Solved Exercises on Word Problems with Subtracting Decimals Year 5

Master decimal subtraction through word problems: money calculations, measurements, weight problems, and practical applications through these 10 detailed exercises.

Solution: Exercises 1 to 3
1 Shopping Change Problem
Exercise 1
Emma bought a toy for £15.75 and gave the cashier £20.00. How much change did she receive?
Definition:

Subtracting decimals in word problems: Real-life situations requiring subtraction of decimal numbers to find differences, remaining amounts, or change.

Subtraction method:
  1. Identify the larger number (amount given) and smaller number (cost)
  2. Align decimal points vertically
  3. Perform subtraction with borrowing if needed
  4. Verify the answer makes sense in the context
Amount Given
£20.00
Cost
£15.75
Change
£20.00 - £15.75
Step 1: Set up the subtraction

£20.00 - £15.75

Align decimal points:
   20.00
- 15.75
------

Step 2: Perform the subtraction

Start from the rightmost digit, borrowing as needed:
   20.00
- 15.75
------
    4.25

Step 3: Verify the answer

Cost (£15.75) + Change (£4.25) = Amount given (£20.00) ✓

Emma received £4.25 in change
Final answer:

Emma received £4.25 in change

Applied rules:

• Decimal alignment: Always align decimal points when subtracting

• Borrowing: When subtracting, borrow from higher place values as needed

• Verification: Check that the sum of parts equals the original amount

2 Measurement Problem
Exercise 2
A rope was 12.5 meters long. After cutting off 3.75 meters, how much rope remains?
Definition:

Measurement subtraction: Finding the remaining length after a portion has been removed from an original measurement.

Original Length
12.5 m
Cut Off
3.75 m
Remaining
12.5 - 3.75
Step 1: Align decimal points

12.5 - 3.75

Rewrite 12.5 as 12.50 to match decimal places:
   12.50
- 3.75
------

Step 2: Perform the subtraction

Starting from the right, borrow when needed:
   12.50
- 3.75
------
    8.75

Step 3: Verify the result

Remaining (8.75m) + Cut off (3.75m) = Original (12.50m) ✓

8.75 meters of rope remains
Final answer:

8.75 meters of rope remains

Applied rules:

• Decimal alignment: Align decimal points before subtracting

• Equivalent decimals: Rewrite 12.5 as 12.50 to match decimal places

• Verification: Confirm that the sum of remaining and cut portions equals original

3 Weight Problem
Exercise 3
A bag of rice weighed 5.2 kg initially. After cooking, it weighed 2.85 kg. How much rice was used?
Definition:

Weight subtraction problems: Calculating the difference between initial and final weights to find the amount consumed or removed.

Initial Weight
5.2 kg
Final Weight
2.85 kg
Used Weight
5.2 - 2.85
Step 1: Align decimal points

5.2 - 2.85

Rewrite 5.2 as 5.20 to match decimal places:
   5.20
- 2.85
------

Step 2: Perform the subtraction

Borrowing is needed in multiple places:
   5.20
- 2.85
------
   2.35

Step 3: Verify the calculation

Used (2.35kg) + Remaining (2.85kg) = Initial (5.20kg) ✓

2.35 kg of rice was used
Final answer:

2.35 kg of rice was used

Applied rules:

• Decimal alignment: Match decimal places before subtracting

• Borrowing across zeros: When borrowing from zero, continue to the next non-zero digit

• Context verification: Ensure the answer makes sense in the real-world scenario

Solution: Exercises 4 to 6
4 Temperature Problem
Exercise 4
The temperature was 25.6°C in the morning. It dropped to 18.45°C in the evening. How much did the temperature decrease?
Definition:

Temperature change problems: Calculating the difference between two temperatures to find the increase or decrease.

