Anthony’s quilt has an area of 81 square feet and is made out of square patches with a length of nine inches. How many patches are there?
So Anthony’s quilt must be a square if it’s made out of square patches. And its area is 81 square feet. And we know that each patch has a length of nine inches. Now the patch is square, so each side of the part will be nine inches. We wanna know how many patches are there.
Well the problem is the patches are in terms of inches and the area is in terms of feet. So let’s go ahead and change the inches to feet. We know that there are 12 inches in one foot. So if we have nine inches, how many feet is that? We can call it 𝑥. So we can set up a proportion and solve for the number of feet.
So we can take nine times one and set it equal to 12 times 𝑥, which is 12𝑥. And nine times one is nine. Now to solve for 𝑥, we’ll need to divide both sides of the equation by 12. Now notice that we didn’t put inches or feet. This is because we know that 𝑥 should be in terms of feet. So we will add feet at the end. So the 12s cancel, and nine divided by 12 is equal to 0.75.
So this means that nine inches represents 0.75 feet. So the square, the patch, is 0.75 by 0.75. So if we want to know how many patches actually can take up this entire quilt, we need to find the area of one of the small patches. And the area of a square is equal to the length times the width. Now the length and the width are of the exact same.
So we could say length times length, or we could say length squared, 0.75 feet times 0.75 feet. This will make the area of the patch 0.5625 feet squared. So now that we know the area of one of the patches, we can find the number of patches by taking the area of the entire quilt, 81 square feet, and dividing it by the area of a small patch 0.5625 square feet. And we get 144. This means there are 144 patches in this quilt.