Arrange four-sixths, four-eighths, four-tenths, and four-halves in descending order.
This problem gives us four fractions to put in order. And they are four-sixths, four-eighths, four-tenths, and four-halves. And we’ve been asked to arrange these in descending order. Now, we know that when something descends, it goes downwards. And so if we’re asked to put the fractions in descending order, the value of the fractions needs to go downwards. In other words, we need to start with the largest fraction and end with the smallest.
So let’s take a moment to look at our four fractions. What do we notice about them? Or perhaps the first thing that we can see, and this is actually the most important thing we can see, is that they all have the same numerator. The number on the top of each fraction is four. Let’s remind ourselves what each number in a fraction represents. The top number, the numerator, represents the number of chosen parts. And the bottom number, the denominator, shows the number of equal parts that one whole has been split into. In other words, the numerator tells us the number of parts that we’re talking about. But the denominator tells us the size of those parts.
Let’s imagine that each of these fractions represents parts of a chocolate bar. Each one tells us that four parts have been eaten. But to work out the size of each fraction, we need to know how large those parts are. We could’ve eaten four very small parts or four large parts. So we need to find the answer here and to put these in order by looking at the denominators. Now, it’ll be very easy to look at these fractions and think, ten is a large number. Four-tenths must be the largest fraction. So let’s start with four-tenths. Two is a very small number, so let’s end with four-halves. That must be the smallest fraction.
But if we think like that, we’re not really thinking about what the denominator means. If a fraction has two as a denominator, it means it’s been split up into two equal parts. And if, for example, the denominator is 10, much larger than two, this means the whole has been split into 10 equal parts. We can see that the larger the denominator, the smaller the part. And the opposite is true. The smaller the denominator, the larger the part. And so we can say that our largest fraction is going to be the one with the smallest denominator, in other words, the biggest part. And this is four-halves. If we look at all four fractures, we could see that four-halves is the only fraction that is an improper fraction. It’s more than one. All the others are less than one whole.
Which fraction has the next largest parts? Remember, we’re looking for the smallest denominator. Well, after halves, the next smallest denominator is sixths. So the second fraction in our order is four-sixths. We only have four-eighths and four-tenths left. And we know the larger of these two fractions is four-eighths, so that comes next. And we need to end with the smallest of the four fractions, which is four-tenths.
This problem has taught us that, with fractions, a large number doesn’t always mean a large fraction. It depends where that large number is. If all the numerators in a set of fractions are the same, then we need to compare the denominators. And the larger the denominator, the smaller the part. So if the numerators are the same, a large denominator means a smaller fraction. And that’s why the fractions in descending order are four-halves, four-sixths, four-eighths, and four-tenths.