Addition of Decimals
When you buy a notebook for Rs 25.50 and a pen for Rs 12.75, how much do you pay? You add the two amounts: 25.50 + 12.75 = Rs 38.25. This is addition of decimals.
Adding decimals is very similar to adding whole numbers. The only extra step is to line up the decimal points. Once the decimal points are aligned, you add column by column just like regular addition.
In this chapter, we will learn how to add two or more decimal numbers, including cases where the numbers have a different number of decimal places.
What is Addition of Decimals - Grade 6 Maths (Decimals)?
Definition: Addition of decimals means finding the total of two or more decimal numbers.
Key rule: When adding decimals, line up the decimal points one below the other. This ensures that tenths are added to tenths, hundredths to hundredths, and so on.
Important points:
- If the numbers have different decimal places, add trailing zeros to make them the same length. For example, 3.5 + 2.75 → write 3.5 as 3.50, then add.
- The decimal point in the answer goes directly below the decimal points in the numbers.
- Carrying over works the same way as in whole number addition.
Addition of Decimals Formula
Steps to add decimals:
Line up decimal points → Add zeros if needed → Add column by column → Place decimal point
- Write the numbers one below the other, with decimal points aligned.
- Add trailing zeros if the numbers have different decimal places.
- Add from right to left, column by column, just like whole numbers.
- Carry over when a column sum is 10 or more.
- Place the decimal point in the answer, directly below the other decimal points.
Derivation and Proof
Let us add 34.56 + 8.7 step by step:
- Align decimal points and add zeros:
34.56
+ 8.70 (added a zero to make both have 2 decimal places) - Add hundredths column: 6 + 0 = 6.
- Add tenths column: 5 + 7 = 12. Write 2, carry 1.
- Add ones column: 4 + 8 + 1 (carry) = 13. Write 3, carry 1.
- Add tens column: 3 + 0 + 1 (carry) = 4.
- Place the decimal point.
Answer: 34.56 + 8.7 = 43.26.
Another example: 0.6 + 0.45
- Align: 0.60 + 0.45.
- Hundredths: 0 + 5 = 5.
- Tenths: 6 + 4 = 10. Write 0, carry 1.
- Ones: 0 + 0 + 1 = 1.
Answer: 0.6 + 0.45 = 1.05.
Types and Properties
Types of decimal addition problems:
- Type 1: Same decimal places — Both numbers have the same number of digits after the decimal. Example: 3.25 + 4.13.
- Type 2: Different decimal places — Add trailing zeros to match. Example: 5.6 + 3.42 → 5.60 + 3.42.
- Type 3: Whole number + decimal — Treat the whole number as having a decimal point. Example: 7 + 2.35 → 7.00 + 2.35.
- Type 4: Adding more than two decimals — Align all decimal points and add column by column. Example: 1.5 + 2.75 + 0.8.
- Type 5: Word problems — Real-life problems involving money, length, or weight. Example: total bill, total distance.
Solved Examples
Example 1: Example 1: Adding Decimals with Same Places
Problem: Add 12.34 + 5.62.
Solution:
- Align: 12.34 + 5.62.
- Hundredths: 4 + 2 = 6.
- Tenths: 3 + 6 = 9.
- Ones: 2 + 5 = 7.
- Tens: 1.
Answer: 12.34 + 5.62 = 17.96.
Example 2: Example 2: Different Decimal Places
Problem: Add 7.8 + 3.25.
Solution:
- Write 7.8 as 7.80 (add a trailing zero).
- Hundredths: 0 + 5 = 5.
- Tenths: 8 + 2 = 10. Write 0, carry 1.
- Ones: 7 + 3 + 1 = 11. Write 1, carry 1.
- Tens: 1.
Answer: 7.8 + 3.25 = 11.05.
Example 3: Example 3: Whole Number + Decimal
Problem: Add 9 + 4.75.
Solution:
- Write 9 as 9.00.
- Hundredths: 0 + 5 = 5.
- Tenths: 0 + 7 = 7.
- Ones: 9 + 4 = 13. Write 3, carry 1.
- Tens: 1.
Answer: 9 + 4.75 = 13.75.
Example 4: Example 4: Adding with Carrying
Problem: Add 6.78 + 5.46.
Solution:
- Hundredths: 8 + 6 = 14. Write 4, carry 1.
- Tenths: 7 + 4 + 1 = 12. Write 2, carry 1.
- Ones: 6 + 5 + 1 = 12. Write 2, carry 1.
- Tens: 1.
Answer: 6.78 + 5.46 = 12.24.
Example 5: Example 5: Adding Three Decimals
Problem: Add 2.5 + 3.75 + 1.8.
Solution:
- Write as: 2.50 + 3.75 + 1.80.
- Hundredths: 0 + 5 + 0 = 5.
- Tenths: 5 + 7 + 8 = 20. Write 0, carry 2.
- Ones: 2 + 3 + 1 + 2 = 8.
Answer: 2.5 + 3.75 + 1.8 = 8.05.
Example 6: Example 6: Money Problem
Problem: Riya bought a book for Rs 145.50 and a pen for Rs 32.75. How much did she spend in total?
Solution:
Given:
- Book = Rs 145.50
- Pen = Rs 32.75
Total = 145.50 + 32.75
- Hundredths: 0 + 5 = 5.
- Tenths: 5 + 7 = 12. Write 2, carry 1.
