Compare Fractions and Mixed Numbers | Grade 5 Math | LevelKite

You already know how to compare fractions with benchmarks such as $\frac{1}{2}$ and $1$ . Sometimes benchmarks are not enough. Look at $\frac{2}{3}$ and $\frac{3}{4}$ . Both are more than $\frac{1}{2}$ . Both are less than $1$ . So which one is greater?

In this lesson you will learn a way that always works. Then you will use it on numbers greater than $1$ .

Picture two paper strips of the same length. Cut the first strip into 3 equal pieces and shade 2 of them. The shaded part is $\frac{2}{3}$ of the strip. Cut the second strip into 4 equal pieces and shade 3 of them. The shaded part is $\frac{3}{4}$ of the strip.

The denominator is the bottom number of a fraction. It tells you how many equal pieces make one whole, so it also tells you how big each piece is. More pieces in one whole means smaller pieces. The numerator is the top number. It tells you how many pieces you have.

Thirds and fourths are different sizes. So you cannot just count the shaded pieces to see which strip has more shading. First you need pieces of the same size.

Here is how to get pieces of the same size. Cut each third of the first strip into 4 smaller equal pieces. The strip now has $3 \times 4 = 12$ pieces, called twelfths. The 2 shaded pieces have become $2 \times 4 = 8$ shaded pieces. The shaded amount did not change. Only its name changed, from $\frac{2}{3}$ to $\frac{8}{12}$ .

Fractions that name the same amount are called equivalent fractions. To rename a fraction means to write an equivalent fraction for it. You do that by multiplying the numerator and the denominator by the same number, such as 2, 3 or 4.

Now cut each fourth of the second strip into 3 smaller equal pieces. That strip has $4 \times 3 = 12$ pieces too, and $\frac{3}{4}$ is renamed as $\frac{9}{12}$ . Both fractions are now counted in twelfths. A denominator that two fractions share is called a common denominator. Here the common denominator is $12$ .

The same two amounts, renamed in twelfths

Fraction	In words	Multiply numerator and denominator by	Equivalent fraction	New name in words
$\frac{2}{3}$	2 thirds	$4$	$\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$	8 twelfths
$\frac{3}{4}$	3 fourths	$3$	$\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$	9 twelfths

Now the pieces are the same size, so you can simply count them. 8 twelfths is less than 9 twelfths. That means 2 thirds is less than 3 fourths.

In symbols, $\frac{8}{12} < \frac{9}{12}$ , so $\frac{2}{3} < \frac{3}{4}$ . The symbol $<$ means is less than. The symbol $>$ means is greater than.

Key concept

Same-size pieces first, then compare

A numerator only counts pieces. A count is fair to compare only when the pieces are the same size. So before you compare two fractions, rename them with a common denominator. Then the fraction with the greater numerator is the greater fraction.

You will use this idea all through the lesson: first write both numbers the same way, then compare.

Worked example

Compare two fractions

Which is greater, $\frac{5}{8}$ or $\frac{7}{12}$ ?

Find a common denominator. Renaming multiplies a denominator, so the common denominator must be a multiple of 8 and a multiple of 12. The multiples of a number are the numbers you say when you count by it. Count by 8s: 8, 16, 24. Count by 12s: 12, 24. Both lists reach 24, so use twenty-fourths.
Rename 5 eighths in twenty-fourths. To turn 8 into 24 you multiply by 3, so multiply the numerator by 3 too. The amount stays the same because each eighth is cut into 3 equal pieces.
$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$
Rename 7 twelfths in twenty-fourths. To turn 12 into 24 you multiply by 2, so multiply the numerator by 2 too. That keeps the amount the same.
$\frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24}$
Now both fractions count twenty-fourths, so compare the numerators. 15 pieces is more than 14 pieces of the same size.
$\frac{15}{24} > \frac{14}{24}$

Answer

$\frac{5}{8}$ is greater than $\frac{7}{12}$ , so $\frac{5}{8} > \frac{7}{12}$ .

A mixed number has a whole number part and a fraction part written together. For example, $3\frac{1}{8}$ means 3 wholes and 1 eighth more. The fraction part of a mixed number is less than 1 whole.

