Ordering Fractions
🎯 In this topic you will
- Compare and order fractions
- Compare and order positive and negative fractions
🧠 Key Words
- advantages
- common denominator
- compare
- denominator
- disadvantages
- fractional part
- improper fraction
- improve
- mixed number
- order of size
- whole-number part
Show Definitions
- advantages: Positive aspects or benefits of a method or solution.
- common denominator: A shared multiple of the denominators of two or more fractions.
- compare: To examine numbers or quantities to determine their relationship in size or value.
- denominator: The bottom number in a fraction showing into how many equal parts the whole is divided.
- disadvantages: Negative aspects or drawbacks of a method or solution.
- fractional part: The portion of a number that represents a value less than one, expressed as a fraction.
- improper fraction: A fraction in which the numerator is equal to or larger than the denominator.
- improve: To make something better or more effective.
- mixed number: A number made up of a whole number and a fraction.
- order of size: Arranging numbers or objects from smallest to largest or vice versa.
- whole-number part: The integer part of a mixed number, found before the fraction.
When you write fractions in order of size, you must first compare them. You can compare fractions in two ways:
- Write them as fractions that have the same denominator.
- Write them as decimals.
Sometimes when you change a fraction to a decimal, you will get a decimal that goes on forever; for example, $\frac{1}{7} = 0.1428\ldots$
🔎 Reasoning Tip
Ellipses in decimals: The three dots (called ellipses) at the end of a decimal show that it goes on forever.
Sometimes there are repeating numbers in the decimal; for example, $\frac{2}{3} = 0.6666\ldots$ and $\frac{21}{37} = 0.567567567\ldots$
These decimals are called recurring decimals.
You can write $0.6666\ldots$ as $0.\overline{6}$.
You can write $0.567567567\ldots$ as $0.\overline{567}$.
🔎 Reasoning Tip
Recurring decimals: The dot above the 6 shows that the 6 is recurring (or repeating).
🔎 Reasoning Tip
Recurring decimals: The dots above the 5 and 7 show that the 567 is recurring (or repeating).
❓ EXERCISES
Use the common denominator method to answer Questions 1–2.
1. Write the correct symbol, $=$ or $\ne$, between each pair of fractions.
a. $\dfrac{2}{3}\ \square\ \dfrac{10}{15}$
b. $\dfrac{3}{5}\ \square\ \dfrac{13}{20}$
👀 Show answers
b. $\ne$ because $\dfrac{3}{5}=\dfrac{12}{20}\ne\dfrac{13}{20}$.
2. Write the correct symbol, $<$ or $>$, between each pair of fractions.
🔎 Reasoning Tip
Comparing fractions: First, change any improper fractions to mixed numbers. When the whole number parts are the same, compare the fractional parts. Use a common denominator if needed.
a. $\dfrac{21}{5}\ \square\ 3\dfrac{4}{5}$
b. $4\dfrac{8}{9}\ \square\ \dfrac{46}{9}$
c. $\dfrac{37}{4}\ \square\ 9\dfrac{3}{4}$
d. $7\dfrac{2}{3}\ \square\ \dfrac{22}{3}$
e. $\dfrac{17}{3}\ \square\ 5\dfrac{5}{6}$
f. $\dfrac{25}{12}\ \square\ 2\dfrac{1}{4}$
g. $3\dfrac{5}{7}\ \square\ \dfrac{67}{21}$
h. $9\dfrac{3}{4}\ \square\ \dfrac{77}{8}$
👀 Show answers
b. $<$ since $4\dfrac{8}{9}=\dfrac{44}{9}<\dfrac{46}{9}$.
c. $<$ since $\dfrac{37}{4}<\dfrac{39}{4}=9\dfrac{3}{4}$.
d. $>$ since $7\dfrac{2}{3}=\dfrac{23}{3}>\dfrac{22}{3}$.
e. $<$ since $\dfrac{17}{3}=\dfrac{34}{6}<\dfrac{35}{6}=5\dfrac{5}{6}$.
f. $<$ since $\dfrac{25}{12}\approx2.083<2\dfrac{1}{4}=2.25$.
g. $>$ since $3\dfrac{5}{7}=\dfrac{26}{7}=\dfrac{78}{21}>\dfrac{67}{21}$.
h. $>$ since $9\dfrac{3}{4}=\dfrac{39}{4}=9.75>\dfrac{77}{8}=9.625$.
