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Comparing fractions that share a top number or a bottom number, why a bigger bottom number makes a smaller fraction, putting three in order, and using a half as a landmark.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you work out which of two fractions is greater — and, more importantly, why. When the bottom numbers match the pieces are the same size, so more pieces wins. When the top numbers match you have the same number of pieces, so it comes down to how big a piece is: cutting a whole into more parts makes every part smaller. That last one is the fact about fractions that feels wrong at first, and being able to say why is what stops it being used backwards.
You can name fractions as counts of equal parts, and you can place them on a number line. You know that the bottom number sets the size of a part and the top number counts the parts. Comparing two fractions means deciding which is the greater amount, and every comparison in this lesson comes from those two facts about the numbers.
| Term | What it means |
|---|---|
| Greater than (>) | Names the larger amount first: 3/4 > 1/4. |
| Less than (<) | Names the smaller amount first: 1/4 < 3/4. |
| Equal (=) | The two amounts are the same size. |
| Same denominator | Two fractions with the same bottom number, so their parts match in size. |
| Same numerator | Two fractions with the same top number, so they count the same number of parts. |
A fraction tells you two things: how big each part is and how many parts there are. When two fractions have the same bottom number, the parts are the same size, so the one with more parts is greater: 5/8 > 3/8. When two fractions have the same top number, they have the same number of parts, so the one with bigger parts is greater, and bigger parts come from a smaller bottom number: 3/4 > 3/8. Both fractions must be parts of the same whole.
Another way: picture
Two bars of the same length. One is cut into fourths with three shaded; the other is cut into eighths with three shaded. The fourths cover more of the bar.
When two fractions share a bottom number, they are counted in the same unit. 2/6 and 5/6 are both made of sixths. Every sixth is the same size, so the question becomes a counting question: which has more sixths? Five sixths is more than two sixths, so 5/6 > 2/6.
This works only because the parts are equal. Imagine two trays of muffins baked in the same pan, each cut into six equal pieces. Taking five pieces from one tray gives you more muffin than taking two from the other. The pieces are interchangeable, so the count decides.
If the top numbers also match, the fractions are equal. 4/6 and 4/6 name exactly the same amount. Watch for the opposite mistake, too: a shared bottom number does not make two fractions equal. 1/6 and 5/6 share sixths, yet one is a single small part and the other is almost the whole. Always look at both numbers before you decide.
When two fractions share a top number, they count the same number of parts. 3/4 and 3/8 both count three parts. Now the question is which parts are bigger. A whole cut into 4 equal parts has bigger parts than the same whole cut into 8 equal parts, because sharing among more parts leaves less for each one. Three big parts beat three small parts, so 3/4 > 3/8.
This feels backwards at first. With whole numbers, 8 is more than 4. With fractions that share a top number, the bigger bottom number means the smaller fraction. The reason is not a trick about digits. It is about sharing: the more people who share a pizza, the smaller each slice.
Saying why protects you from using the rule the wrong way round. If you only remember that the bottom numbers do something odd, you might apply it to fractions with the same bottom number, where it does not belong. Remember the reason instead. Same-size parts: count them. Same count: compare the sizes.
Mathematicians write comparisons with three symbols. The symbol > means greater than, and < means less than. The open side of the symbol faces the greater amount. So 5/8 > 3/8 reads five eighths is greater than three eighths, and 3/8 < 5/8 reads three eighths is less than five eighths. Both sentences say the same thing from opposite ends.
The equal sign = means the two amounts are the same size. You can write 4/4 = 1 because four fourths fill one whole. You can also write 6/6 = 4/4, since both are one whole, even though they look different.
Before writing a symbol, decide in words which fraction is greater. Then point the open side of the symbol toward it. Read your sentence aloud to check it makes sense. If you wrote 2/5 > 4/5, reading it aloud as two fifths is greater than four fifths should sound wrong at once, because two parts cannot be more than four of the same parts.
To order three fractions that share a bottom number, order their top numbers. For 4/9, 1/9 and 7/9, the parts are all ninths, so the order from least to greatest is 1/9, 4/9, 7/9. Ordering them is exactly like ordering 4, 1 and 7.
To order three fractions that share a top number, order their bottom numbers the other way. For 2/3, 2/10 and 2/5, the counts are all two parts, so the biggest parts win. Thirds are the biggest parts and tenths the smallest. From least to greatest the order is 2/10, 2/5, 2/3.
Check an ordering by placing the fractions on a number line in your head. The least should be nearest to 0 and the greatest nearest to 1. If your list puts a fraction that is almost the whole before one that is barely past zero, swap them and look again at what the numbers mean.
Sometimes neither number matches, as with 3/8 and 4/6. The rules about counting parts or comparing part sizes do not apply directly. A good strategy is to compare each fraction with a benchmark you know well: one half.
A fraction is more than one half when double its top number is bigger than its bottom number. For 3/8, double 3 is 6, which is less than 8, so 3/8 is less than one half. For 4/6, double 4 is 8, which is more than 6, so 4/6 is more than one half. One fraction is below the landmark and the other above it, so 4/6 > 3/8.
This method needs no common bottom number and no drawing. It only works when the two fractions land on opposite sides of the landmark. If both are above one half, you need another way to decide, which you will learn in later grades. Knowing when a method works is as important as knowing how to use it.
When you are unsure, draw two bars of exactly the same length, one under the other. The equal lengths matter: they make sure both fractions are parts of the same whole. Cut the top bar into the parts named by the first bottom number and the lower bar into the parts named by the second. Shade the number of parts each top number asks for, then look at which shading reaches further along.
