Ordering decimals
🎯 In this topic you will
- Compare and order decimals
🧠 Key Words
- compare
- decimal number
- order
- term-to-term rule
- fourth
- hundredths
- tenths
- whole-number part
Show Definitions
- compare: To examine numbers or quantities to determine which is greater, smaller, or if they are equal.
- decimal number: A number that includes a decimal point to show values smaller than one.
- order: To arrange numbers or items according to size, value, or position.
- term-to-term rule: A rule describing how to move from one term in a sequence to the next.
- fourth: One of four equal parts of a whole; also called a quarter.
- hundredths: Parts formed when a whole is divided into 100 equal sections.
- tenths: Parts formed when a whole is divided into 10 equal sections.
- whole-number part: The integer part of a decimal number, found before the decimal point.
To order decimal numbers, you write them from the smallest to the largest.
Different whole-number parts
| First, compare the whole-number part of the numbers. | |
| Consider these three decimal numbers: | $8.9$, $14.639$, $6.45$ |
| If you highlight just the whole-number parts, you get: | $8$$.9$, $14$$.639$, $6$$.45$ |
| You can see that $14$ is the biggest of the whole numbers and $6$ is the smallest. | |
| So, in order of size, the numbers are: | $6.45$, $8.9$, $14.639$ |
Same whole-number parts
| When you have to put in order numbers with the same whole-number part, you must first compare the tenths, then the hundredths, and so on. | |
| Consider these three decimal numbers: | $2.82$, $2.6$, $2.816$ |
| They all have the same whole number of $2$. | $2$$.82$, $2$$.6$, $2$$.816$ |
| If you highlight just the tenths, you get: | $2.$$8$$2$, $2.$$6$, $2.$$8$$16$ |
🔎 Reasoning TipOrdering numbers: Write \( 2.6 \) at the start because you know it is the smallest number. You can see that $2.6$ is the smallest, but the other two numbers both have $8$ tenths, so highlight the hundredths. |
|
| You can see that $2.816$ is smaller than $2.82$. | $2.$$6$, $2.8$$2$, $2.8$$1$$6$ |
| So, in order of size, the numbers are: | $2.6$, $2.816$, $2.82$ |
❓ EXERCISES
1. For each pair, write down which is the smallest decimal number.
a) $13.5$, $9.99$
b) $4.32$, $3.67$
c) $12.56$, $21.652$
d) $127.06$, $246.9$
e) $0.67$, $0.72$
f) $3.4$, $3.21$
g) $18.54$, $18.45$
h) $0.05$, $0.043$
i) $0.09$, $0.1$
👀 Show answer
b) $3.67$
c) $12.56$
d) $127.06$
e) $0.67$
f) $3.21$
g) $18.45$
h) $0.043$
i) $0.09$
2. The table shows six of the fastest times run by women in the $100\text{ m}$ race.
| Name | Country | Year | Time (seconds) |
|---|---|---|---|
| Kerron Stewart | Jamaica | 2009 | $10.75$ |
| Marion Jones | USA | 1998 | $10.65$ |
| Merlene Ottey | Jamaica | 1996 | $10.74$ |
| Carmelita Jeter | USA | 2009 | $10.64$ |
| Shelly-Ann Fraser | Jamaica | 2009 | $10.73$ |
| Florence Griffith-Joyner | USA | 1988 | $10.49$ |
a) Write the times in order of size.
b) Which woman has the fourth fastest time?
👀 Show answer
Order of times:
$10.49$, $10.64$, $10.65$, $10.73$, $10.74$, $10.75$
Fourth fastest: Shelly-Ann Fraser ($10.73$)
3. Marcus is comparing the numbers $8.27$ and $8.4$. He says: “$8.27$ is greater than $8.4$ because $27$ is greater than $4$.” Is Marcus correct? Explain your answer.
👀 Show answer
No, Marcus is incorrect. When comparing decimals, you compare digits from left to right. $8.27$ is less than $8.4$ because at the tenths place, $2 < 4$.
