Comparing Decimals
Which is greater: 0.5 or 0.35? Many students think 0.35 is greater because 35 is bigger than 5. But that is wrong! 0.5 = 0.50, which is greater than 0.35. To avoid such mistakes, you need to learn the correct way to compare decimals.
Comparing decimals means figuring out which decimal number is greater, which is smaller, or whether two decimals are equal. The trick is to compare them place by place, starting from the left.
This skill is used every day — when comparing prices (Rs 12.50 vs Rs 12.75), measurements (2.3 m vs 2.28 m), or scores (9.8 vs 9.75).
In this chapter, you will learn the step-by-step method to compare decimals, arrange them in ascending and descending order, and avoid common mistakes.
What is Comparing Decimals - Grade 6 Maths (Decimals)?
Definition: Comparing decimals means determining which of two or more decimal numbers is greater, smaller, or if they are equal.
Key idea: To compare decimals, we compare digits at each place value from left to right — first the whole number part, then the tenths, then the hundredths, and so on.
Important rules:
- You can add trailing zeros to a decimal without changing its value: 0.5 = 0.50 = 0.500.
- Always align the decimals by their decimal point before comparing.
- Start comparing from the leftmost place value.
- The first place where the digits differ tells you which number is greater.
Symbols used:
- > means "greater than" (5.7 > 5.3)
- < means "less than" (3.14 < 3.41)
- = means "equal to" (0.50 = 0.5)
Comparing Decimals Formula
Steps to compare two decimals:
Step 1: Compare whole parts → Step 2: Equalise decimal places → Step 3: Compare place by place
- Compare the whole number parts. The number with the larger whole part is greater. Example: 5.3 > 4.9 (because 5 > 4).
- If the whole parts are equal, make the decimal places equal by adding trailing zeros. Example: Compare 3.5 and 3.42 → write as 3.50 and 3.42.
- Compare digit by digit from left to right (tenths, hundredths, etc.). The first place where digits differ decides the result. Example: 3.50 vs 3.42 → tenths: 5 > 4 → so 3.50 > 3.42.
Ascending and Descending Order:
- Ascending order: From smallest to largest (e.g., 1.2, 1.5, 1.8, 2.1).
- Descending order: From largest to smallest (e.g., 2.1, 1.8, 1.5, 1.2).
Derivation and Proof
Why does the place-by-place method work?
Each place in a decimal has a value that is 10 times the place to its right:
- Ones place = 1
- Tenths place = 0.1 (one-tenth of 1)
- Hundredths place = 0.01 (one-tenth of 0.1)
- Thousandths place = 0.001 (one-tenth of 0.01)
So a difference in the tenths place is worth more than a difference in the hundredths place.
Example: Compare 0.7 and 0.68.
- 0.7 = 0.70 (adding a trailing zero)
- Now compare 0.70 and 0.68
- Tenths: 7 vs 6 → 7 > 6
- So 0.70 > 0.68, which means 0.7 > 0.68
Even though 68 looks bigger than 7, in decimal place value, 0.7 (seven-tenths) is greater than 0.68 (sixty-eight hundredths = six-tenths and eight-hundredths).
Common mistake to avoid:
- Do NOT compare the digits after the decimal point as if they are whole numbers.
- 0.5 is NOT less than 0.35 (0.5 = 0.50, and 50 > 35).
- Always equalise the number of decimal places first.
Types and Properties
Types of decimal comparison problems:
Type 1: Compare two decimals
- Use >, <, or = to compare two decimal numbers.
- Example: 4.56 ___ 4.6 → 4.56 < 4.60
Type 2: Arrange in ascending order
- Given a set of decimals, arrange from smallest to largest.
- Example: 3.5, 3.05, 3.55, 3.15 → 3.05, 3.15, 3.5, 3.55
Type 3: Arrange in descending order
- Given a set of decimals, arrange from largest to smallest.
Type 4: Find the greatest/smallest
- From a given set, identify the greatest or smallest decimal.
Type 5: Decimals between two numbers
- Find a decimal between 3.4 and 3.5 → Answer: 3.45
Type 6: Real-life comparison
- Compare prices, heights, weights, or times given as decimals.
