A mirror’s length is two-thirds of a foot, and its width is three-eighths of a foot less than its length. Determine the width of the mirror in fraction form.
So here, we have our mirror with the length and the width labelled. We’re told the length is two-thirds of a foot. So this entire question references a foot, so let’s go ahead and let 𝑥 represent a foot. So if the length is two-thirds of a foot, the length would be equal to two-thirds times 𝑥 because the word of, in mathematical word problems, means to multiply. So the length is two-thirds 𝑥.
Now the width is three-eighths of a foot less than its length. So if it’s less than its length, we need to subtract away from the length. So 𝑊 equals the length minus three-eighths of a foot. And it asks us to determine the width of the mirror in fraction form. So this is our equation for the width. Now we can replace 𝐿 with two-thirds 𝑥, and now it’s all in terms of 𝑥, which is a foot.
So now we need to combine like terms. So the fraction part can make it seem a little tricky. If we were to subtract five 𝑥 and three 𝑥, we would take five minus three to get two, and then we bring down our 𝑥. So we need to do the exact same thing with the fractions; we need to subtract the fractions and then bring down the 𝑥. So to subtract our fractions, we need a common denominator, the smallest number that three and eight both go into, which will be 24.
So now that we have our denominator, we have to change our numerators because we can’t just change the denominators without changing the numerator. So how do we get from three to 24? We multiply it by eight. So we need to do the same thing to the numerator. And two times eight is 16. Now to get from eight to 24, we multiply it by three. So on the numerator, three times three is nine. So now we keep our common denominator and subtract the numerators. So 16 minus nine is seven, and then we bring down our 𝑥. So it says determine the width of the mirror in fraction form, which we’ve done. Now the only thing we should do is — we introduced 𝑥, we let 𝑥 represent a foot, so the question uses the words “of a foot” — so let’s replace 𝑥 with “of a foot”. So our final answer would be: seven twenty-fourths of a foot.