4. Write the correct sign, < or >, between each pair of numbers.
The symbol < means ‘is smaller than’. The symbol > means ‘is bigger than’.
a) $6.03 \; \square \; 6.24$
b) $9.35 \; \square \; 9.41$
c) $0.49 \; \square \; 0.51$
d) $18.05 \; \square \; 18.02$
e) $9.2 \; \square \; 9.01$
f) $2.19 \; \square \; 2.205$
g) $0.072 \; \square \; 0.06$
h) $29.882 \; \square \; 29.88$
👀 Show answer
d) $18.05 > 18.02$ e) $9.2 > 9.01$ f) $2.19 < 2.205$
g) $0.072 > 0.06$ h) $29.882 > 29.88$
5. Write the correct sign, $=$ or $\neq$, between each pair of numbers.
The symbol = means ‘is equal to’. The symbol ≠ means ‘is not equal to’.
a) $4.2 \; \square \; 4.20$
b) $3.75 \; \square \; 3.57$
c) $0.340 \; \square \; 0.304$
d) $9.58 \; \square \; 9.580$
e) $128.00 \; \square \; 128$
f) $0.0034 \; \square \; 0.034$
👀 Show answer
d) $9.58 = 9.580$ e) $128.00 = 128$ f) $0.0034 \neq 0.034$
6. For each set, use Ulrika’s method to write the decimal numbers in order of size.
Question
Write the decimal numbers $4.23$, $4.6$ and $4.179$ in order of size.
Solution
$4.179$ has the most decimal places, so give all the other numbers three decimal places by adding zeros at the end:
$4.230$, $4.600$, $4.179$
Now compare $230$, $600$ and $179$: $179$ is the smallest, then $230$, and $600$ is the biggest.
In order of size, the numbers are: $4.179$, $4.23$, $4.6$
a) $2.7$, $2.15$, $2.009$
b) $3.45$, $3.342$, $3.2$
c) $17.05$, $17.1$, $17.125$, $17.42$
d) $0.71$, $0.52$, $0.77$, $0.59$
e) $5.212$, $5.2$, $5.219$, $5.199$
f) $9.08$, $9.7$, $9.901$, $9.03$, $9.99$
g) Critique Ulrika’s method by explaining the advantages and disadvantages of her method. Can you improve Ulrika’s method?
👀 Show answer
b) Add zeros: $3.450$, $3.342$, $3.200$. Order: $3.2$, $3.342$, $3.45$.
c) Add zeros: $17.050$, $17.100$, $17.125$, $17.420$. Order: $17.05$, $17.1$, $17.125$, $17.42$.
d) Add zeros: $0.710$, $0.520$, $0.770$, $0.590$. Order: $0.52$, $0.59$, $0.71$, $0.77$.
e) Add zeros: $5.212$, $5.200$, $5.219$, $5.199$. Order: $5.199$, $5.2$, $5.212$, $5.219$.
f) Add zeros: $9.080$, $9.700$, $9.901$, $9.030$, $9.990$. Order: $9.03$, $9.08$, $9.7$, $9.901$, $9.99$.
g)Advantages: Easy to compare when all numbers have the same number of decimal places; avoids confusion with unequal decimal lengths.
Disadvantages: Adding zeros is unnecessary if you understand place value; may be slower for large sets.
Improvement: Compare numbers digit-by-digit from left to right without adding zeros explicitly, recognising place value directly.
7. Write these amounts in order of size. Use the tip box to help you.
🔎 Reasoning Tip
Metric conversions:
- 10 mm = 1 cm
- 100 cm = 1 m
- 1000 m = 1 km
- 10 mL = 1 cL
- 100 cL = 1 L
- 1000 mL = 1 L
- 1000 mg = 1 g
- 1000 g = 1 kg
- 1000 kg = 1 t
a) $38.1\ \text{cL}$, $300\ \text{mL}$, $0.385\ \text{L}$
b) $725\ \text{mm}$, $7.3\ \text{cm}$, $0.705\ \text{m}$
c) $5.12\ \text{kg}$, $530\ \text{g}$, $0.0058\ \text{t}$, $519000\ \text{mg}$
d) $461.5\ \text{cm}$, $0.0046\ \text{km}$, $0.45\ \text{m}$, $4450\ \text{mm}$
👀 Show answer
b) Convert all to metres: $0.725$, $0.073$, $0.705$. Order: $7.3\ \text{cm}$, $0.705\ \text{m}$, $725\ \text{mm}$.