Solved Examples
Example 1: Example 1: Comparing Two Decimals
Problem: Which is greater: 6.3 or 6.25?
Solution:
- Whole parts: 6 = 6 (equal, move to decimals).
- Equalise decimal places: 6.30 and 6.25.
- Tenths: 3 vs 2 → 3 > 2.
Answer: 6.3 > 6.25
Example 2: Example 2: Common Mistake
Problem: Which is greater: 0.8 or 0.45?
Solution:
- Whole parts: 0 = 0 (equal).
- Equalise: 0.80 and 0.45.
- Tenths: 8 vs 4 → 8 > 4.
Answer: 0.8 > 0.45 (even though 45 looks bigger than 8, the place value of 8 in the tenths position makes 0.80 greater).
Example 3: Example 3: Equal Decimals
Problem: Are 2.40 and 2.4 equal?
Solution:
- 2.40 and 2.4 → adding trailing zero to 2.4 gives 2.40.
- 2.40 = 2.40.
Answer: Yes, 2.40 = 2.4. Trailing zeros do not change the value.
Example 4: Example 4: Ascending Order
Problem: Arrange in ascending order: 5.6, 5.06, 5.66, 5.16
Solution:
- Equalise: 5.60, 5.06, 5.66, 5.16
- All have whole part 5. Compare tenths: 0, 1, 6, 6.
- Smallest tenths = 0 → 5.06 is smallest.
- Next: 1 → 5.16.
- Two have tenths = 6: compare hundredths: 0 vs 6 → 5.60 < 5.66.
Answer: 5.06, 5.16, 5.6, 5.66
Example 5: Example 5: Descending Order
Problem: Arrange in descending order: 3.2, 3.02, 3.22, 3.12
Solution:
- Equalise: 3.20, 3.02, 3.22, 3.12
- Compare tenths: 2, 0, 2, 1.
- Largest tenths = 2: compare hundredths of 3.20 and 3.22 → 2 > 0 → 3.22 > 3.20.
- Next: tenths = 1 → 3.12.
- Smallest: tenths = 0 → 3.02.
Answer: 3.22, 3.2, 3.12, 3.02
Example 6: Example 6: Comparing Prices
Problem: A pencil costs Rs 8.50 and a pen costs Rs 8.75. Which costs more?
Solution:
- Whole parts: 8 = 8 (equal).
- Tenths: 5 vs 7 → 7 > 5.
Answer: The pen costs more (Rs 8.75 > Rs 8.50).
Example 7: Example 7: Finding a Decimal Between Two Numbers
Problem: Find a decimal between 4.3 and 4.4.
Solution:
- 4.3 = 4.30 and 4.4 = 4.40
- Any number between 4.30 and 4.40 works.
- Examples: 4.31, 4.35, 4.39
Answer: One such decimal is 4.35.
Example 8: Example 8: Three Decimal Places
Problem: Which is greater: 1.205 or 1.21?
Solution:
- Equalise: 1.205 and 1.210
- Whole parts: 1 = 1. Tenths: 2 = 2. Hundredths: 0 vs 1 → 1 > 0.
Answer: 1.21 > 1.205
Example 9: Example 9: Comparing Heights
Problem: Three friends have heights: Aman — 1.52 m, Bina — 1.5 m, Charu — 1.55 m. Arrange from shortest to tallest.
Solution:
- Equalise: 1.52, 1.50, 1.55
- Compare: 1.50 < 1.52 < 1.55
Answer: Shortest to tallest: Bina (1.5 m), Aman (1.52 m), Charu (1.55 m).
Example 10: Example 10: Largest and Smallest
Problem: Find the largest and smallest: 7.1, 7.01, 7.101, 7.11, 7.011
Solution:
- Equalise to 3 decimal places: 7.100, 7.010, 7.101, 7.110, 7.011
- Compare: 7.010 < 7.011 < 7.100 < 7.101 < 7.110
Answer: Smallest = 7.01. Largest = 7.11.
Real-World Applications
Comparing decimals in real life:
- Shopping: Comparing prices of two products (Rs 45.50 vs Rs 45.25) to find the cheaper one.
