Ordering and comparing fractions
🎯 In this topic you will
- Use a 0–1 number line to order fractions.
- Compare fractions using a number line or fraction strips, and record comparisons using < and >.
- Identify equivalent fractions and record them using the equals sign (=).
🧠 Key Words
- inequality
- multiple
Show Definitions
- inequality: A mathematical statement that compares two values using symbols such as <, >, ≤, or ≥, showing that they are not equal.
- multiple: A number you get by multiplying a given whole number by another whole number (for example, 12 is a multiple of 3 because 3 × 4 = 12).
🛒 Choosing the best deal
Y ou need to know how to order and compare fractions. This will help you to choose the best deal. Would you rather have $\dfrac{1}{3}$ of $\$\!30$ or $\dfrac{1}{5}$ of $\$\!40$?
❓ EXERCISES
$1.$ Mark $\dfrac{1}{5}$, $\dfrac{2}{5}$, $\dfrac{3}{5}$ and $\dfrac{4}{5}$ on this number line.
👀 Show answer
Divide the distance from $0$ to $1$ into $5$ equal parts.
From left to right, mark: $\dfrac{1}{5}$, $\dfrac{2}{5}$, $\dfrac{3}{5}$, $\dfrac{4}{5}$ (at $0.2$, $0.4$, $0.6$, $0.8$).
$2.$ Mark $\dfrac{1}{3}$, $\dfrac{2}{3}$ and $\dfrac{2}{10}$ on this number line.
👀 Show answer
$\dfrac{2}{10}=\dfrac{1}{5}=0.2$, so it is closest to $0$.
$\dfrac{1}{3}\approx 0.333$, so it is about one-third of the way along.
$\dfrac{2}{3}\approx 0.667$, so it is about two-thirds of the way along. Left to right: $\dfrac{2}{10}$, $\dfrac{1}{3}$, $\dfrac{2}{3}$.
$3.$ Which fractions would be marked on a number line between $\dfrac{3}{4}$ and $1$?
👀 Show answer
Any fraction greater than $\dfrac{3}{4}$ and less than $1$ would be between them on the number line.
Examples include: $\dfrac{4}{5}$, $\dfrac{5}{6}$, $\dfrac{7}{8}$, $\dfrac{9}{10}$.
$4.$ Use the fraction strips to help you compare $\dfrac{1}{4}$ and $\dfrac{1}{5}$.
👀 Show answer
When a whole is split into fewer equal parts, each part is bigger. So $\dfrac{1}{4}$ is bigger than $\dfrac{1}{5}$.
$\dfrac{1}{5}$ is less than $\dfrac{1}{4}$.
$\dfrac{1}{4}$ is greater than $\dfrac{1}{5}$.
$\dfrac{1}{5} < \dfrac{1}{4}$ and $\dfrac{1}{4} > \dfrac{1}{5}$.
$5.$ Use $<$, $>$ or $=$ to complete each statement.
a. $\dfrac{2}{4}$ □ $\dfrac{1}{2}$
b. $\dfrac{2}{5}$ □ $\dfrac{2}{10}$
c. $\dfrac{1}{3}$ □ $\dfrac{2}{3}$
d. $\dfrac{7}{10}$ □ $\dfrac{3}{4}$
e. $\dfrac{1}{5}$ □ $\dfrac{2}{10}$
f. $\dfrac{3}{10}$ □ $\dfrac{1}{3}$
👀 Show answer
a.$\dfrac{2}{4}=\dfrac{1}{2}$, so $=$
b.$\dfrac{2}{5}=\dfrac{4}{10}$ and $\dfrac{4}{10}>\dfrac{2}{10}$, so $>$
c.$\dfrac{1}{3}<\dfrac{2}{3}$, so $<$
d.$\dfrac{7}{10}=0.7$ and $\dfrac{3}{4}=0.75$, so $\dfrac{7}{10}<\dfrac{3}{4}$ and the symbol is $<$
e.$\dfrac{1}{5}=\dfrac{2}{10}$, so $=$
f.$\dfrac{3}{10}=0.3$ and $\dfrac{1}{3}\approx 0.333$, so $\dfrac{3}{10}<\dfrac{1}{3}$ and the symbol is $<$
$6.$ Would you rather have $\dfrac{3}{4}$ or $\dfrac{3}{5}$ of $\$\!100$?
👀 Show answer
$\dfrac{3}{4}$ of $\$\!100$ is $\dfrac{3}{4}\times 100=75$, so that is $\$\!75$.
$\dfrac{3}{5}$ of $\$\!100$ is $\dfrac{3}{5}\times 100=60$, so that is $\$\!60$.
Since $75>60$, you would rather have $\dfrac{3}{4}$ of $\$\!100$.
🧠 Think like a Mathematician
If $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4}$, $\dfrac{1}{5}$ and $\dfrac{1}{10}$ of five different whole numbers all have the same value, what can you say about the set of whole numbers? Give some examples.
👀 show answer
Let the common value be $k$. Then: $\dfrac{1}{2}A=k$, $\dfrac{1}{3}B=k$, $\dfrac{1}{4}C=k$, $\dfrac{1}{5}D=k$, $\dfrac{1}{10}E=k$.
So the whole numbers must be: $A=2k,\; B=3k,\; C=4k,\; D=5k,\; E=10k$. That means the set of whole numbers is always in the ratio $2:3:4:5:10$ (and each is a multiple of its denominator).
Examples:
- If $k=6$, the numbers can be $12, 18, 24, 30, 60$.
- If $k=10$, the numbers can be $20, 30, 40, 50, 100$.
- If $k=1$, the numbers can be $2, 3, 4, 5, 10$.
❓ EXERCISES
$7.$ Continue the patterns of the equivalent fractions.
a. $\dfrac{\square}{\square}=\dfrac{2}{4}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}$
b. $\dfrac{1}{10}=\dfrac{2}{20}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}$
c. $\dfrac{1}{4}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}=\dfrac{\square}{\square}$
👀 Show answer
a.$\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{3}{6}=\dfrac{4}{8}=\dfrac{5}{10}=\dfrac{6}{12}$
b.$\dfrac{1}{10}=\dfrac{2}{20}=\dfrac{3}{30}=\dfrac{4}{40}=\dfrac{5}{50}=\dfrac{6}{60}$
c.$\dfrac{1}{4}=\dfrac{2}{8}=\dfrac{3}{12}=\dfrac{4}{16}=\dfrac{5}{20}$
$8.$ Draw a diagram to show one of the sets of equivalent fractions in question $6$.
👀 Show answer
Example (one possible diagram): show $\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{3}{6}$.
Draw the same rectangle three times:
- Divide the first into $2$ equal parts and shade $1$ part.
- Divide the second into $4$ equal parts and shade $2$ parts.
- Divide the third into $6$ equal parts and shade $3$ parts.
The shaded area is the same each time, so the fractions are equivalent.
$9.$ Which unit fraction is equivalent to $\dfrac{9}{45}$?
👀 Show answer
Simplify $\dfrac{9}{45}$ by dividing the top and bottom by $9$:
$\dfrac{9}{45}=\dfrac{9\div 9}{45\div 9}=\dfrac{1}{5}$, so the unit fraction is $\dfrac{1}{5}$.