c) Convert all to kilograms: $5.12$, $0.53$, $5.8$, $0.519$. Order: $519000\ \text{mg}$, $530\ \text{g}$, $5.12\ \text{kg}$, $0.0058\ \text{t}$.
d) Convert all to metres: $4.615$, $4.6$, $0.45$, $4.45$. Order: $0.45\ \text{m}$, $4450\ \text{mm}$, $0.0046\ \text{km}$, $461.5\ \text{cm}$.
8. Brad puts these decimal number cards in order of size. There is a mark covering part of the number on the middle card.
a) Write down three possible numbers that could be on the middle card.
b) How many different numbers with three decimal places do you think could be on the middle card?
c) Show how you can convince others that your answer to part b is correct.
👀 Show answer
b) Any number between $3.07$ and $3.083$ with three decimal places. The hundredths digit must be $7$ and the thousandths digit between $1$ and $8$, so $8$ possibilities.
c) By listing: $3.071$, $3.072$, $3.073$, $3.074$, $3.075$, $3.076$, $3.077$, $3.078$ — all satisfy $3.07 < \text{number} < 3.083$.
🍬 Learning Bridge
Now that you’ve mastered comparing and ordering decimal numbers by looking at whole parts first and then decimal places, you’re ready to refine those skills further. In the next section, you’ll apply the same strategies to more varied examples — including measurements and negative numbers — where careful attention to units, decimal places, and signs makes all the difference.
To order decimal numbers, compare the whole-number part first.
When the numbers you are ordering have the same whole-number part, look at the decimal part and compare the tenths, then the hundredths, and so on.
🔎 Reasoning Tip
Decimal places: The number of digits after the decimal point is the number of decimal places (d.p.) in the number.
| Look at the three decimal numbers on the right. | $8.56$ | $7.4$ | $8.518$ |
| 1 Highlight the whole numbers. You can see that $7.4$ is the smallest number, so $7.4$ goes first. |
$8$.56 | $7$.4 | $8$.518 |
| 2 The other two numbers both have $8$ units, so highlight the tenths. | $7.4$ | $8$.$5$6 | $8$.$5$18 |
| 3 They both have the same number of tenths, so highlight the hundredths. | $7.4$ | $8.5$$6$ | $8.51$$8$ |
| 4 You can see that $8.518$ is smaller than $8.56$, so in order of size the numbers are: | $7.4$ | $8.518$ | $8.56$ |
🔎 Reasoning Tip
Comparing decimal numbers: You can use these symbols when comparing decimal numbers:
- = means “is equal to”
- ≠ means “is not equal to”
- > means “is bigger than”
- < means “is smaller than”
When you order decimal measurements, you must make sure they are all in the same units.
You need to remember these conversion factors.
| Length | Mass | Capacity |
| $10\ \text{mm} = 1\ \text{cm}$ | $1000\ \text{g} = 1\ \text{kg}$ | $1000\ \text{ml} = 1\ \text{l}$ |
| $100\ \text{cm} = 1\ \text{m}$ | $1000\ \text{kg} = 1\ \text{t}$ | |
| $1000\ \text{m} = 1\ \text{km}$ |
❓ EXERCISES
9. Write these decimal numbers in order of size, starting with the smallest.
They have all been started for you.
a) $5.49$, $2.06$, $7.99$, $5.91$ Started: $2.06$, $\square$, $\square$, $\square$
b) $3.09$, $2.87$, $3.11$, $2.55$ Started: $2.55$, $\square$, $\square$, $\square$
c) $12.1$, $11.88$, $12.01$, $11.82$ Started: $11.82$, $\square$, $\square$, $\square$
