Arrange seven-halves, seven-quarters, seven-thirds, seven-eighths in descending order.
So in this problem we’re given four fractions. And we’re asked to arrange them in descending order. That’s from largest to smallest now. And often, when we compare a set of fractions, it’s good to have the denominators the same. It’s very quick to compare them if the denominators are all the same. So one way to find the answer could be to convert these fractions so that they all have the same denominator. But in this particular question, we can find out the answer just by using our knowledge of fractions. Let’s have a look at the four fractions and see what we can spot about them.
Firstly, nearly all of the fractions are improper. In other words, the numerator is larger than the denominator. This means that the fraction represents more than one whole. Seven-halves is more than one. Seven-quarters is more than one. Seven-thirds is more than one. In fact, the only fraction where the numerator is smaller than the denominator is seven-eighths. Seven-eighths is the only fraction we’re looking at that’s less than one. It’s the smallest fraction out of our list of four. So we haven’t had to convert anything, but we’ve already worked out that the smallest fraction is seven-eighths.
Let’s cross it off our list and look at the remaining three fractions. We know that two-halves equal one whole. So how many wholes and how many halves do we have if we have seven-halves. Two-halves equal to whole. Four-halves equal two wholes. Six-halves equal three wholes. And so seven-halves equal three and a half. Now, let’s convert our next improper fraction into a mixed number. We know that four-quarters are in one whole. So how many wholes and how many quarters are in seven-quarters? Seven-quarters must be equal to one and three-quarters.
Finally, how many wholes and how many thirds are in seven-thirds? We know three-thirds equal one whole. So six-thirds must be the same as two wholes. And so seven-thirds is the same as two and a third. We’re now in a position to be able to compare all of the fractions. We’ve converted them all into mixed numbers. So we can see what their values worth. We can also see why we didn’t need to convert them to have the same denominator. We can simply compare the number of ones because they’re all different. The mixed number with the greatest number of one’s is three and a half. This is the largest mixed number. And so it’s also the largest fraction. The largest fraction is seven and a half.
The next largest number of ones is two, in the mixed number, two and a third. So we know the second largest fraction is seven-thirds. Because it’s the only one remaining, we know the third largest fraction must be seven-quarters. Look at how we could have predicted this before we started. The numerator in all of these fractions is the same, seven parts every time. But the denominator changes. Remember, the smaller the denominator with a fraction, the larger the parts. So if we’ve got seven parts, and they’re large parts, it’s going to be a larger number, a larger amount.
And the opposite is true; the larger the denominator, the smaller the parts. So if we have seven parts and they’re small parts, because the denominators is a large number, then it’s going to be a smaller fraction. And that’s why these denominators go in order — two, three, four, eight. Because the numerator stays the same, all we have to do is to look at the denominators and think to ourselves, a smaller number means a larger piece, and then just put the denominators in order. And so the four fractions arranged in descending order are seven-halves, seven-thirds, seven-quarters, seven-eighths.