Equivalence and comparison
🎯 In this topic you will
- Find equivalent improper fractions and mixed numbers
- Find equivalent proper fractions, decimals and percentages
- Order and compare proper fractions, decimals and percentages
🧠 Key Words
- improper fraction
- mixed number
Show Definitions
- improper fraction: A fraction where the numerator is greater than or equal to the denominator.
- mixed number: A number consisting of a whole number and a proper fraction combined.
Reviewing Equivalent Fractions 🍕
In earlier stages, you worked with equivalent proper fractions.
Moving to Improper Fractions ➕
In this section, you will work with equivalent improper fractions and mixed numbers.
Comparing Fractions, Decimals & Percentages 🔍
You will also order fractions, decimals and percentages.
Making Numbers Comparable ✏️
You need to write them all in the same way to compare them.
❓ EXERCISES
1. What do these diagrams show? Write your answer as a mixed number and as an improper fraction.
a. Circle diagrams divided into quarters.
b. Rectangle diagrams divided into fifths.
👀 Show answer
1a. There are $6$ quarters altogether, so the mixed number is $1\dfrac{1}{2}$ and the improper fraction is $\dfrac{3}{2}$ (or $\dfrac{6}{4}$).
1b. There are $14$ fifths altogether, so the mixed number is $2\dfrac{4}{5}$ and the improper fraction is $\dfrac{14}{5}$.
2. Convert these improper fractions to mixed numbers.
a. $\dfrac{9}{4}$
b. $\dfrac{12}{7}$
c. $\dfrac{16}{3}$
d. $\dfrac{37}{10}$
👀 Show answer
2a. $\dfrac{9}{4} = 2\dfrac{1}{4}$ because $4 \times 2 = 8$ and $9 - 8 = 1$.
2b. $\dfrac{12}{7} = 1\dfrac{5}{7}$ because $7 \times 1 = 7$ and $12 - 7 = 5$.
2c. $\dfrac{16}{3} = 5\dfrac{1}{3}$ because $3 \times 5 = 15$ and $16 - 15 = 1$.
2d. $\dfrac{37}{10} = 3\dfrac{7}{10}$ because $10 \times 3 = 30$ and $37 - 30 = 7$.
3. Find the odd one out.
$1\dfrac{1}{4},\ \dfrac{9}{4},\ \dfrac{5}{4},\ 3\dfrac{1}{4},\ 2\dfrac{1}{4}$
Explain your answer.
👀 Show answer
Convert everything to improper fractions or decimals:
$1\dfrac{1}{4} = \dfrac{5}{4}$, $\dfrac{9}{4} = 2\dfrac{1}{4}$, $\dfrac{5}{4} = 1\dfrac{1}{4}$, $3\dfrac{1}{4} = \dfrac{13}{4}$, $2\dfrac{1}{4} = \dfrac{9}{4}$.
Odd one out: $3\dfrac{1}{4}$ because all the others are equal to either $1\dfrac{1}{4}$ or $2\dfrac{1}{4}$, but $3\dfrac{1}{4}$ is a different value.
4. Which of these fractions are equivalent to $40\%$?
$\dfrac{4}{10},\ \dfrac{1}{40},\ \dfrac{40}{100},\ \dfrac{1}{4}$
👀 Show answer
$40\% = \dfrac{40}{100} = 0.4$.
$\dfrac{4}{10} = 0.4$, $\dfrac{40}{100} = 0.4$, $\dfrac{1}{40} = 0.025$, and $\dfrac{1}{4} = 0.25$.
So the fractions equivalent to $40\%$ are $\dfrac{4}{10}$ and $\dfrac{40}{100}$.
5. Look at the group of fractions, decimals and percentages.
$\dfrac{1}{2},\ 20\%,\ \dfrac{1}{5},\ 0.2$
Find the odd one out.
Explain why it is the odd one out.
👀 Show answer
$\dfrac{1}{2} = 0.5$, $20\% = 0.2$, $\dfrac{1}{5} = 0.2$, and $0.2 = 0.2$.
Odd one out: $\dfrac{1}{2}$ because it is equal to $0.5$ while all the others are equal to $0.2$.
6. Find the missing number.
a. $\dfrac{9}{\square} = 75\%$
b. $\dfrac{2}{\square} = 25\%$
c. $\dfrac{\square}{50} = 50\%$
👀 Show answer
6a. $75\% = 0.75 = \dfrac{3}{4}$. So $\dfrac{9}{\square} = \dfrac{3}{4}$ gives $\square = 12$, and the fraction is $\dfrac{9}{12}$.
6b. $25\% = 0.25 = \dfrac{1}{4}$. So $\dfrac{2}{\square} = \dfrac{1}{4}$ gives $\square = 8$, and the fraction is $\dfrac{2}{8}$.
6c. $50\% = 0.5 = \dfrac{1}{2}$. So $\dfrac{\square}{50} = \dfrac{1}{2}$ gives $\square = 25$, and the fraction is $\dfrac{25}{50}$.
7. Use one of the symbols $<$, $>$ or $=$ to complete these statements.
a. $\dfrac{3}{5} \ \square\ 30\%$
b. $0.4 \ \square\ \dfrac{2}{5}$
c. $25\% \ \square\ \dfrac{1}{3}$
d. $\dfrac{1}{4} \ \square\ 0.4$
e. $0.7 \ \square\ \dfrac{3}{4}$
f. $90\% \ \square\ 0.9$
👀 Show answer
7a. $\dfrac{3}{5} = 0.6$ and $30\% = 0.3$, so $\dfrac{3}{5} > 30\%$.
7b. $0.4 = \dfrac{2}{5}$, so $0.4 = \dfrac{2}{5}$.
7c. $25\% = 0.25$ and $\dfrac{1}{3} \approx 0.33$, so $25\% < \dfrac{1}{3}$.
7d. $\dfrac{1}{4} = 0.25$ and $0.4 = 0.4$, so $\dfrac{1}{4} < 0.4$.
7e. $0.7 = 0.7$ and $\dfrac{3}{4} = 0.75$, so $0.7 < \dfrac{3}{4}$.
7f. $90\% = 0.9$, so $90\% = 0.9$.
8. Write these fractions, decimals and percentages in order starting with the smallest.
a. $70\%,\ \dfrac{2}{5},\ 0.1,\ \dfrac{3}{5},\ 50\%$
b. $0.7,\ \dfrac{4}{5},\ 75\%,\ \dfrac{3}{5},\ 65\%$
👀 Show answer
8a. Convert to decimals: $70\% = 0.7$, $\dfrac{2}{5} = 0.4$, $0.1 = 0.1$, $\dfrac{3}{5} = 0.6$, $50\% = 0.5$.
Ordered from smallest to largest: $0.1,\ \dfrac{2}{5},\ 50\%,\ \dfrac{3}{5},\ 70\%$.
8b. Convert to decimals: $0.7 = 0.7$, $\dfrac{4}{5} = 0.8$, $75\% = 0.75$, $\dfrac{3}{5} = 0.6$, $65\% = 0.65$.
Ordered from smallest to largest: $\dfrac{3}{5},\ 65\%,\ 0.7,\ 75\%,\ \dfrac{4}{5}$.