To compare two mixed numbers, start with the biggest parts: the whole numbers. Look at $3\frac{1}{8}$ and $2\frac{7}{8}$ . The number $2\frac{7}{8}$ is 2 wholes and a part that is less than 1 whole, so it has not reached $3$ . The number $3\frac{1}{8}$ has already passed $3$ . So $3\frac{1}{8} > 2\frac{7}{8}$ . The whole numbers settled it, even though $\frac{7}{8}$ is greater than $\frac{1}{8}$ .

What if the whole numbers are equal? Then the wholes tie, and the fraction parts decide. That is the only time you need to compare the fraction parts.

Worked example

Compare two mixed numbers

Which is greater, $4\frac{2}{3}$ or $4\frac{3}{5}$ ?

Compare the whole number parts first. Both numbers have 4 wholes, so the wholes tie. The fraction parts, 2 thirds and 3 fifths, will decide.
$4 = 4$
Find a common denominator for thirds and fifths. Count by 3s: 3, 6, 9, 12, 15. Count by 5s: 5, 10, 15. Both lists reach 15, so use fifteenths.
Rename each fraction part in fifteenths. Multiply the numerator and denominator of 2 thirds by 5, and of 3 fifths by 3. Multiplying both by the same number keeps each amount the same.
$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \qquad \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$
Compare the numerators. 10 fifteenths is more than 9 fifteenths, so 2 thirds is the greater fraction part. The wholes are equal, so 4 and 2 thirds is the greater number.
$4\frac{10}{15} > 4\frac{9}{15}$

Answer

$4\frac{2}{3}$ is greater than $4\frac{3}{5}$ , so $4\frac{2}{3} > 4\frac{3}{5}$ .

An improper fraction is a fraction whose numerator is equal to or greater than its denominator, such as $\frac{17}{6}$ . It names an amount of 1 whole or more.

A mixed number such as $2\frac{3}{4}$ and an improper fraction such as $\frac{17}{6}$ are written in different forms. That makes them hard to compare as they are. So first write both numbers in the same form. You can change the improper fraction into a mixed number, or you can change the mixed number into an improper fraction. Both ways give the same answer.

To change forms, use the denominator. It tells you how many pieces make 1 whole.

Worked example

Compare a mixed number and an improper fraction

Which is greater, $2\frac{3}{4}$ or $\frac{17}{6}$ ?

Write both numbers in the same form. Change 17 sixths into a mixed number. 6 sixths make 1 whole, so 12 sixths make 2 wholes. Take those 12 sixths out of the 17 sixths, and 5 sixths are left.
$\frac{17}{6} = \frac{12}{6} + \frac{5}{6} = 2 + \frac{5}{6} = 2\frac{5}{6}$
Now both numbers are mixed numbers, so compare the whole number parts first. Both have 2 wholes, so the wholes tie. The fraction parts, 3 fourths and 5 sixths, will decide.
$2 = 2$
Rename the fraction parts with a common denominator. Count by 4s: 4, 8, 12. Count by 6s: 6, 12. Both lists reach 12, so use twelfths. Multiply the numerator and denominator of 3 fourths by 3, and of 5 sixths by 2.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \qquad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Compare the numerators. 9 twelfths is less than 10 twelfths, so 3 fourths is the smaller fraction part. The wholes are equal, so 2 and 3 fourths is the smaller number.
$2\frac{9}{12} < 2\frac{10}{12}$
Check with the other form. Change 2 and 3 fourths into an improper fraction: 2 wholes are 8 fourths, and 3 more fourths make 11 fourths. Rename in twelfths as before. Multiply the numerator and denominator by 3 to turn 11 fourths into 33 twelfths, and by 2 to turn 17 sixths into 34 twelfths. 33 is less than 34, so the result is the same.
$2\frac{3}{4} = \frac{11}{4} = \frac{33}{12} \qquad \frac{17}{6} = \frac{34}{12} \qquad \frac{33}{12} < \frac{34}{12}$

Answer

$\frac{17}{6}$ is greater than $2\frac{3}{4}$ , so $2\frac{3}{4} < \frac{17}{6}$ .

Takeaway: write both numbers the same way before you compare them. Two fractions need a common denominator. Two mixed numbers need their whole numbers compared first, and their fraction parts only when the wholes tie. A mixed number and an improper fraction need one shared form.

Next, you will use these tools to round mixed numbers to the nearest whole number and to put three or more numbers in order.