3. Marcus and Arun compare the methods they use to work out which fraction is larger: $\dfrac{25}{4}$ or $\dfrac{63}{10}$.
a. Critique their methods by explaining the advantages and disadvantages of each method.
b. Can you improve either of their methods? If you can, write down your method(s).
👀 Show answers
a. Critique
Method 1 (mixed numbers, then compare fractional parts): • Pros: strong concept focus; highlights that the whole parts are both $6$ and only the fractions need comparing. • Cons: extra steps (convert to mixed numbers, then find a common denominator for $\dfrac{1}{4}$ and $\dfrac{3}{10}$). Slower and error-prone.
Method 2 (common denominator $40$ for the improper fractions): • Pros: direct comparison: $\dfrac{25}{4}=\dfrac{250}{40}$ and $\dfrac{63}{10}=\dfrac{252}{40}$; fewer conversions once you choose $40$. • Cons: still requires finding an LCM and multiplying both numerators and denominators.
b. Improvements
Cross-multiplication (no LCM needed): Compare $25/4$ and $63/10$ by checking $25\times10$ vs $63\times4$. $25\times10=250$ and $63\times4=252$; since $252>250$, we have $\dfrac{63}{10}>\dfrac{25}{4}$. This is the quickest exact method.
Or decimals (if allowed): $6.3>6.25$.
4. Work out which fraction is larger.
- $\dfrac{47}{6}$ or $\dfrac{31}{4}$
- $\dfrac{33}{4}$ or $\dfrac{42}{5}$
- $\dfrac{49}{15}$ or $\dfrac{33}{10}$
👀 Show answers
b. $33\times5 = 165$, $42\times4 = 168$. Since $168 > 165$, $\dfrac{42}{5}$ is larger.
c. $49\times10 = 490$, $33\times15 = 495$. Since $495 > 490$, $\dfrac{33}{10}$ is larger.
🧠 Think like a Mathematician
Question 5:
What method would you use to answer this question?
Put these fraction cards in order of size, starting with the smallest:
$\dfrac{13}{10},\ \dfrac{7}{12},\ \dfrac{7}{5},\ \dfrac{1}{4}$
Question 6:
When you compare fractions by converting them to decimals, how many decimal places do you need to look at?
👀 show answer
To compare fractions, one useful method is to convert them into decimals (or percentages).
$\dfrac{13}{10} = 1.3$
$\dfrac{7}{12} \approx 0.583$
$\dfrac{7}{5} = 1.4$
$\dfrac{1}{4} = 0.25$
So, in order from smallest to largest: $\dfrac{1}{4},\ \dfrac{7}{12},\ \dfrac{13}{10},\ \dfrac{7}{5}$.
✅ You usually only need to look at 2 or 3 decimal places to compare fractions accurately, unless the decimals are very close together.
❓ EXERCISES
7a. Copy and complete the workings to write each of these improper fractions as a decimal.
- $\dfrac{11}{6}$
- $\dfrac{19}{11}$
- $\dfrac{17}{9}$
7b. Write the fractions $\dfrac{11}{6}$, $\dfrac{19}{11}$ and $\dfrac{17}{9}$ in order of size, starting with the smallest.
8a. Match each of these fractions to its correct decimal: $ \dfrac{7}{3}, \dfrac{16}{7}, \dfrac{58}{25}, \dfrac{9}{4}$ Decimals: 2.25, 2.28…, 2.32, 2.33…
8b. Write the fractions $\dfrac{7}{3}, \dfrac{16}{7}, \dfrac{58}{25}, \dfrac{9}{4}$ in order of size, starting with the smallest.
9. Write these fractions in order of size, starting with the smallest: $3\dfrac{4}{5}, \dfrac{15}{4}, \dfrac{37}{10}, 3\dfrac{5}{7}$
10. Yasmeen has five improper fraction cards. She puts them in order, starting with the smallest. There are marks on two of the cards: $\dfrac{8}{5}, \dfrac{7}{4}, \dfrac{17}{9}$ plus two missing cards.
What fractions could be under the marks? Give two examples for each card. Explain how you worked out your answer.
👀 Show answers
7a.