Morning Temp
25.6°C
Evening Temp
18.45°C
Decrease
25.6 - 18.45
Step 1: Align decimal points

25.6 - 18.45

Rewrite 25.6 as 25.60 to match decimal places:
   25.60
- 18.45
------

Step 2: Perform the subtraction

Subtract starting from the right:
   25.60
- 18.45
------
    7.15

Step 3: Verify the result

Evening temp (18.45°C) + Decrease (7.15°C) = Morning temp (25.60°C) ✓

Temperature decreased by 7.15°C
Final answer:

Temperature decreased by 7.15°C

Applied rules:

• Decimal alignment: Align decimal points for accurate subtraction

• Temperature context: Understand that decrease means subtracting the lower from the higher value

• Verification: Check that the calculated change returns to the original value

5 Volume Problem
Exercise 5
A container had 8.5 liters of water. After pouring out some water, 3.75 liters remained. How much water was poured out?
Definition:

Volume subtraction problems: Determining how much liquid was removed by subtracting the remaining amount from the initial amount.

Initial Volume
8.5 L
Remaining Volume
3.75 L
Poured Out
8.5 - 3.75
Step 1: Align decimal points

8.5 - 3.75

Rewrite 8.5 as 8.50 to match decimal places:
   8.50
- 3.75
------

Step 2: Perform the subtraction

Borrowing may be needed:
   8.50
- 3.75
------
   4.75

Step 3: Verify the calculation

Poured out (4.75L) + Remaining (3.75L) = Initial (8.50L) ✓

4.75 liters of water was poured out
Final answer:

4.75 liters of water was poured out

Applied rules:

• Decimal alignment: Match decimal places before subtracting

• Volume conservation: Initial volume = Removed + Remaining

• Verification: Always check that parts sum to the whole

6 Distance Problem
Exercise 6
Sam ran 6.8 km on Monday. On Tuesday, he ran 1.95 km less than Monday. How far did he run on Tuesday?
Definition:

Distance reduction problems: Calculating a reduced distance by subtracting a specified amount from an original distance.

Monday Distance
6.8 km
Reduction
1.95 km
Tuesday Distance
6.8 - 1.95
Step 1: Align decimal points

6.8 - 1.95

Rewrite 6.8 as 6.80 to match decimal places:
   6.80
- 1.95
------

Step 2: Perform the subtraction

Borrowing is needed:
   6.80
- 1.95
------
   4.85

Step 3: Verify the result

Tuesday distance (4.85km) + Reduction (1.95km) = Monday distance (6.80km) ✓

Sam ran 4.85 km on Tuesday
Final answer:

Sam ran 4.85 km on Tuesday

Applied rules:

• Decimal alignment: Align decimal points for accurate subtraction

• Distance relationships: Reduced distance = Original - Reduction

• Verification: Confirm the relationship between the original and reduced distances

Solution: Exercises 7 to 9
7 Money Savings Problem
Exercise 7
Lisa had £32.50 saved. She spent £14.75 on a birthday gift. How much money does she have left?
Definition:

Money subtraction problems: Calculating remaining money after purchases by subtracting expenses from initial amounts.

Initial Amount
£32.50
Amount Spent
£14.75
Remaining
£32.50 - £14.75
Step 1: Align decimal points

£32.50 - £14.75

Both numbers already have the same number of decimal places:
   32.50
- 14.75
------

Step 2: Perform the subtraction

Borrowing may be needed in multiple places:
   32.50
- 14.75
------
   17.75

Step 3: Verify the calculation

Spent (£14.75) + Remaining (£17.75) = Initial (£32.50) ✓

Lisa has £17.75 left
Final answer:

Lisa has £17.75 left

Applied rules:

• Decimal alignment: Ensure decimal points are aligned before subtracting

• Money context: Express results with appropriate currency symbols

• Verification: Check that the sum of spent and remaining equals the initial amount

8 Fruit Weight Problem
Exercise 8
A basket contained 4.6 kg of apples. After eating some, 1.85 kg remained. How many kilograms of apples were eaten?
Definition:

Consumption subtraction problems: Finding the amount consumed by subtracting the remaining quantity from the initial quantity.

Initial Weight
4.6 kg
Remaining Weight
1.85 kg
Eaten Weight
4.6 - 1.85
Step 1: Align decimal points

4.6 - 1.85

Rewrite 4.6 as 4.60 to match decimal places:
   4.60
- 1.85
------

Step 2: Perform the subtraction

Borrowing is needed:
   4.60
- 1.85
------
   2.75

Step 3: Verify the result

Eaten (2.75kg) + Remaining (1.85kg) = Initial (4.60kg) ✓

2.75 kg of apples were eaten
Final answer:

2.75 kg of apples were eaten

Applied rules:

• Decimal alignment: Match decimal places before subtracting

• Conservation principle: Initial = Consumed + Remaining

• Verification: Always check that the parts sum to the original amount

9 Fuel Consumption Problem
Exercise 9
A car's fuel tank had 45.2 liters of petrol. After a journey, 18.75 liters remained. How much petrol was used during the journey?
Definition:

Fuel consumption problems: Calculating fuel used by subtracting the remaining fuel from the initial fuel amount.

