Compare fractions with benchmarks and number lines | Grade 5 Math | LevelKite

To compare two fractions, you decide which one is greater, or whether they are equal, which means they are the same amount. Comparing amounts only makes sense when both fractions refer to the same whole, or to wholes of equal size. Half of a small pizza is less food than half of a large pizza.

In this lesson you compare fractions in two ways. You use benchmarks: numbers you know well, such as $0$ , $\frac{1}{2}$ and $1$ , that help you judge other numbers. You also use a number line: a line where each number has its own point, starting from $0$ . On one number line, every fraction refers to the same whole, the length from $0$ to $1$ .

Points A, B and C sit at

\frac{1}{8}

,

\frac{4}{8}

and

\frac{7}{8}

on a line from 0 to 1 cut into 8 equal parts.

On a number line, the denominator (the bottom number) tells how many equal parts the whole from 0 to 1 is cut into. The numerator (the top number) tells how many of those parts to count from 0 to reach the fraction. On this line the whole is cut into 8 equal parts, so each part is $\frac{1}{8}$ . Each small mark on the line is a tick.

A point farther right is farther from 0, so it stands for more of the same whole. That is why the fraction farther to the right is the greater one.

The tick labeled $\frac{4}{8}$ is 4 parts from 0, and 4 more parts reach 1. It is exactly halfway, so $\frac{4}{8}$ and $\frac{1}{2}$ are equal.

Point A is 1 part from 0, so it is $\frac{1}{8}$ . It sits to the left of the halfway tick, so $\frac{1}{8} < \frac{1}{2}$ . With just 1 part between it and 0, it is close to 0.
Point B is on the halfway tick, so it is $\frac{4}{8}$ , which is equal to $\frac{1}{2}$ .
Point C is 7 parts from 0, so it is $\frac{7}{8}$ . It sits to the right of the halfway tick, so $\frac{7}{8} > \frac{1}{2}$ . With just 1 part between it and 1, it is close to 1.

You can also judge a fraction from its numbers alone. The halfway point comes after half of the parts, so compare the numerator with half the denominator. For eighths, half of 8 is 4. The checks $1 < 4$ , $4 = 4$ and $7 > 4$ match what the line shows for $\frac{1}{8}$ , $\frac{4}{8}$ and $\frac{7}{8}$ . A match, like $4 = 4$ , means the fraction is equal to $\frac{1}{2}$ .

When the denominator is odd, half of it is not a whole number. Then compare twice the numerator with the denominator. This is the same check: if the numerator were exactly half the denominator, twice the numerator would equal the denominator. For $\frac{3}{7}$ , $2 \times 3 = 6$ and $6 < 7$ , so $\frac{3}{7} < \frac{1}{2}$ .

A fraction is close to 0 only when its numerator is small next to its denominator. It is close to 1 only when its numerator is nearly its denominator. In the table, $\frac{5}{9}$ is just above $\frac{1}{2}$ , because 5 is only a little more than $4\frac{1}{2}$ . But it is 4 parts short of 1, so it is not close to 1. In every check, the fraction and the benchmark refer to the same whole.

Two checks against one half of the same whole, and where each fraction sits

Fraction	Numerator and half the denominator	Twice the numerator and the denominator	Compared with one half	Where it sits
$\frac{1}{10}$	$1 < 5$	$2 < 10$	less than $\frac{1}{2}$	close to $0$
$\frac{4}{9}$	$4 < 4\frac{1}{2}$	$8 < 9$	less than $\frac{1}{2}$	just below $\frac{1}{2}$
$\frac{6}{12}$	$6 = 6$	$12 = 12$	equal to $\frac{1}{2}$	exactly at $\frac{1}{2}$
$\frac{5}{9}$	$5 > 4\frac{1}{2}$	$10 > 9$	greater than $\frac{1}{2}$	just above $\frac{1}{2}$ , not close to $1$
$\frac{11}{12}$	$11 > 6$	$22 > 12$	greater than $\frac{1}{2}$	close to $1$

Key concept

Where a fraction sits tells how big it is

When fractions refer to the same whole, or to wholes of equal size, you can compare them on one number line. The fraction farther to the right is the greater one, and equal fractions share one point. Benchmarks tell you where a fraction sits from its numbers alone. To check against $\frac{1}{2}$ , compare the numerator with half the denominator. To check against $1$ , compare the numerator with the denominator. If a benchmark lies between two fractions, the fraction above the benchmark is greater. If no benchmark lies between them, you need another method, such as reading both points on one number line.

Worked example

Compare $\frac{2}{5}$ and $\frac{7}{10}$ . Both fractions refer to the same whole.