- $\dfrac{11}{6} = 1.8333\ldots$
- $\dfrac{19}{11} = 1.7272\ldots$
- $\dfrac{17}{9} = 1.8888\ldots$
7b. Order: $\dfrac{19}{11} \lt \dfrac{11}{6} \lt \dfrac{17}{9}$
8a.
- $\dfrac{7}{3} = 2.33\ldots$
- $\dfrac{16}{7} = 2.28\ldots$
- $\dfrac{58}{25} = 2.32$
- $\dfrac{9}{4} = 2.25$
8b. Order: $\dfrac{9}{4} \lt \dfrac{16}{7} \lt \dfrac{58}{25} \lt \dfrac{7}{3}$
9.
- $3\dfrac{4}{5} = \dfrac{19}{5} = 3.8$
- $\dfrac{15}{4} = 3.75$
- $\dfrac{37}{10} = 3.7$
- $3\dfrac{5}{7} = \dfrac{26}{7} \approx 3.714$
10. Known values: $\dfrac{8}{5} = 1.6$, $\dfrac{7}{4} = 1.75$, $\dfrac{17}{9} \approx 1.89$. Possible missing fractions must be less than 1.6 and greater than 1.89.
Examples: $\dfrac{5}{4} = 1.25$, $\dfrac{3}{2} = 1.5$ could go before $\dfrac{8}{5}$, and $\dfrac{11}{5} = 2.2$, $\dfrac{23}{10} = 2.3$ could go after $\dfrac{17}{9}$.
🍬 Learning Bridge
Now that you know how to compare and order fractions using common denominators or by converting them into decimals, it’s time to extend these skills. Next, you’ll see how the same ideas apply when fractions involve negative numbers or recurring decimals, where careful attention to signs and repeating patterns helps you decide which values are greater or smaller.
When you write fractions in order of size, you must first compare them.
You can compare fractions in two ways.
- 1 Write them as fractions which have the same denominator.
- 2 Write them as decimals.
❓ EXERCISES
For Questions 1–2, use the common denominator method.
11. Write the correct sign, $=$ or $\ne$, between each pair of fractions.
- $\dfrac{11}{4}\ \square\ 2\dfrac{16}{20}$
- $\dfrac{45}{6}\ \square\ 7\dfrac{1}{2}$
- $-\dfrac{15}{8}\ \square\ -2\dfrac{1}{8}$
- $-8\dfrac{4}{5}\ \square\ -\dfrac{132}{15}$
👀 Show answers
b. $=$ ($\dfrac{45}{6}=7.5=7\dfrac{1}{2}$)
c. $\ne$ ($-\dfrac{15}{8}=-1.875$, $-2\dfrac{1}{8}=-2.125$)
d. $=$ ($-8\dfrac{4}{5}=-\dfrac{44}{5}$ and $-\dfrac{132}{15}=-\dfrac{44}{5}$)
12. Write the correct symbol, $<$ or $>$, between each pair of fractions.
🔎 Reasoning Tip
Comparing fractions: Change the improper fractions to mixed numbers first. Then compare the fractions by using a common denominator.
- $\dfrac{13}{2}\ \square\ 6\dfrac{5}{8}$
- $\dfrac{7}{3}\ \square\ 6\dfrac{7}{12}$
- $5\dfrac{3}{5}\ \square\ \dfrac{82}{15}$
- $\dfrac{19}{4}\ \square\ 4\dfrac{4}{5}$
- $-\dfrac{17}{4}\ \square\ -4\dfrac{5}{12}$
- $-\dfrac{7}{3}\ \square\ -2\dfrac{5}{9}$
- $-\dfrac{21}{5}\ \square\ -4\dfrac{2}{15}$
- $-\dfrac{8}{5}\ \square\ -1\dfrac{5}{7}$
👀 Show answers
b. $<$ ($\dfrac{7}{3}\approx2.33 < 6\dfrac{7}{12}$)
c. $>$ ($5.6 > 5.466\ldots$)
d. $<$ ($4.75 < 4.8$)
e. $>$ ($-4.25 > -4.416\ldots$)
f. $>$ ($-2.333\ldots > -2.555\ldots$)
g. $<$ ($-4.2 < -4.133\ldots$)
h. $>$ ($-1.6 > -1.714\ldots$)
13. Zara and Sofia compare the methods they use to work out which is larger, $-\dfrac{8}{3}$ or $-2\dfrac{4}{7}$. They both start by rewriting as mixed numbers and using a common denominator of 21:
- $-\dfrac{8}{3} = -2\dfrac{2}{3} = -2\dfrac{14}{21}$
- $-2\dfrac{4}{7} = -2\dfrac{12}{21}$
a) Write the advantages and disadvantages of each method (Zara’s symbolic comparison vs Sofia’s number line).
b) Can you improve either method?
c) What is your preferred method for comparing negative fractions? Explain why.