For 2/3 and 2/6, the top bar is cut into three parts and the bottom bar into six. Shading two thirds covers most of the top bar. Shading two sixths covers only a third of the bottom bar. The drawing shows 2/3 > 2/6 before you say any rule, and it shows why: each third is as long as two sixths.
A drawing is a check, not a replacement for thinking. Rough drawings with uneven parts can mislead you, especially when two fractions are close. If your drawing and your reasoning disagree, trust neither until you have found the reason. Usually one of the bars was cut unevenly or the two bars were not the same length to begin with.
Every comparison so far has used the same whole. That is not a small detail. One half of a large watermelon is more fruit than one half of a small one, even though both are 1/2. And 1/4 of a large watermelon can be more fruit than 1/2 of a tiny one. The fractions alone cannot tell you which amount is bigger unless the wholes match.
So before comparing, ask what each fraction is a fraction of. In a story problem, look for the whole: the same spool of ribbon, the same size of pitcher, the same length of trail. If the wholes are different, the comparison needs more information, such as the actual lengths or amounts.
In this lesson every comparison uses the same whole, so the rules apply. Keeping that condition in mind will help you later, when you compare fractions of measured amounts and need to change everything into the same unit first.
Two students are making bookmarks for a class fair, and each bookmark needs ribbon. Maria has 3/4 of a yard of ribbon. Leo has 3/8 of a yard. Both pieces were cut from the same kind of yard-long spool, so the wholes match. Both have three parts, so the question is whose parts are bigger. Fourths of a yard are bigger than eighths of a yard, so Maria has more ribbon: 3/4 > 3/8.
The class also has three spools that have been partly used. One spool has 2/6 of its ribbon left, another 5/6, and the third 4/6. All three are measured in sixths of a spool, so the teacher orders them by their top numbers: 2/6 < 4/6 < 5/6. The spool with 5/6 left should be used first for long bookmarks.
Finally, a student asks whether 3/8 of a yard is enough for a bookmark that needs half a yard. Double 3 is 6, which is less than 8, so 3/8 is less than one half. Leo's ribbon is too short on its own. The comparison does not say how much more he needs, but it tells him he needs more before he starts cutting. If he borrows another 1/8 of a yard, he will have 4/8, which is exactly one half.
When the top numbers match, the fraction with the bigger bottom number is smaller, because its whole was cut into more, smaller parts. Only when the bottom numbers match do you compare the top numbers directly. Before choosing a rule, say which number matches, and say what that match means about the size and number of the parts.
Check the bottom numbers.
8 and 8
They match, so the parts are the same size.
Name the parts.
eighths
Both fractions count eighths of the same whole.
Compare the counts.
5 > 3
With equal parts, more parts is more.
Write the comparison.
5/8 > 3/8
The open side faces the greater amount.
Read it back.
five eighths is greater than three eighths
The sentence matches the counts.
Check the top numbers.
3 and 3
They match, so the counts of parts are the same.
Name the part sizes.
fourths and eighths
The bottom numbers set the part sizes.
Compare the part sizes.
1/4 > 1/8
Cutting into fewer parts makes bigger parts.
Compare the amounts.
3 big parts and 3 small parts
The same count of bigger parts is more.
Write the comparison.
3/4 > 3/8
Three fourths is the greater amount.
Check with one half.
3/4 > 1/2 > 3/8
Double 3 is 6: more than 4, less than 8.
Check the bottom numbers.
6, 6 and 6
All three are counted in sixths.
Order the top numbers.
2 < 4 < 5
With equal parts, the counts decide.
Write the order.
2/6 < 4/6 < 5/6
Least to greatest.
Name the landmark.
1/2
Half a yard is what the bookmark needs.
Double the top number of 3/8.
3 × 2 = 6
A fraction is one half when this equals the bottom number.
Compare with the bottom number.
6 < 8
Double the top is less than the bottom.
State the result.
3/8 < 1/2
Three eighths of a yard is not enough on its own.
Check the top numbers.
2 and 2
The counts of parts match.
Compare the part sizes.
1/5 > 1/9
Fifths are bigger parts than ninths.
Write the comparison.
Check with one half.
Which is greater, 4/9 or 2/9?
Complete the steps that compare 4/5 with 4/9.
Compare the counts of parts.
4 parts each
The top numbers match.
Name the bigger part.
1/big > 1/9
Cutting the whole into fewer parts makes each part bigger.
Write the greater fraction.
4/winner > 4/9
The same number of bigger parts is more.
Which is greater, 2/5 or 2/11?
2/3 is greater than 2/6. Why?
Put 4/8, 2/8 and 6/8 in order, smallest first. Write the three top numbers.
p, q, r
Write the greater fraction: 2/10 or 2/6.
n/m
Maria has 2/3 of a yard of ribbon and Leo has 2/8 of a yard. Write the length of the longer piece, as a fraction of a yard.
n/m of a yard
Which is greater, 2/5 or 5/8? Neither the top numbers nor the bottom numbers match.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Two same-size pitchers hold lemonade. Pitcher A is 3/8 full and pitcher B is 5/6 full. Write the fraction for the fuller pitcher.
n/m full
You can compare two fractions that share a top number or a bottom number, and say why. Without looking: which is greater, 2/5 or 2/7 — and what would you compare 3/8 and 5/9 against instead?
17. Your turn: compare 2/5 and 2/9, step 3
2/5 > 2/9
Two bigger parts are more.
17. Your turn: compare 2/5 and 2/9, step 4
2/5 < 1/2 and 2/9 < 1/2
Both are less than one half, so the halves test alone cannot decide.