Write these fractions in order, starting with the smallest. Six-fifths, three-fifths, three-quarters.
So here, we’re given three different fractions and we’re asked to write them in order, from smallest to largest. Let’s spend a moment just looking at the fractions. What do we notice? The first two fractions have the same denominator; they’re both fifths. This could be useful. The second and third fractions both have the same numerator. Perhaps, we could use this to compare them.
But a good place to start is with the first fraction. It’s different from the other two for one reason: the numerator, the number on the top, is larger than the denominator. This tells us that it’s an improper fraction. It’s larger than one. The other two fractions three-fifths and three-quarters are both less than one. But because six-fifths is an improper fraction, it’s greater than one. It must be the largest fraction.
And so, if we’re writing our fractions in order from smallest to largest, six-fifths is going to come at the end. Let’s cross it through to show we’ve used it. Now what we’ll need to do is compare three-fifths and three-quarters to decide which is smaller and which is larger.
Remember we said the numerators were both the same. This means the number of chosen parts in the fraction is the same. But the size of those parts is different. It might be very easy to think that five is larger than four. So three-fifths must be larger than three-quarters.
But we need to think about what the denominator in a fraction means. It’s the number of equal parts that a whole has been divided into. Here is one whole divided into fifths and here’s the same size whole divided into four equal parts or quarters. One-quarter is larger than one-fifth. And so, we know three-quarters must be greater than three-fifths.
The smallest fraction is three-fifths. And so, the middle fraction must be three-quarters. We found the correct order by recognising that six-fifths was an improper fraction. And so it was the only fraction that was greater than one whole. It was the largest fraction. We then recognised that one-fifth was smaller than a quarter. And so, three-fifths must be smaller than three-quarters.
So starting with the smallest, the fractions in order are three-fifths, three-quarters, and six-fifths.