👀 Show discussion
Zara’s method: ✅ Advantage: Quick symbolic reasoning by comparing without negatives first. ❌ Disadvantage: Can be confusing when re-applying negatives, risk of error in sign logic.
Sofia’s method: ✅ Advantage: Visual, number line makes negative comparison clear. ❌ Disadvantage: Slower and less practical for larger/complex fractions.
Improvement: Combine both: Use common denominators first, then check on a number line if unsure.
Preferred method: Common denominator works best—clear, systematic, and avoids confusion. Conclusion: $-2\dfrac{14}{21} < -2\dfrac{12}{21}$, so $-\dfrac{8}{3} < -2\dfrac{4}{7}$.
14. Work out which fraction is larger.
- $-\dfrac{7}{4}$ or $-1\dfrac{13}{16}$
- $-\dfrac{21}{5}$ or $-\dfrac{83}{20}$
- $-6\dfrac{2}{9}$ or $-\dfrac{37}{6}$
👀 Show answers
b. $-4.2 \gt -4.15$, so $-\dfrac{83}{20}$ is larger.
c. $-6.222\ldots \gt -6.166\ldots$, so $-\dfrac{37}{6}$ is larger.
🧠 Think like a Mathematician
Question 15:
a) With each pair of fractions, decide which is larger.
i. $\dfrac{1}{5}$ or $\dfrac{3}{5}$
ii. $\dfrac{7}{9}$ or $\dfrac{5}{9}$
iii. $\dfrac{13}{11}$ or $\dfrac{19}{11}$
b) Discuss your answers to part a.
Copy and complete this sentence. Use either ‘larger’ or ‘smaller’:
c) With each pair of fractions, decide which is larger.
i. $\dfrac{1}{5}$ or $\dfrac{1}{7}$
ii. $\dfrac{2}{9}$ or $\dfrac{2}{3}$
iii. $\dfrac{13}{4}$ or $\dfrac{13}{7}$
d) Discuss your answers to part c.
Copy and complete this sentence. Use either ‘larger’ or ‘smaller’:
👀 show answer
a) Comparing fractions with the same denominator:
i. $\dfrac{3}{5} > \dfrac{1}{5}$
ii. $\dfrac{7}{9} > \dfrac{5}{9}$
iii. $\dfrac{19}{11} > \dfrac{13}{11}$
b) When denominators are equal, the bigger numerator gives the bigger fraction. ✅
c) Comparing fractions with the same numerator:
i. $\dfrac{1}{5} > \dfrac{1}{7}$
ii. $\dfrac{2}{3} > \dfrac{2}{9}$
iii. $\dfrac{13}{4} > \dfrac{13}{7}$
d) When numerators are equal, the bigger denominator gives the smaller fraction. ✅
❓ EXERCISE
16. Write the correct symbol, < or >, between each pair of fractions.
- $\dfrac{3}{11} \; \Box \; \dfrac{5}{11}$
- $\dfrac{7}{18} \; \Box \; \dfrac{5}{18}$
- $\dfrac{12}{7} \; \Box \; \dfrac{10}{7}$
- $\dfrac{8}{17} \; \Box \; \dfrac{8}{19}$
- $\dfrac{9}{13} \; \Box \; \dfrac{9}{10}$
- $\dfrac{15}{4} \; \Box \; \dfrac{15}{7}$
👀 Show answers
b. $\dfrac{7}{18} \gt \dfrac{5}{18}$
c. $\dfrac{12}{7} \gt \dfrac{10}{7}$
d. $\dfrac{8}{17} \gt \dfrac{8}{19}$
e. $\dfrac{9}{13} \lt \dfrac{9}{10}$
f. $\dfrac{15}{4} \gt \dfrac{15}{7}$
🧠 Think like a Mathematician
Question 17:
Work with a partner. Discuss different methods you could use to answer this question.
