Solved Exercises on Adding and Subtracting Decimals Year 5

Master decimal operations through comprehensive exercises: addition, subtraction, mixed operations, and detailed answers with step-by-step explanations.

Solution: Exercises 1 to 3
1 Simple Addition
Exercise 1
Calculate: 12.45 + 3.7
Definition:

Adding decimals: The process of combining two or more decimal numbers to find their sum, requiring alignment of decimal points.

Addition method:
  1. Align decimal points vertically
  2. Add zeros to match decimal places if needed
  3. Add column by column from right to left
  4. Place decimal point in the answer directly below the aligned points
Expression
12.45 + 3.7
Aligned
12.45 + 3.70
Result
16.15
Step 1: Align decimal points

12.45 + 3.7

Add zero to match decimal places:
   12.45
+ 3.70
------

Step 2: Add column by column

Hundredths: 5 + 0 = 5
Tenths: 4 + 7 = 11 (write 1, carry 1)
Units: 2 + 3 + 1 (carry) = 6
Tens: 1 + 0 = 1

Step 3: Place decimal point

Decimal point goes directly below aligned points:
   12.45
+ 3.70
------
   16.15

12.45 + 3.7 = 16.15
Final answer:

12.45 + 3.7 = 16.15

Applied rules:

• Decimal alignment: Always align decimal points before adding

• Carrying: When sum exceeds 9, carry the excess to the next column

• Zero padding: Add zeros to make decimal places equal

2 Simple Subtraction
Exercise 2
Calculate: 15.8 - 4.35
Definition:

Subtracting decimals: The process of taking one decimal number away from another to find the difference, requiring alignment of decimal points.

Expression
15.8 - 4.35
Aligned
15.80 - 4.35
Result
11.45
Step 1: Align decimal points

15.8 - 4.35

Add zero to match decimal places:
   15.80
- 4.35
------

Step 2: Subtract column by column

Hundredths: 0 - 5 (borrow from tenths)
Tenths: 7 - 3 = 4 (after borrowing)
Units: 5 - 4 = 1
Tens: 1 - 0 = 1

Step 3: Place decimal point

Decimal point goes directly below aligned points:
   15.80
- 4.35
------
   11.45

15.8 - 4.35 = 11.45
Final answer:

15.8 - 4.35 = 11.45

Applied rules:

• Decimal alignment: Align decimal points before subtracting

• Borrowing: When top digit is smaller, borrow from the next column

• Zero padding: Add zeros to make decimal places equal

3 Mixed Operations
Exercise 3
Calculate: 23.65 + 4.8 - 12.37
Definition:

Mixed decimal operations: Problems involving both addition and subtraction of decimals, performed from left to right following order of operations.

Expression
23.65 + 4.8 - 12.37
Step 1
28.45 - 12.37
Result
16.08
Step 1: Perform addition first

23.65 + 4.8

Add zero to match decimal places:
   23.65
+ 4.80
------
   28.45

Step 2: Perform subtraction

28.45 - 12.37

Align decimal points:
   28.45
- 12.37
------
   16.08

Step 3: Verify the calculation

Work backwards: 16.08 + 12.37 - 4.8 = 23.65 ✓

23.65 + 4.8 - 12.37 = 16.08
Final answer:

23.65 + 4.8 - 12.37 = 16.08

Applied rules:

• Order of operations: Perform operations from left to right

• Decimal alignment: Align decimal points for each operation

• Verification: Check by reversing the operations

Solution: Exercises 4 to 6
4 Multiple Addition
Exercise 4
Calculate: 5.2 + 3.75 + 1.48
Definition:

Multiple decimal addition: Adding three or more decimal numbers together, still requiring alignment of decimal points.

Expression
5.2 + 3.75 + 1.48
Aligned
5.20 + 3.75 + 1.48
Result
10.43
Step 1: Align decimal points

5.2 + 3.75 + 1.48

Add zeros to match decimal places:
   5.20
   3.75
+ 1.48
------

Step 2: Add column by column

Hundredths: 0 + 5 + 8 = 13 (write 3, carry 1)
Tenths: 2 + 7 + 4 + 1 (carry) = 14 (write 4, carry 1)
Units: 5 + 3 + 1 + 1 (carry) = 10 (write 0, carry 1)
Tens: 0 + 0 + 0 + 1 (carry) = 1

Step 3: Place decimal point

Decimal point goes directly below aligned points:
   5.20
   3.75
+ 1.48
------
10.43

5.2 + 3.75 + 1.48 = 10.43
Final answer:

5.2 + 3.75 + 1.48 = 10.43

Applied rules:

• Decimal alignment: Align decimal points for all numbers

• Carrying: Handle carries carefully when summing multiple numbers

• Zero padding: Add zeros to make all numbers have the same decimal places

5 Multiple Subtraction
Exercise 5
Calculate: 20.5 - 8.37 - 4.25
Definition:

Multiple decimal subtraction: Taking away two or more decimal numbers sequentially from an initial number.