Initial Fuel
45.2 L
Remaining Fuel
18.75 L
Fuel Used
45.2 - 18.75
Step 1: Align decimal points

45.2 - 18.75

Rewrite 45.2 as 45.20 to match decimal places:
  45.20
- 18.75
------

Step 2: Perform the subtraction

Borrowing may be needed in multiple places:
  45.20
- 18.75
------
  26.45

Step 3: Verify the calculation

Fuel used (26.45L) + Remaining (18.75L) = Initial (45.20L) ✓

26.45 liters of petrol was used
Final answer:

26.45 liters of petrol was used

Applied rules:

• Decimal alignment: Align decimal points before subtracting

• Fuel conservation: Initial fuel = Used fuel + Remaining fuel

• Verification: Confirm that the sum of used and remaining equals initial amount

Solution: Exercise 10
10 Budget Problem
Exercise 10
A school had a budget of £125.50 for supplies. They spent £48.75 on stationery and £32.40 on books. How much of the budget remains?
Definition:

Multi-step budget problems: Complex subtraction problems requiring multiple operations to find the remaining balance.

Initial Budget
£125.50
Total Spent
£48.75 + £32.40
Remaining
£125.50 - Total
Step 1: Calculate total spent

£48.75 + £32.40

Align decimal points:
   48.75
+ 32.40
------
   81.15

Step 2: Subtract from initial budget

£125.50 - £81.15

Align decimal points:
  125.50
- 81.15
------
   44.35

Step 3: Verify the calculation

Remaining (£44.35) + Total spent (£81.15) = Initial budget (£125.50) ✓

£44.35 of the budget remains
Final answer:

£44.35 of the budget remains

Applied rules:

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

• Sequential calculations: Results from one step feed into the next

• Verification: Check that all expenditures and remaining equal initial amount

Rules and methods, laws,...
\( \text{Remaining} = \text{Initial} - \text{Amount Used} \)
Subtraction Formula
Basic Subtraction
\(a - b = c\)
Find difference
Money Problems
\(G - C = Ch\)
Given - Cost = Change
Multi-step
\(a - (b + c) = d\)
Sequential operations
Key definitions:

Decimal subtraction: The process of taking away one decimal number from another

Word problems: Real-life scenarios requiring mathematical operations to solve

Remainder: The amount left after subtraction

Complete methodology:
  1. Read the problem: Identify what is being asked
  2. Find key information: Locate the numbers and what they represent
  3. Determine operation: Recognize that subtraction is needed
  4. Align decimals: Line up decimal points vertically
  5. Perform calculation: Subtract with borrowing if necessary
  6. Verify answer: Check if it makes sense in context
Tip 1: Always align decimal points when subtracting decimals.
Tip 2: Rewrite whole numbers as decimals (e.g., 5 as 5.00) to make alignment easier.
Tip 3: For multi-step problems, solve one operation at a time.
Tip 4: Always verify your answer by adding the result back to the subtracted amount.
Common errors: Misaligning decimal points, forgetting to borrow in subtraction, misinterpreting the problem question.
Exam preparation: Practice various contexts (money, measurement, weight), focus on decimal alignment, master borrowing in subtraction.
Rules to remember:

• Decimal alignment: Always line up decimal points when subtracting

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

• Units: Keep consistent units throughout the problem

• Verification: Check if your answer is reasonable in the context

• Multi-step problems: Solve one operation at a time

Comprehensive Summary: Word Problems with Subtracting Decimals

Key Concepts & Definitions

Decimal Subtraction: The mathematical operation of taking one decimal number away from another to find the difference.

Word Problems: Mathematical problems presented in a real-world context that require translation into mathematical operations.

Decimal Point Alignment: The process of lining up decimal points vertically when performing addition or subtraction to maintain place value.