- Ones: 5 + 2 + 1 = 8.
- Tens: 4 + 3 = 7.
- Hundreds: 1.
Answer: Riya spent Rs 178.25.
Example 7: Example 7: Length Problem
Problem: A ribbon is cut into two pieces of length 2.45 m and 3.8 m. What was the total length of the ribbon?
Solution:
- Total = 2.45 + 3.80
- Hundredths: 5 + 0 = 5.
- Tenths: 4 + 8 = 12. Write 2, carry 1.
- Ones: 2 + 3 + 1 = 6.
Answer: Total length = 6.25 m.
Example 8: Example 8: Adding Small Decimals
Problem: Add 0.05 + 0.9 + 0.125.
Solution:
- Write as: 0.050 + 0.900 + 0.125.
- Thousandths: 0 + 0 + 5 = 5.
- Hundredths: 5 + 0 + 2 = 7.
- Tenths: 0 + 9 + 1 = 10. Write 0, carry 1.
- Ones: 0 + 0 + 0 + 1 = 1.
Answer: 0.05 + 0.9 + 0.125 = 1.075.
Example 9: Example 9: Weight Problem
Problem: A bag of rice weighs 5.5 kg and a bag of flour weighs 2.75 kg. What is the total weight?
Solution:
- 5.50 + 2.75
- Hundredths: 0 + 5 = 5.
- Tenths: 5 + 7 = 12. Write 2, carry 1.
- Ones: 5 + 2 + 1 = 8.
Answer: Total weight = 8.25 kg.
Example 10: Example 10: Adding Decimals to Whole Number Result
Problem: Add 3.6 + 4.4.
Solution:
- Tenths: 6 + 4 = 10. Write 0, carry 1.
- Ones: 3 + 4 + 1 = 8.
Answer: 3.6 + 4.4 = 8.0 (which is simply 8).
Real-World Applications
Where do we add decimals?
- Money — Adding prices of items to find the total bill. Rs 45.50 + Rs 32.25 = Rs 77.75.
- Measurements — Adding lengths of ribbons, ropes, or cloth pieces. 2.5 m + 3.75 m = 6.25 m.
- Weight — Adding weights of ingredients in a recipe. 0.5 kg + 0.25 kg = 0.75 kg.
- Distance — Adding distances covered in parts of a journey. 12.6 km + 8.4 km = 21.0 km.
- Temperature — Finding temperature changes. If temperature rises by 2.5°C and then by 1.3°C, total rise = 3.8°C.
- Sports — Adding scores in gymnastics, diving, or figure skating where scores are decimals.
Key Points to Remember
- To add decimals, line up the decimal points vertically.
- If the numbers have different decimal places, add trailing zeros to match.
- Add column by column from right to left, just like whole numbers.
- Carry over when a column total is 10 or more.
- The decimal point in the answer is placed directly below the other decimal points.
- A whole number can be written with a decimal: 7 = 7.00.
- Adding trailing zeros does NOT change the value: 3.5 = 3.50 = 3.500.
- The sum of two decimals can be a whole number: 2.5 + 7.5 = 10.0 = 10.
- Always double-check by estimating: 3.8 + 5.1 is about 4 + 5 = 9 (actual: 8.9).
- Decimal addition is used in money, measurement, weight, and distance calculations.
Practice Problems
- Add 4.56 + 3.21.
- Add 8.3 + 5.67.
- Add 15 + 3.75.
- Add 0.08 + 0.6 + 0.125.
- A bottle holds 1.5 litres and another holds 0.75 litres. Find the total.
- Add 23.45 + 6.8 + 12.375.
- Aman walked 3.25 km in the morning and 2.8 km in the evening. How far did he walk in total?
- Find the sum: 99.9 + 0.1. What do you notice?
Frequently Asked Questions
Q1. How do you add decimal numbers?
Line up the decimal points vertically. Add trailing zeros if needed to make the decimal places equal. Then add column by column from right to left, carrying over when needed. Place the decimal point directly below the others.
Q2. Why do we line up the decimal points?
To make sure we add matching place values: tenths with tenths, hundredths with hundredths, and so on. Without alignment, we would mix up the place values and get a wrong answer.
Q3. Can I add zeros at the end of a decimal?
Yes. Adding trailing zeros does not change the value. 3.5 = 3.50 = 3.500. This is useful when adding numbers with different decimal places.
Q4. Is adding decimals the same as adding whole numbers?
Almost. The only extra step is aligning the decimal points. After that, you add column by column with carrying, just like whole numbers. The decimal point stays in its position.
Q5. What if I forget to align the decimal points?
You will get a wrong answer. For example, 3.5 + 2.25: without alignment you might add 35 + 225 = 260, which is completely wrong. The correct answer is 5.75.
Q6. Can the sum of two decimals be a whole number?
Yes. For example, 3.5 + 6.5 = 10.0 = 10. When the decimal parts add up to a whole number, the result has no fractional part.
Q7. How do I add a whole number and a decimal?
Write the whole number with a decimal point and zeros. For example, 5 + 3.25: write 5 as 5.00, then add 5.00 + 3.25 = 8.25.
Q8. How can I check my answer?
Estimate first. Round each number to the nearest whole number and add. For example, 4.8 + 3.3 is about 5 + 3 = 8. Your exact answer should be close to this (8.1). If it is very different, recheck your work.