d) $9.09$, $8.9$, $9.53$, $9.4$ Started: $8.9$, $\square$, $\square$, $\square$
👀 Show answer
b) $2.55,\ 2.87,\ 3.09,\ 3.11$
c) $11.82,\ 11.88,\ 12.01,\ 12.1$
d) $8.9,\ 9.09,\ 9.4,\ 9.53$
10. Write the correct sign, $<$ or $>$, between each pair of numbers.
a) $4.23\ \boxed{\ \ } \ 4.54$
b) $6.71\ \boxed{\ \ } \ 6.03$
c) $0.27\ \boxed{\ \ } \ 0.03$
d) $27.9\ \boxed{\ \ } \ 27.85$
e) $8.55\ \boxed{\ \ } \ 8.508$
f) $5.055\ \boxed{\ \ } \ 5.505$
👀 Show answer
b) $6.71\ >\ 6.03$
c) $0.27\ >\ 0.03$
d) $27.9\ >\ 27.85$
e) $8.55\ >\ 8.508$
f) $5.055\ <\ 5.505$
🧠 Think like a Mathematician
Scenario: Maya uses this method to order decimals:
Question: Write these numbers in order of size, starting with the smallest: 26.5, 26.41, 26.09, 26.001, 26.92
Answer:
The greatest number of decimal places in the numbers is 3.
Step 1: Write all the numbers with 3 decimal places: 26.500, 26.410, 26.090, 26.001, 26.920
Step 2: Compare only the numbers after the decimal point: 500, 410, 090, 001, 920
Step 3: Write these in order of size: 001, 090, 410, 500, 920
Step 4: Now write the decimal numbers in order: 26.001, 26.09, 26.41, 26.5, 26.92
- a) Do you understand how Maya’s method works?
- b) Do you like Maya’s method?
- c) Do you prefer Maya’s method to the method shown in Worked Example 4.1? Explain your answer.
Follow-up prompts:
👀 show answer
- a: Yes — Maya standardises all numbers to the same number of decimal places by adding zeros. This makes comparison easier because we can then compare digit by digit.
- b: This method is clear and avoids confusion when decimals have different lengths. It works well for teaching place value.
- c: Preference may vary — Maya’s method is very visual, while other methods (like converting to fractions or using a calculator) may be faster for advanced learners. However, Maya’s approach helps reinforce understanding of decimal place value.
- Note: Adding zeros to the end of a decimal does not change its value, but helps align numbers for easier comparison.
❓ EXERCISES
12. Use your preferred method to write these decimal numbers in order of size, starting with the smallest.
a) $23.66$, $23.592$, $23.6$, $23.605$
b) $0.107$, $0.08$, $0.1$, $0.009$
c) $6.725$, $6.78$, $6.007$, $6.71$
d) $11.02$, $11.032$, $11.002$, $11.1$
👀 Show answer
b) $0.009,\ 0.08,\ 0.1,\ 0.107$
c) $6.007,\ 6.71,\ 6.725,\ 6.78$
d) $11.002,\ 11.02,\ 11.032,\ 11.1$
13. Write the correct sign, $=$ or $\neq$, between each pair of measurements.
🔎 Reasoning Tip
Unit consistency: Start by converting one of the measurements so that both measurements are in the same units.