Put these fraction cards in order of size, starting with the smallest:
$-\dfrac{17}{8}$, $-3\dfrac{1}{4}$, $-\dfrac{13}{6}$, $-\dfrac{7}{13}$
What do you think is the best method to use? Explain why.
👀 show answer
Convert each mixed number or fraction into a decimal (or improper fraction) to compare them clearly:
$-3\dfrac{1}{4} = -3.25$
$-\dfrac{17}{8} = -2.125$
$-\dfrac{13}{6} \approx -2.166...$
$-\dfrac{7}{13} \approx -0.538...$
Now order them from smallest (most negative) to largest:
$-3.25 \; < \; -2.166... \; < \; -2.125 \; < \; -0.538...$
✅ Final order: $-3\dfrac{1}{4},\; -\dfrac{13}{6},\; -\dfrac{17}{8},\; -\dfrac{7}{13}$
❓ EXERCISE
18. Put these fraction cards in order of size, starting with the smallest:
$-3\dfrac{1}{6}, \; -\dfrac{9}{11}, \; -\dfrac{19}{5}, \; -4\dfrac{2}{5}$
👀 Show answer
Step 1: Convert to improper fractions or decimals.
- $-3\dfrac{1}{6} = -\dfrac{19}{6} \approx -3.17$
- $-\dfrac{9}{11} \approx -0.82$
- $-\dfrac{19}{5} = -3.8$
- $-4\dfrac{2}{5} = -\dfrac{22}{5} = -4.4$
Step 2: Order them from smallest (most negative) to largest (least negative).
Final Order: $-4\dfrac{2}{5} \; \lt \; -\dfrac{19}{5} \; \lt \; -3\dfrac{1}{6} \; \lt \; -\dfrac{9}{11}$
19. Three sisters sat a maths test on the same day. Adele scored $\dfrac{16}{25}$, Belle scored $\dfrac{13}{20}$ and Catrina scored $63\%$. Who had the highest percentage score?
🔎 Reasoning Tip
Fractions to percentages: Change fractions into percentages by writing equivalent fractions with a denominator of 100.
20. Two driving instructors compare the pass rates for their students in January. Steffan had $34$ out of $40$ students pass. Irena had $87\%$ of students pass. Who had the higher pass rate? Show how you worked it out.
21a. Using division, write each fraction as a decimal (first four decimal places).
- $-\dfrac{11}{7}$
- $-\dfrac{14}{9}$
- $-\dfrac{19}{12}$
Also find the decimals (four d.p.) for: $\dfrac{4}{7}$, $\dfrac{5}{9}$, $\dfrac{7}{12}$.
21b. Write $-\dfrac{11}{7}$, $-\dfrac{14}{9}$, $-\dfrac{19}{12}$ in order of size, starting with the smallest.
22a. Match each fraction with the correct decimal: $-\dfrac{37}{9},\; -\dfrac{25}{6},\; -\dfrac{209}{50},\; -\dfrac{47}{11}$ Decimals: $-4.18,\; -4.27\ldots,\; -4.16\ldots,\; -4.11\ldots$
22b. Write $-\dfrac{37}{9},\; -\dfrac{25}{6},\; -\dfrac{209}{50},\; -\dfrac{47}{11}$ in order of size, starting with the smallest.
23. Write these fractions in order of size, starting with the smallest: $-\dfrac{107}{20},\; -\dfrac{37}{7},\; -5\dfrac{3}{8},\; -\dfrac{82}{15}$
👀 Show answers
9. Convert to %: $\dfrac{16}{25}=64\%$, $\dfrac{13}{20}=65\%$, Catrina $=63\%$. Highest: Belle ($65\%$).
10. Steffan: $\dfrac{34}{40}=0.85=85\%$. Irena: $87\%$. Higher: Irena.
11a. $-\dfrac{11}{7}=-1.5714\ldots$ → first four d.p. $-1.5714$
$-\dfrac{14}{9}=-1.5555\ldots$ → $-1.5555$
$-\dfrac{19}{12}=-1.5833\ldots$ → $-1.5833$
$\dfrac{4}{7}=0.5714\ldots$ → $0.5714$; $\dfrac{5}{9}=0.5555\ldots$ → $0.5555$; $\dfrac{7}{12}=0.5833\ldots$ → $0.5833$
11b. More negative = smaller: $-\dfrac{19}{12} \lt -\dfrac{11}{7} \lt -\dfrac{14}{9}$.