Check 2/5 against the benchmark 1/2. The denominator 5 is odd, so compare twice the numerator with the denominator. Twice 2 is 4, and 4 is less than 5, so 2/5 is less than 1/2.
$2 \times 2 = 4 < 5, \text{ so } \frac{2}{5} < \frac{1}{2}$
Check 7/10 against 1/2. Half of the denominator 10 is 5. The numerator 7 is greater than 5, so 7/10 is greater than 1/2.
$7 > 5, \text{ so } \frac{7}{10} > \frac{1}{2}$
So the benchmark 1/2 lies between the two fractions. They refer to the same whole, so the fraction above 1/2 is the greater one.
$\frac{2}{5} < \frac{1}{2} < \frac{7}{10}$
Check on the number line below. Its whole is cut into 10 equal parts. The tick labeled 5/10 is 5 parts from 0 and 5 parts from 1, so 5/10 = 1/2. The 10 parts make 5 groups of 2, so each fifth is 2 parts. Point A is 2/5: 2 groups of 2 parts, 4 parts from 0. Point B is 7/10: 7 parts from 0. The point for 7/10 is farther right, so 7/10 is greater.

Answer

For the same whole, $\frac{2}{5}$ is less than $\frac{7}{10}$ : $\frac{2}{5} < \frac{7}{10}$ .

Point A is

\frac{2}{5}

and point B is

\frac{7}{10}

; the tick

\frac{5}{10}

, equal to

\frac{1}{2}

, lies between them.

Worked example

Compare $\frac{5}{4}$ and $\frac{7}{8}$ . Both fractions refer to the same whole.

Try the benchmark 1/2 first. Half of 4 is 2, and 5 > 2. Half of 8 is 4, and 7 > 4. Both fractions are greater than 1/2, so 1/2 does not lie between them.
$\frac{5}{4} > \frac{1}{2} \text{ and } \frac{7}{8} > \frac{1}{2}$
Try the benchmark 1. A fraction is equal to 1 when its numerator and denominator match, greater than 1 when the numerator is greater, and less than 1 when it is less. Since 5 > 4, 5/4 is greater than 1: it is 1 fourth more than 4 fourths, which make a whole. Since 7 < 8, 7/8 is less than 1: it is 1 eighth short of 8 eighths.
$5 > 4, \text{ so } \frac{5}{4} > 1; \quad 7 < 8, \text{ so } \frac{7}{8} < 1$
The benchmark 1 lies between them. They refer to the same whole, so the fraction above 1 is the greater one.
$\frac{7}{8} < 1 < \frac{5}{4}$
Check on the number line below. It runs from 0 to 2, and each whole is cut into 8 equal parts. Point A is 7/8: 7 parts from 0. The 8 parts of a whole make 4 groups of 2, so each fourth is 2 parts. Point B is 5/4: 5 groups of 2 parts, 10 parts from 0, which is 2 parts past the tick for 1. The point for 5/4 is farther right, so 5/4 is greater.

Answer

For the same whole, $\frac{5}{4}$ is greater than $\frac{7}{8}$ , even though $\frac{7}{8}$ is written with bigger numbers.

Point A is

\frac{7}{8}

, one part left of 1, and point B is

\frac{5}{4}

, two parts right of 1.

Worked example

Order $\frac{5}{6}$ , $\frac{1}{4}$ and $\frac{7}{12}$ from least to greatest. All three fractions refer to the same whole.

Check each fraction against 1/2. For 1/4, 1 is less than 2, half of 4, so 1/4 is less than 1/2. For 5/6, 5 is greater than 3, half of 6. For 7/12, 7 is greater than 6, half of 12. Both of these are greater than 1/2. So 1/2 lies between 1/4 and each of the others, which makes 1/4 the least.
$\frac{1}{4} < \frac{1}{2}, \quad \frac{5}{6} > \frac{1}{2}, \quad \frac{7}{12} > \frac{1}{2}$
Both 5/6 and 7/12 are also less than 1, because 5 < 6 and 7 < 12. Each sits between 1/2 and 1, so no benchmark lies between them. Another method is needed: read their points on one number line.
$\frac{1}{2} < \frac{5}{6} < 1, \quad \frac{1}{2} < \frac{7}{12} < 1$
The number line below cuts the whole into 12 equal parts. The tick labeled 6/12 is 6 parts from 0 and 6 parts from 1, so 6/12 = 1/2. The 12 parts make 4 groups of 3, so 1/4 is 1 group of 3 parts, 3 parts from 0: point A. 7/12 is 7 parts from 0: point B. The 12 parts make 6 groups of 2, so 5/6 is 5 groups of 2 parts, 10 parts from 0: point C.
On one line, all three fractions refer to the same whole, so position decides. The point for 5/6 is farther right than the point for 7/12, so 5/6 is greater than 7/12. The point for 1/4 is farthest left, so 1/4 is the least, as the benchmark showed.
$\frac{1}{4} < \frac{7}{12} < \frac{5}{6}$

Answer

From least to greatest, the order is $\frac{1}{4}$ , $\frac{7}{12}$ , $\frac{5}{6}$ .

Point A is

\frac{1}{4}

, point B is

\frac{7}{12}

and point C is

\frac{5}{6}

; the tick

\frac{6}{12}

equals

\frac{1}{2}

.

Takeaway: For fractions of the same whole, greater means farther right on one number line. A benchmark between two fractions tells you which is greater without a picture. When no benchmark lies between them, another method is needed, such as comparing their points on one line.

Next, you will compare fractions by renaming them so they have the same denominator, and you will compare mixed numbers.