Expression
20.5 - 8.37 - 4.25
Step 1
12.13 - 4.25
Result
7.88
Step 1: Perform first subtraction

20.5 - 8.37

Add zero to match decimal places:
   20.50
- 8.37
------
   12.13

Step 2: Perform second subtraction

12.13 - 4.25

Align decimal points:
   12.13
- 4.25
------
    7.88

Step 3: Verify the calculation

Work backwards: 7.88 + 4.25 + 8.37 = 20.50 ✓

20.5 - 8.37 - 4.25 = 7.88
Final answer:

20.5 - 8.37 - 4.25 = 7.88

Applied rules:

• Sequential operations: Perform subtractions from left to right

• Decimal alignment: Align decimal points for each operation

• Verification: Add the subtrahends to the result to get the original number

6 Mixed Operations
Exercise 6
Calculate: 18.75 - 5.3 + 2.68
Definition:

Mixed decimal operations: Problems involving addition and subtraction in the same expression, performed from left to right.

Expression
18.75 - 5.3 + 2.68
Step 1
13.45 + 2.68
Result
16.13
Step 1: Perform subtraction first

18.75 - 5.3

Add zero to match decimal places:
   18.75
- 5.30
------
   13.45

Step 2: Perform addition

13.45 + 2.68

Align decimal points:
   13.45
+ 2.68
------
   16.13

Step 3: Verify the calculation

Work backwards: 16.13 - 2.68 + 5.3 = 18.75 ✓

18.75 - 5.3 + 2.68 = 16.13
Final answer:

18.75 - 5.3 + 2.68 = 16.13

Applied rules:

• Order of operations: Perform operations from left to right

• Decimal alignment: Align decimal points for each operation

• Verification: Check by reversing the operations

Solution: Exercises 7 to 9
7 Large Number Addition
Exercise 7
Calculate: 45.678 + 12.34
Definition:

Adding decimals with different places: Adding decimal numbers with different numbers of decimal places, requiring careful alignment.

Expression
45.678 + 12.34
Aligned
45.678 + 12.340
Result
58.018
Step 1: Align decimal points

45.678 + 12.34

Add zeros to match decimal places:
   45.678
+ 12.340
-------

Step 2: Add column by column

Thousandths: 8 + 0 = 8
Hundredths: 7 + 4 = 11 (write 1, carry 1)
Tenths: 6 + 3 + 1 (carry) = 10 (write 0, carry 1)
Units: 5 + 2 + 1 (carry) = 8
Tens: 4 + 1 = 5

Step 3: Place decimal point

Decimal point goes directly below aligned points:
   45.678
+ 12.340
-------
   58.018

45.678 + 12.34 = 58.018
Final answer:

45.678 + 12.34 = 58.018

Applied rules:

• Decimal alignment: Align decimal points before adding

• Zero padding: Add zeros to make decimal places equal

• Carrying: Handle carries carefully when summing

8 Large Number Subtraction
Exercise 8
Calculate: 63.45 - 28.785
Definition:

Subtracting decimals with different places: Subtracting decimal numbers with different numbers of decimal places, requiring careful alignment.

Expression
63.45 - 28.785
Aligned
63.450 - 28.785
Result
34.665
Step 1: Align decimal points

63.45 - 28.785

Add zero to match decimal places:
   63.450
- 28.785
-------

Step 2: Subtract column by column

Thousandths: 0 - 5 (borrow from hundredths)
Hundredths: 4 - 8 (borrow from tenths)
Tenths: 3 - 7 (borrow from units)
Units: 2 - 8 (borrow from tens)
Tens: 5 - 2 = 3

Step 3: Complete the calculation

After borrowing:
   63.450
- 28.785
-------
   34.665

63.45 - 28.785 = 34.665
Final answer:

63.45 - 28.785 = 34.665

Applied rules:

• Decimal alignment: Align decimal points before subtracting

• Zero padding: Add zeros to make decimal places equal

• Borrowing: Borrow across multiple columns when needed

9 Complex Mixed Operations
Exercise 9
Calculate: 50.25 - 18.4 + 7.65 - 3.28
Definition:

Complex mixed operations: Problems with multiple additions and subtractions, performed from left to right following order of operations.