a) $6.71\ \boxed{\ \ }\ 670\ \text{ml}$
b) $4.05\ \text{t}\ \boxed{\ \ }\ 4500\ \text{kg}$
c) $0.85\ \text{km}\ \boxed{\ \ }\ 850\ \text{m}$
d) $0.985\ \text{m}\ \boxed{\ \ }\ 985\ \text{cm}$
e) $14.5\ \text{cm}\ \boxed{\ \ }\ 145\ \text{mm}$
f) $2300\ \text{g}\ \boxed{\ \ }\ 0.23\ \text{kg}$
👀 Show answer
b) $4.05\ \text{t} = 4500\ \text{kg}$
c) $0.85\ \text{km} = 850\ \text{m}$
d) $0.985\ \text{m} \neq 985\ \text{cm}$
e) $14.5\ \text{cm} = 145\ \text{mm}$
f) $2300\ \text{g} \neq 0.23\ \text{kg}$
14. Write the correct sign, $<$ or $>$, between each pair of measurements.
a) $4.51\ \boxed{\ \ }\ 2700\ \text{ml}$
b) $0.45\ \text{t}\ \boxed{\ \ }\ 547\ \text{kg}$
c) $3.5\ \text{cm}\ \boxed{\ \ }\ 345\ \text{mm}$
d) $0.06\ \text{kg}\ \boxed{\ \ }\ 550\ \text{g}$
e) $7800\ \text{m}\ \boxed{\ \ }\ 0.8\ \text{km}$
f) $0.065\ \text{m}\ \boxed{\ \ }\ 6.7\ \text{cm}$
👀 Show answer
b) $0.45\ \text{t} < 547\ \text{kg}$
c) $3.5\ \text{cm} < 345\ \text{mm}$
d) $0.06\ \text{kg} < 550\ \text{g}$
e) $7800\ \text{m} > 0.8\ \text{km}$
f) $0.065\ \text{m} < 6.7\ \text{cm}$
15. Write these measurements in order of size, starting with the smallest.
🔎 Reasoning Tip
Consistent units for ordering: Make sure all the measurements are in the same units before you start to order them.
a) $2.3\ \text{kg}$, $780\ \text{g}$, $2.18\ \text{kg}$, $1950\ \text{g}$
b) $5.4\ \text{cm}$, $12\ \text{mm}$, $0.8\ \text{cm}$, $9\ \text{mm}$
c) $12\ \text{m}$, $650\ \text{cm}$, $0.5\ \text{m}$, $53\ \text{cm}$
d) $0.551\ \text{L}$, $95\ \text{ml}$, $0.91\ \text{L}$, $450\ \text{ml}$
e) $6.55\ \text{km}$, $780\ \text{m}$, $6.4\ \text{km}$, $1450\ \text{m}$
f) $0.08\ \text{t}$, $920\ \text{kg}$, $0.15\ \text{t}$, $50\ \text{kg}$
👀 Show answer
b) Convert to cm: $5.4$, $1.2$, $0.8$, $0.9$. Order: $0.8\ \text{cm}$, $9\ \text{mm}$, $12\ \text{mm}$, $5.4\ \text{cm}$.
c) Convert to m: $12$, $6.5$, $0.5$, $0.53$. Order: $0.5\ \text{m}$, $53\ \text{cm}$, $650\ \text{cm}$, $12\ \text{m}$.
d) Convert to ml: $551$, $95$, $910$, $450$. Order: $95\ \text{ml}$, $450\ \text{ml}$, $0.551\ \text{L}$, $0.91\ \text{L}$.
e) Convert to km: $6.55$, $0.78$, $6.4$, $1.45$. Order: $780\ \text{m}$, $1450\ \text{m}$, $6.4\ \text{km}$, $6.55\ \text{km}$.
f) Convert to kg: $80$, $920$, $150$, $50$. Order: $50\ \text{kg}$, $0.08\ \text{t}$, $0.15\ \text{t}$, $920\ \text{kg}$.
🧠 Think like a Mathematician
Scenario: Arun’s task is to order the following decimal numbers from smallest to largest: $-4.52, \ -4.31, \ -4.05, \ -4.38$
Arun’s reasoning: “All the numbers start with -4, so I will just compare the decimal parts: 52, 31, 05, and 38. In order, they are 05, 31, 38, 52. So the order is $-4.05, \ -4.31, \ -4.38, \ -4.52$.”
- a) Is Arun correct? Explain your answer.
- b) What do you think is the best method to use to order negative decimal numbers?
Follow-up prompts:
👀 show answer
- a: No — Arun’s reasoning is incorrect. For negative numbers, a smaller decimal part actually means the number is closer to zero and therefore larger. The correct order from smallest to largest is: $-4.52, \ -4.38, \ -4.31, \ -4.05$.
- b: The best method is to visualise the numbers on a number line or compare them directly as whole negative values, remembering that for negative numbers, greater absolute values represent smaller numbers.
- Tip: When ordering negative decimals, reverse the normal comparison rule for their decimal parts.
