# Comparing fractions

This lesson teaches several methods for comparing fractions: when the denominators are the same, when the numerators are the same, comparing to 1/2 or to 1, and using equivalent fractions. It also includes word problems and a puzzle. The lesson is meant for fifth grade.

In the video below, I explain several methods for comparing fractions. When comparing two fractions, there are three methods that sometimes work:
- If the denominators are the same, compare the amount of pieces
- If the numerators are the same, think of the size of the parts.
- Sometimes you can easily compare to 1/2

If none of those "work", you can always convert the two fractions so they have the same denominator, and then compare.

Sometimes it is easy to know which fraction is the greater of the two. Study the examples below!
 7 9 > 2 9

With like fractions, all you
need to do is to check which
fraction has more “slices,”
and that fraction is greater.

 3 9 < 3 8

If both fractions have the
same number of pieces,
then the one with bigger
pieces is greater.

 4 7 > 5 12

Sometimes you can compare
to 1/2
. Here, 4/7 is clearly
more than 1/2, and 5/12 is
clearly less than 1/2.

 6 5 > 9 10

Any fraction that is bigger than one must also be bigger than any fraction that is less than one. Here, 6/5 is more than 1, and 9/10 is less than 1.

 2 5 > 1 4

If you can imagine the pie pictures in your mind,
you can sometimes “see” which fraction is
bigger. For example, it is easy to see that 2/5 is
more than 1/4.

1. Compare the fractions, and write > ,  < , or = .

 a. 1 8 1 10
 b. 4 9 1 2
 c. 6 10 1 2
 d. 3 9 3 7
 e. 8 11 4 11
 f. 7 4 7 6
 g. 5 14 5 9
 h. 4 20 2 20
 i. 2 11 2 5
 j. 1 2 5 8
 k. 3 6 1 2
 l. 1 20 1 8
 m. 1 2 3 4
 n. 8 7 3 3
 o. 49 100 61 100
 p. 7 8 8 7
 q. 9 10 3 4
 r. 6 5 3 4
 s. 4 4 9 11
 t. 1 3 3 9

Sometimes none of the “tricks” explained in the previous
page work, but we do have one more up our sleeve!

Convert both fractions into like fractions. Then compare.

In the picture on the right, it is hard to be sure if 3/5 is really
more than 5/9. Convert both into 45th parts, and then it is easy
to see that 27/45 is more than 25/45. Not by much, though!

 3 5 5 9 ↓ ↓ 27 45 > 25 45

2. Convert the fractions into like fractions, and then compare them.

 a. 2 3 5 8 ↓ ↓
 b. 5 6 7 8 ↓ ↓
 c. 1 3 3 10 ↓ ↓
 d. 8 12 7 10 ↓ ↓
 e. 5 8 7 12 ↓ ↓
 f. 11 8 14 10 ↓ ↓
 g. 6 10 58 100 ↓ ↓
 h. 6 5 11 9 ↓ ↓
 i. 7 10 5 7 ↓ ↓

 j. 43 100 3 10 ↓ ↓

 k. 9 8 8 7 ↓ ↓

 l. 7 10 2 3 ↓ ↓

3. One cookie recipe calls for 1/2 cup of sugar. Another one calls for 2/3 cup of sugar.
Which uses more sugar, a triple batch of the first recipe, or a double batch of the second?

How much more?

4. Compare the fractions using any method.

 a. 5 12 3 8

 b. 5 12 4 11
 c. 3 10 1 5
 d. 3 8 4 7
 e. 4 15 1 3
 f. 5 6 11 16
 g. 7 6 10 8

 h. 5 12 5 8
 i. 3 4 4 11
 j. 13 10 9 8
 k. 2 13 1 5
 l. 1 10 1 11

5. A coat costs \$40. Which is a bigger discount:
1/4 off the normal price, or 3/10 off the normal price?

of the coat was \$60 instead? Why or why not?

6. Here are three number lines that are divided respectively into halves, thirds, and fifths. Use them to
help you put the given fractions in order, from the least to the greatest.

 a. 1 3 , 2 5 , 2 3 , 1 5 , 1 2
 b. 7 5 , 3 2 , 4 3 , 6 5 , 2 2
___< ___< ___ < ___ < ___       ___< ___< ___< ___ < ___

7. Write the three fractions in order.

 a. 7 8 , 9 10 , 7 9

___< ___< ___

 b. 1 3 , 4 10 , 2 9

___< ___< ___

8. Rebecca made a survey of a group of 600 women. She found that 1/3 of
them never exercised, that 22/100 of them swam regularly, 1/5 of them
jogged regularly, and the rest were involved in other sports.

a. Which was a bigger group, the women who jogged or the women who swam?

b. What fraction of this group of women exercise?

c. How many women in this group exercise?

d. How many women in this group swim?

 The seven dwarfs could not divide a pizza into seven equal slices.  The oldest suggested, “Let’s cut it into eight slices, let each dwarf have one piece, and give the last piece to the dog.” Then another dwarf said, “No! Let’s cut it into 12 slices instead, and give each of us 1 ½ of those pieces, and the dog gets the 1 ½ pieces left over.”     Which suggestion would give more pizza to the dog?

This lesson is taken from my book Math Mammoth Fractions 1.

#### Math Mammoth Fractions 1

A self-teaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.