Expression
50.25 - 18.4 + 7.65 - 3.28
Step 1
31.85 + 7.65 - 3.28
Step 2
39.50 - 3.28
Result
36.22
Step 1: Perform first operation

50.25 - 18.4

Add zero to match decimal places:
   50.25
- 18.40
------
   31.85

Step 2: Perform second operation

31.85 + 7.65

Align decimal points:
   31.85
+ 7.65
------
   39.50

Step 3: Perform third operation

39.50 - 3.28

Align decimal points:
   39.50
- 3.28
------
   36.22

Step 4: Verify the calculation

Work backwards: 36.22 + 3.28 - 7.65 + 18.4 = 50.25 ✓

50.25 - 18.4 + 7.65 - 3.28 = 36.22
Final answer:

50.25 - 18.4 + 7.65 - 3.28 = 36.22

Applied rules:

• Order of operations: Perform operations from left to right

• Decimal alignment: Align decimal points for each operation

• Verification: Check by reversing the operations

Solution: Exercise 10
10 Multi-step Problem
Exercise 10
A store had $75.50 in the register. During the day, they made $42.75 in sales and spent $18.30 on supplies. What was the final amount in the register?
Definition:

Multi-step decimal problems: Real-world scenarios requiring multiple operations with decimals to solve practical problems.

Initial Amount
$75.50
Operations
+ $42.75 - $18.30
Result
$99.95
Step 1: Identify the operations

Start with $75.50, add $42.75 (sales), subtract $18.30 (expenses)

Expression: 75.50 + 42.75 - 18.30

Step 2: Perform addition first

75.50 + 42.75

Align decimal points:
   75.50
+ 42.75
------
118.25

Step 3: Perform subtraction

118.25 - 18.30

Align decimal points:
 118.25
- 18.30
------
  99.95

Step 4: Verify the calculation

Work backwards: $99.95 + $18.30 - $42.75 = $75.50 ✓

The final amount in the register was $99.95
Final answer:

The final amount in the register was $99.95

Applied rules:

• Multi-step approach: Handle one operation at a time

• Decimal alignment: Align decimal points for each operation

• Real-world context: Interpret results in the context of the problem

Rules and methods, laws,...
\( a + b = c \)
Addition Formula
\( a - b = c \)
Subtraction Formula
Addition
\(a + b = c\)
Combine quantities
Subtraction
\(a - b = c\)
Find difference
Mixed Ops
\(a + b - c = d\)
Sequential operations
Key definitions:

Decimal: A number that contains a decimal point representing fractional parts

Decimal point: The dot that separates the whole number part from the fractional part

Place value: The value of a digit based on its position in the number

Complete methodology:
  1. Align decimals: Line up decimal points vertically
  2. Equalize places: Add zeros to make all numbers have the same decimal places
  3. Perform operation: Add or subtract column by column from right to left
  4. Handle carries/borrows: Carry over when adding, borrow when subtracting
  5. Place decimal: Put decimal point in answer directly below aligned points
  6. Verify: Check by reversing the operation
Tip 1: Always align decimal points when adding or subtracting decimals.
Tip 2: Add zeros to make decimal places equal (e.g., 5.2 becomes 5.20).
Tip 3: For mixed operations, perform from left to right.
Tip 4: Always check your answer by performing the inverse operation.
Common errors: Misaligning decimal points, forgetting to carry/borrow, not adding zeros for place value.
Exam preparation: Practice decimal alignment, master carrying and borrowing, work on mixed operations.
Rules to remember:

• Decimal alignment: Always line up decimal points when adding/subtracting

• Carrying: When adding, if sum ≥ 10, carry the excess to the next column

• Borrowing: When subtracting, if top digit is smaller, borrow from next column

• Zero padding: Add zeros to make decimal places equal

• Order of operations: For mixed operations, go from left to right

Comprehensive Summary: Adding and Subtracting Decimals

Key Concepts & Definitions

Decimal: A number that contains a decimal point representing fractional parts of a whole number.

Decimal Point: The dot that separates the whole number part from the fractional part (e.g., in 3.14, the decimal point is between 3 and 14).

Place Value: The value of a digit based on its position in the number (units, tenths, hundredths, thousandths, etc.).

Adding Decimals: The process of combining two or more decimal numbers to find their sum.

Subtracting Decimals: The process of taking one decimal number away from another to find the difference.

Carrying: The process of moving an excess amount from one column to the next when adding (e.g., when 7 + 6 = 13, write 3 and carry 1).

Borrowing: The process of taking value from a higher place value column when subtracting (e.g., when subtracting 7 from 2, borrow 1 from the next column).