Borrowing (Regrouping): The process of taking value from a higher place value column when the top digit is smaller than the bottom digit during subtraction.

Remainder: The amount left after a subtraction operation has been completed.

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.
Borrowing Principle: When subtracting, if the top digit is smaller than the bottom digit in any column, borrow 1 from the next higher place value column (which reduces that column by 1 and adds 10 to the current column).
Conservation Principle: In subtraction word problems, the initial amount equals the amount used/subtracted plus the remaining amount: A = B + C.
Context Recognition: Look for key phrases that indicate subtraction: "left," "remaining," "how much more," "difference," "decrease," "after," "used."
Step-by-Step Method
  1. Read Carefully: Read the entire problem to understand what is being asked
  2. Identify Information: Circle or highlight the important numbers and what they represent
  3. Determine Operation: Decide whether subtraction is needed based on key words or the situation described
  4. Set Up Calculation: Write the larger number above the smaller number, aligning decimal points
  5. Equalize Decimal Places: Add zeros to make all numbers have the same number of decimal places if needed
  6. Perform Subtraction: Subtract starting from the rightmost digit, borrowing when necessary
  7. Verify Answer: Check if the answer makes sense in the context and verify by adding back
Examples (Simple to Advanced)

Simple Example: £5.00 - £2.50 = £2.50 (money change)

Moderate Example: 12.75 - 8.4 = 12.75 - 8.40 = 4.35 (measurement)

Advanced Example: Budget of £100.00 - Expenses of £35.75 and £24.60 = £100.00 - £60.35 = £39.65

Complex Example: Temperature drops from 25.6°C to 18.35°C: 25.60 - 18.35 = 7.25°C decrease

Real-world Example: Tank with 45.2L - Used 18.75L = 26.45L remaining

Common Pitfalls & Tips
Common Mistake: Misaligning decimal points, leading to incorrect place values.
Tip: Rewrite whole numbers as decimals (e.g., 5 becomes 5.00) to make alignment easier.
Common Mistake: Forgetting to borrow when the top digit is smaller than the bottom digit.
Tip: Always check your work by adding the result back to the subtracted amount to see if you get the original number.
Memory Aid: "Decimal points must line up straight!" Remember to align before operating.
Memorization Notes
Phrase to Remember: "Before you subtract, 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.
Key Words: Look for "remaining," "left," "change," "difference," "less than" for subtraction.

Questions & Answers

Question: How do I know when to subtract decimals in a word problem? What words should I look for?

Answer: Great question! Look for these key words and phrases that indicate subtraction:

  • Change: "How much change did he receive?"
  • Remaining/Left: "How much money is left?"
  • Difference: "What is the difference in heights?"
  • Less/More: "1.5 kg less than", "how much more"
  • After: "After spending..."
  • Used: "How much was used?"
  • Spent: "How much did she spend?"

Also, subtraction is often used when you need to find what's left after something is taken away, or when comparing two amounts to see how much bigger or smaller one is compared to the other.

Question: When I subtract decimals, I sometimes get confused about where to put the decimal point in my answer. How can I make sure I get it right?

Answer: The decimal point in your answer goes directly below the aligned decimal points in your problem! Here's how to do it correctly:

  • First: Line up the decimal points of the numbers you're subtracting vertically
  • Second: Put the decimal point in your answer directly below the aligned decimal points
  • Third: Fill in zeros if needed to make the decimal places match

Example: 15.75 - 8.3
   15.75
- 8.30 (added zero)
------
    7.45

Notice how the decimal points are perfectly aligned in a vertical line!

Question: I often make mistakes when I need to borrow from zero in decimal subtraction. How do I handle this?

Answer: Borrowing from zero is tricky but follows the same principle! Here's the method:

  • When you encounter zero: Go to the next non-zero digit to the left
  • Borrow from that digit: Reduce it by 1 and bring 10 to the zero
  • Then continue: Take 1 from the 10 you just created to use in your subtraction

Example: 5.00 - 2.75
   5.00
- 2.75
------
   2.25

To subtract 5 from 0 in the hundredths place, you need to borrow. Since the tenths place is also 0, you go to the units place (5), reduce it to 4, bring 10 to the tenths place, then borrow from there.