- Sports: In athletics, race times are compared in decimals (9.58 seconds vs 9.63 seconds). The smaller time wins.
- Measurements: Comparing heights (1.72 m vs 1.68 m), weights (52.5 kg vs 52.3 kg), or distances.
- Temperature: Comparing temperatures (36.8 degrees C vs 37.2 degrees C) to check for fever.
- Marks and grades: CGPA or percentage scores are often in decimals (8.75 vs 8.5).
- Money: Bank balances, interest rates, and exchange rates all use decimal comparison.
Key Points to Remember
- To compare decimals, first compare the whole number parts.
- If whole parts are equal, compare tenths, then hundredths, then thousandths.
- Add trailing zeros to make the number of decimal places equal before comparing.
- Trailing zeros do NOT change the value: 3.5 = 3.50 = 3.500.
- Do NOT compare digits after the decimal as whole numbers (0.5 is NOT less than 0.35).
- The first place (from left) where the digits differ decides which number is greater.
- Ascending order = smallest to largest. Descending order = largest to smallest.
- There are infinitely many decimals between any two decimal numbers.
- Always align numbers at the decimal point when comparing.
- Use the symbols >, <, and = correctly when writing comparisons.
Practice Problems
- Compare using >, <, or =: (a) 4.7 ___ 4.07 (b) 0.50 ___ 0.5 (c) 3.14 ___ 3.41
- Arrange in ascending order: 2.5, 2.05, 2.55, 2.15, 2.51
- Arrange in descending order: 0.9, 0.09, 0.99, 0.19, 0.91
- Which is the smallest: 6.1, 6.01, 6.001, 6.11?
- Find two decimals between 5.3 and 5.4.
- A ribbon is 2.75 m long. Another is 2.7 m long. Which is longer? By how much?
- Compare: 0.125 ___ 0.13
- Arrange the following marks in descending order: 87.5, 87.05, 87.55, 87.15
Frequently Asked Questions
Q1. How do I compare two decimal numbers?
First compare the whole number parts. If they are equal, add trailing zeros to make the decimal places equal, then compare digit by digit from left to right (tenths, hundredths, thousandths). The first place where digits differ decides the answer.
Q2. Is 0.5 greater than 0.35?
Yes. 0.5 = 0.50, and 0.50 > 0.35 because the tenths digit 5 is greater than 3. Do not compare 5 and 35 as whole numbers — compare place by place.
Q3. Does adding zeros after the last decimal digit change the number?
No. Trailing zeros after the decimal point do not change the value. 3.5 = 3.50 = 3.500. Adding zeros helps when comparing decimals by making them the same length.
Q4. What does ascending order mean?
Ascending order means arranging numbers from smallest to largest. For example: 1.2, 1.5, 1.8, 2.1 is in ascending order.
Q5. What does descending order mean?
Descending order means arranging numbers from largest to smallest. For example: 2.1, 1.8, 1.5, 1.2 is in descending order.
Q6. Can there be a decimal between 3.4 and 3.5?
Yes, there are infinitely many decimals between any two decimal numbers. Between 3.4 and 3.5, you have 3.41, 3.42, ..., 3.49, and even 3.401, 3.455, etc.
Q7. Is 7.0 equal to 7?
Yes. 7.0 and 7 represent the same value. The .0 just shows that the measurement is precise to the tenths place, but the numerical value is the same.
Q8. How do I compare decimals with different number of decimal places?
Add trailing zeros to make them the same length. For example, to compare 2.3 and 2.15, write 2.30 and 2.15. Now compare: tenths 3 > 1, so 2.30 > 2.15.
Q9. What is the most common mistake when comparing decimals?
The most common mistake is treating the digits after the decimal point as a whole number. Students think 0.35 > 0.5 because 35 > 5. The correct approach is to compare place by place: tenths first, then hundredths.
Q10. Is comparing decimals the same as comparing fractions?
The idea is similar, but the method is different. With decimals, you compare place by place. With fractions, you need to find a common denominator first. You can also convert fractions to decimals and then compare.