Core Rules & Principles
Decimal Alignment Rule: Always align decimal points vertically when adding or subtracting decimals. This ensures that digits in the same place value are operated on together.
Zero Padding Principle: Add zeros to the right of the last decimal digit to make all numbers have the same number of decimal places, simplifying the operation.
Carrying Rule: When adding, if the sum in any column is 10 or greater, write the units digit and carry the tens digit to the next column.
Borrowing Rule: When subtracting, if the top digit is smaller than the bottom digit, borrow 1 from the next higher place value column.
Order of Operations: For mixed operations (addition and subtraction), perform operations from left to right.
Step-by-Step Method
  1. Align Decimals: Write the numbers vertically with decimal points lined up
  2. Equalize Places: Add zeros to the right of shorter decimals to make all numbers have the same decimal places
  3. Perform Operation: Add or subtract column by column starting from the rightmost digit
  4. Handle Carries/Borrows: Carry over when adding (if sum ≥ 10) or borrow when subtracting (if top digit is smaller)
  5. Place Decimal Point: Put the decimal point in the answer directly below the aligned decimal points
  6. Verify Answer: Check by performing the inverse operation
Examples (Simple to Advanced)

Simple Addition: 2.5 + 1.3 = 3.8

Moderate Addition: 12.45 + 3.7 = 12.45 + 3.70 = 16.15

Advanced Addition: 45.678 + 12.34 = 45.678 + 12.340 = 58.018

Simple Subtraction: 8.6 - 3.2 = 5.4

Moderate Subtraction: 15.8 - 4.35 = 15.80 - 4.35 = 11.45

Advanced Subtraction: 63.45 - 28.785 = 63.450 - 28.785 = 34.665

Mixed Operations: 20.5 - 8.37 + 4.25 = 12.13 + 4.25 = 16.38

Common Pitfalls & Tips
Common Mistake: Misaligning decimal points, leading to incorrect place values.
Tip: Rewrite all numbers with the same number of decimal places by adding zeros.
Common Mistake: Forgetting to carry when adding or borrow when subtracting.
Tip: Always check your work by performing the inverse operation.
Common Mistake: Placing the decimal point in the wrong position in the answer.
Memory Aid: "Decimal points must line up straight!" Remember to align before operating.
Memorization Notes
Phrase to Remember: "Before you operate, align the dots!"
Visual Memory: Picture decimal points as "poles" that must be perfectly aligned vertically.
Process Memory: Always add zeros to match decimal places (e.g., 5.2 becomes 5.20).
Verification Memory: "Add back to check" - if A - B = C, then C + B should equal A.
Operation Memory: For mixed operations, always go from left to right.

Questions & Answers

Question: Why is it so important to line up the decimal points when adding or subtracting decimals? Can't I just add the numbers like regular numbers?

Answer: Decimal alignment is crucial because it ensures that you're adding or subtracting digits with the same place value. Without proper alignment:

  • You might add tenths to hundredths, which doesn't make sense mathematically
  • You could get completely wrong answers
  • It becomes difficult to keep track of place values

For example, if you try to add 12.45 + 3.7 without aligning:
12.45
+ 3.7
----- (misaligned)
You might incorrectly add 5 (hundredths) to 7 (tenths), which is wrong!
Proper alignment: 12.45 + 3.70 = 16.15

Question: When I'm subtracting decimals and I need to borrow, especially from zeros, I get confused. How do I handle this?

Answer: Borrowing from zeros requires going to the next non-zero digit. Here's the method:

  • When you encounter zero and need to borrow, look to the left for the next non-zero digit
  • Reduce that non-zero digit by 1 and turn the zeros in between into 9s
  • The immediate zero becomes 10, allowing you to borrow from it

Example: 100.00 - 25.67
To subtract 7 from 0 in hundredths place:
Borrow from the tens place (0), which requires borrowing from the hundreds place (1)
The 100.00 becomes 99.9(10) after borrowing
Now you can subtract: 10-7=3 in hundredths place

The full result: 100.00 - 25.67 = 74.33

Question: I often forget to add zeros when the decimals have different numbers of places. How can I remember to do this?

Answer: Adding zeros (zero padding) is essential for maintaining place value. Here's how to remember:

  • Count the decimal places in each number and make them all equal
  • Think of it as "filling in the gaps" to make the calculation easier
  • Always do this BEFORE starting the addition or subtraction
  • Create a mental checklist: "Align, Equalize, Calculate"

Example: 15.3 + 4.25 + 7.123
Equalize decimal places: 15.300 + 4.250 + 7.123
Now all numbers have 3 decimal places, making alignment and calculation straightforward

This ensures you're always adding tenths to tenths, hundredths to hundredths, etc.