12a. $-\dfrac{209}{50}=-4.18$; $-\dfrac{47}{11}=-4.27\ldots$; $-\dfrac{25}{6}=-4.16\ldots$; $-\dfrac{37}{9}=-4.11\ldots$.
12b. From smallest (most negative): $-\dfrac{47}{11} \lt -\dfrac{209}{50} \lt -\dfrac{25}{6} \lt -\dfrac{37}{9}$.
13. Convert to decimals: $-\dfrac{107}{20}=-5.35$, $-\dfrac{37}{7}\approx-5.2857$, $-5\dfrac{3}{8}=-5.375$, $-\dfrac{82}{15}\approx-5.4666\ldots$ Order (smallest to largest): $-\dfrac{82}{15} \lt -5\dfrac{3}{8} \lt -\dfrac{107}{20} \lt -\dfrac{37}{7}$.
24. One day, a farmer sells 92% of her eggs. The following day, she sells $56$ out of $62$ eggs. Use a calculator to work out on which day she sold the greater percentage of eggs.
🔎 Reasoning Tip
Fraction to percentage: Change \( \tfrac{56}{62} \) into a decimal, then multiply the answer by 100 to get a percentage.
25. Arun takes two English tests. - First test: $\tfrac{65}{72}$ - Second test: $\tfrac{35}{38}$
Arun says: *“If I compare using a common denominator, I’ll need 1368.”* Sofia says: *“It’s easier to convert into decimals or percentages.”*
- Use Arun’s method to compare the scores.
- Use Sofia’s method to compare the scores.
- Which method do you prefer and why?
- In which test did Arun get the better score?
26. In a science experiment, two groups of seeds are planted: - Group A: $175$ planted, $156$ grew. - Group B: $220$ planted, $189$ grew.
Use a calculator to work out which group is better at growing.
🔎 Reasoning Tip
Comparing fractions: Change \( \tfrac{156}{175} \) and \( \tfrac{189}{220} \) into decimals or percentages to compare.
27. Li has 5 improper fraction cards, in order from smallest to largest: $\; -\tfrac{20}{7},\; -\tfrac{25}{9},\; \_\_\_,\; -\tfrac{13}{5},\; \_\_\_ \;$
Two cards are missing. What fractions could be under the marks? Give two examples for each card.
👀 Show answers
14. Day 1: $92\%$. Day 2: $\dfrac{56}{62} \approx 90.32\%$. Greater: Day 1 (92%).
15a (Arun’s method). $\tfrac{65}{72} = \tfrac{1235}{1368}$, $\tfrac{35}{38} = \tfrac{1260}{1368}$. $\tfrac{1260}{1368} > \tfrac{1235}{1368}$ → second test higher.
15b (Sofia’s method). $\tfrac{65}{72} \approx 0.9028 = 90.3\%$, $\tfrac{35}{38} \approx 0.9210 = 92.1\%$. Second test higher.
15c. Sofia’s method is easier (no huge denominators, quick with calculator).
15d. Better score: second test.
16. Group A: $\dfrac{156}{175} \approx 89.14\%$. Group B: $\dfrac{189}{220} \approx 85.91\%$. Better: Group A.
17. Decimal values: $-\tfrac{20}{7} \approx -2.857$, $-\tfrac{25}{9} \approx -2.778$, $-\tfrac{13}{5} = -2.6$. Missing fractions should fit between these values. Examples: Between $-\tfrac{25}{9}$ and $-\tfrac{13}{5}$: $-\tfrac{8}{3}=-2.667$, $-\tfrac{21}{8}=-2.625$. After $-\tfrac{13}{5}$: $-\tfrac{5}{2}=-2.5$, $-\tfrac{19}{8}=-2.375$.
Be Careful
Always check whether the fractions are positive or negative before comparing. With negative numbers, the fraction further to the left on the number line is the smaller one.