Arrange one and two-thirds, two and a half, and two and one-seventh in ascending order.
To arrange a series of numbers or values in ascending order means to arrange them in order of size from smallest to largest. And in this problem, we’re being asked to arrange three mixed numbers. The mixed number is an amount that’s made up of a whole number part and a fraction part. And so, we have one, which is the whole number part, and two-thirds, which is the fraction part. Our second mixed number is two and a half. And thirdly, we have two and one-seventh.
Now, the first thing we can do when comparing these mixed numbers is to look at the whole number parts. What numbers can we see? Well, the first number is one and a fraction, and then we have two and a fraction, and the third number is also two wholes and a fraction. And so, to begin with, we don’t even need to look at the fraction parts. We can see straightaway that the smallest mixed number is going to be one and two-thirds because one one is less than two ones.
Let’s write one and two-thirds as our first fraction. Now, we just need to put two and a half and two and one-seventh in order. Now, they both have two ones, so we can’t split them up by looking at the whole number part. We need to look at and compare the fraction parts. Which is smaller, one-half or one-seventh? An easy mistake to make is to look at the fractions quickly to see that the denominator with a half is two and the denominator for one-seventh is seven and think well, two is smaller than seven. So, one-half must be less than one-seventh.
But we know this isn’t how fractions work. The denominator in a fraction shows us the number of equal parts that the whole has been split into. With the fraction one-half, the denominator is two. This shows us that the whole fraction has been split into two equal parts. This isn’t very many equal parts. So, the size of those parts is fairly large.
Now, we could do the same to find one-seventh. These two fractions are actually quite straightforward to compare because they both have the same numerator. It’s one. So, we’re looking at one part, one-half and one-seventh, this makes it easier to compare them.
So, we’re going to divide our whole now into seven equal parts. We’re looking at one-seventh. And we can see that even though seven is a larger number than two, this means that we’re splitting the whole into more parts. And so, each part is going to be smaller than one-half. The more parts we chop a chocolate bar into, the less of the chocolate bar we’re going to have.
If the numerators are the same, we just need to compare the denominators. And the greater the denominator, the smaller the fraction. And so, we can use this reasoning to say that two and one-seventh is less than two and one-half. And so, we’ve managed to put our mixed numbers in order. We used a combination of comparing the whole number parts. This helped us to see that one and two-thirds was the smallest fraction. And then, with the remaining two mixed numbers, although they both had the same whole number parts, we could compare the fraction parts.
Both one-half and one-seventh had the same numerator, so all we needed to do is to look at the denominator. The larger the denominator, the smaller the fraction. So, two and one-seventh was less than two and one-half. We’ve arranged our fractions in ascending order and that order is one and two-thirds, two and one-seventh, and two and a half.