❓ EXERCISES
17. Write the correct sign, $<$ or $>$, between each pair of numbers.
🔎 Reasoning Tip
Using number lines: Draw a number line to help if you want to.
a) $-4.27\ \boxed{\ \ }\ -4.38$
b) $-6.75\ \boxed{\ \ }\ -6.25$
c) $-0.2\ \boxed{\ \ }\ -0.03$
d) $-8.05\ \boxed{\ \ }\ -8.9$
👀 Show answer
b) $-6.75 < -6.25$
c) $-0.2 < -0.03$
d) $-8.05 > -8.9$
18. Write these decimal numbers in order of size, starting with the smallest.
a) $-4.67$, $-4.05$, $-4.76$, $-4.5$
b) $-11.525$, $-11.91$, $-11.08$, $-11.6$
👀 Show answer
b) $-11.91,\ -11.6,\ -11.525,\ -11.08$
19. Shen and Mia swim every day. They record the distances they swim each day for $10$ days.
These are the distances that Shen swims each day.
a) Shen has written down one distance incorrectly. Which one do you think it is? Explain your answer.
These are the distances that Mia swims each day.
b) Mia says that the longest distance she swam is more than eight times the shortest distance she swam. Is Mia correct? Explain your answer.
Shen and Mia swim in different swimming pools. One pool is $25\ \text{m}$ long. The other pool is $20\ \text{m}$ long. Shen and Mia always swim a whole number of lengths.
c) Who do you think swims in the $25\ \text{m}$ pool? Explain how you made your decision.
👀 Show answer
b) Yes, Mia is correct. Convert to kilometres: $1.2$, $0.24$, $0.4$, $1.64$, $0.82$, $0.64$, $0.2$, $1.42$, $0.96$, $0.88$. Shortest $=0.2\ \text{km}$, longest $=1.64\ \text{km}$. $8 \times 0.2 = 1.6\ \text{km}$ and $1.64\ \text{km} > 1.6\ \text{km}$, so the longest is more than eight times the shortest.
c) Shen swims in the $25\ \text{m}$ pool and Mia swims in the $20\ \text{m}$ pool. Shen’s distances in metres ($250$, $500$, $750$, $1500$, $1750$, $2000$, $2500$, …) are all multiples of $25$. Mia’s distances ($1200$, $240$, $400$, $1640$, $820$, $640$, $200$, $1420$, $960$, $880$) are all multiples of $20$ but some are not multiples of $25$ (e.g., $820$), so she fits the $20\ \text{m}$ pool.
20. Each of the cards describes a sequence of decimal numbers.
a) Work out the fifth term of each sequence.
b) Write the numbers from part a in order of size, starting with the smallest.
👀 Show answer
$A:$ start $0.5$, add $0.5 \Rightarrow$ $0.5, 1.0, 1.5, 2.0, \boxed{2.5}$.
$B:$ start $0.15$, multiply by $2 \Rightarrow$ $0.15, 0.30, 0.60, 1.20, \boxed{2.40}$.
$C:$ start $-1.7$, add $1 \Rightarrow$ $-1.7, -0.7, 0.3, 1.3, \boxed{2.3}$.
$D:$ start $33.6$, divide by $2 \Rightarrow$ $33.6, 16.8, 8.4, 4.2, \boxed{2.1}$.
$E:$ start $1.25$, add $0.25 \Rightarrow$ $1.25, 1.5, 1.75, 2.0, \boxed{2.25}$.
$F:$ start $10.45$, subtract $2 \Rightarrow$ $10.45, 8.45, 6.45, 4.45, \boxed{2.45}$.
Part b — order (smallest first)
$\boxed{2.1\ (D)},\ \boxed{2.25\ (E)},\ \boxed{2.3\ (C)},\ \boxed{2.4\ (B)},\ \boxed{2.45\ (F)},\ \boxed{2.5\ (A)}$.
21. Zara is looking at this inequality: $3.27 \le x < 3.34$
Is Zara correct? Explain your answer.
👀 Show answer
22. $y$ is a number with three decimal places, and $-0.274 < y \le -0.27$. Write all the possible numbers that $y$ could be.
👀 Show answer
⚠️ Be careful!
Don’t compare decimal numbers by looking at the digits after the decimal point without checking the whole-number part first. For example, $8.27$ is less than $8.4$ because $8$ units $27$ hundredths is smaller than $8$ units $4$ tenths, even though $27 > 4$.