 15, plus the one that we’ve exchanged equal 16. 54 times three equals 162.
Now, let’s multiply the fraction part of our mixed number. If we have three lots of two-sevenths, how many sevenths do we have? Well, two times three is six. So, three lots of two-sevenths equals six-sevenths. Now, we can put our whole number part, 162, and our fraction part, six-sevenths, together to show the amount of fencing in feet that Sarah is going to need to fence her entire garden all the way around.
This has been quite an interesting problem to work out. We’ve needed to understand about perimeter, fractions, in particular mixed numbers, also improper fractions. We’ve also had to know how to convert between yards and feet, lots of skills there. If Sarah’s garden is 14 and two-sevenths yards by 12 and six-sevenths yards, she’s going to need 162 and six-sevenths feet of fencing.