Ordering fractions | Math | 4th grade | Khan Academy

Order the fractions from least to greatest.

So we have three fractions and we want to decide which one is the smallest, which one's in the middle, and which is the greatest.

One thing we could do is look at the fractions, think about what they mean, and then estimate.

7/10, let's say maybe that could represent seven of your 10 friends are wearing blue jeans. Well, that's most. Most of your friends are wearing blue jeans.

Then for 1/3, we could say one of your three teachers wears glasses. Well, that's not most. If only one of the three wears glasses, that's not most of the group.

So here's a fraction that represents most of the group; here's one that doesn't. So the most is probably greater.

These two we could compare by estimating and see that this one, 7/10, is probably greater than 1/3.

But then we get over here to 5/6. Well, again, that's most of the group, but is this most of the group greater than the 7/10's most of the group? That gets a lot trickier.

So what we can do is we can try to change these fractions to make them easier to compare. We don't have to compare 10ths to thirds to sixths because those are all different sizes, different size groups, different size pieces. That's tricky to compare.

So we want to change these to be the same size. We need some number, a multiple of 10, 3, and 6. Something we can multiply 10, 3, and 6 by to get a new denominator that will work for all of the fractions.

One way I like to figure this out is I look at the biggest denominator, which is 10, and I think of its multiples. The first multiple is 10; 10 * 1 is 10.

Can we change thirds and sixths to have 10 as a denominator? Is there any whole number you can multiply 3 times to get 10? There's not, so we need to keep going. 10 doesn't work.

The next multiple of 10 is 10 * 2, which is 20. Again, 3 and 6. Is there a whole number we can multiply them by to get 20? Again, no, so 20 doesn't work.

How about 30? Let's see, for 3, we can multiply 3 * 10 to get 30, so 30 works for 3. How about 6? 6 * 5 = 30, so yes, 30 can work to be our common denominator.

30ths. 30 is a multiple of 10, 3, and 6, so let's start converting our fractions to have denominators of 30.

We'll start with 7/10, and we want it to have a denominator of 30. So what do we need to multiply? 10 * 3 is 30.

We always multiply the numerator and denominator by the same number, so 7 * 3 is 21. So, 7/10 is equal to 21/30.

These are equal. We've just changed the size of the group. We've changed the denominator so that they will be easier to compare, but we've not changed what portion of the group we're representing.

7 out of 10 is the same portion as 21 out of 30.

And then let's keep going with 1/3. Again, we want a denominator of 30, so this time we'll multiply 3 * 10 to get 30. Again, numerator also times 10; 1 * 10 is 10.

10 out of 30 is the same as 1/3. If you have 10 of the 30 people, again we'll use the wear glasses example, or 1/3; that is the same size of the group, the same portion.

Finally, 5/6. What do we need to multiply here to get 30? 6 * 5 is 30, so we multiply the numerator by 5, and 5 * 5 is 25.

So now instead of these original fractions that were tricky to compare, we have much easier numbers to compare. We have 21/30, 10/30, and 25/30.

So in this case, the pieces are all 30ths; they're all groups of 30. So this is much easier to compare.

We can simply look at the numerators to see what portion of those 30 the fraction represents.

So the first, 7/10, is the same as 21 out of 30, whereas 1/3 is 10 out of 30.

Well, clearly, 21 out of 30 is a larger portion of the group than 10 out of 30, so we were right when we estimated up here that 7/10 is larger than 1/3.

But then the trickier one over here, now we can see much more clearly: 25 out of 30 is the greatest portion of the group. 25 is more than the 10 or the 21.

So we can list these now from least to greatest. The least, the smallest, is 10/30, which again remember is equal to 1/3, so we can put 1/3 as least and we can cross that off.

Next, it's either 21 out of 30 or 25. 21 is less; that represented 7/10, so we can say 7/10 because 21/30 equals 7/10.

And finally, that leaves us with 25/30, which is equivalent to 5/6.

So from least to greatest, our fractions are 1/3, 7/10, and then 5/6.