Solved Exercises on Word Problems with Decimals Year 5

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

Solution: Exercises 1 to 3
1 Shopping Problem
Exercise 1
Sarah bought a book for £12.45 and a notebook for £3.75. She gave the cashier £20. How much change did she receive?
Definition:

Word problems with decimals: Real-life situations requiring addition, subtraction, or other operations with decimal numbers.

Problem-solving method:
  1. Read the problem carefully and identify key information
  2. Determine what operation is needed (addition, subtraction)
  3. Align decimal points when performing operations
  4. Calculate and verify the answer makes sense
Total Cost
£12.45 + £3.75
Change
£20.00 - Total
Step 1: Add the costs

£12.45 + £3.75 = £16.20

Align decimal points:
   12.45
+ 3.75
------
   16.20

Step 2: Calculate change

£20.00 - £16.20 = £3.80

Align decimal points:
   20.00
- 16.20
------
    3.80

Step 3: Verify the answer

Total cost (£16.20) + Change (£3.80) = £20.00 ✓

Sarah received £3.80 in change
Final answer:

Sarah received £3.80 in change

Applied rules:

• Decimal alignment: Always align decimal points when adding/subtracting

• Money calculations: Express results with appropriate currency symbols

• Verification: Check if the answer makes logical sense

2 Measurement Problem
Exercise 2
A rope was 15.6 meters long. Tom cut off 4.85 meters. How much rope is left?
Definition:

Measurement problems: Situations involving length, weight, volume, etc. expressed in decimal form.

Original Length
15.6 m
Cut Off
4.85 m
Remaining
15.6 - 4.85
Step 1: Set up the subtraction

15.6 - 4.85

Rewrite 15.6 as 15.60 to align decimal places:
   15.60
- 4.85
------

Step 2: Perform the subtraction

Borrowing may be needed:
   15.60
- 4.85
------
   10.75

Step 3: Verify the answer

Remaining rope (10.75m) + Cut piece (4.85m) = Original (15.60m) ✓

10.75 meters of rope is left
Final answer:

10.75 meters of rope is left

Applied rules:

• Decimal alignment: Align decimal points when subtracting

• Regrouping: Borrow when necessary in decimal subtraction

• Units: Include appropriate units in the final answer

3 Money Problem
Exercise 3
James had £25.00. He spent £12.35 on lunch and £6.40 on snacks. How much money does he have left?
Definition:

Multi-step money problems: Problems requiring multiple operations with decimal amounts.

Initial Amount
£25.00
Total Spent
£12.35 + £6.40
Remaining
£25.00 - Total
Step 1: Calculate total spent

£12.35 + £6.40 = £18.75

Align decimal points:
   12.35
+ 6.40
------
   18.75

Step 2: Subtract from initial amount

£25.00 - £18.75 = £6.25

Align decimal points:
   25.00
- 18.75
------
    6.25

Step 3: Verify the calculation

Remaining (£6.25) + Total spent (£18.75) = Initial (£25.00) ✓

James has £6.25 left
Final answer:

James has £6.25 left

Applied rules:

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

• Decimal operations: Carefully align decimal points

• Verification: Always check your work

Solution: Exercises 4 to 6
4 Weight Problem
Exercise 4
A bag of flour weighs 2.5 kg. After using some flour, it weighs 1.75 kg. How much flour was used?
Definition:

Weight subtraction problems: Finding the difference between initial and final weights.

Initial Weight
2.5 kg
Final Weight
1.75 kg
Used
2.5 - 1.75
Step 1: Set up the subtraction

2.5 - 1.75

Rewrite 2.5 as 2.50 to align decimal places:
   2.50
- 1.75
------

Step 2: Perform the subtraction

Borrowing may be needed:
   2.50
- 1.75
------
   0.75

Step 3: Verify the result

Used flour (0.75kg) + Remaining flour (1.75kg) = Initial (2.50kg) ✓

0.75 kg of flour was used
Final answer:

0.75 kg of flour was used

Applied rules:

• Decimal alignment: Align decimal points for accurate subtraction

• Unit consistency: Keep the same units throughout the problem

• Verification: Confirm the answer makes logical sense

5 Distance Problem
Exercise 5
During a race, Emma ran 3.2 km in the morning and 2.85 km in the afternoon. What was her total distance?
Definition:

Distance addition problems: Combining distances traveled in different parts of a journey.

Morning Distance
3.2 km
Afternoon Distance
2.85 km
Total Distance
3.2 + 2.85
Step 1: Align decimal points

3.2 + 2.85

Rewrite 3.2 as 3.20 to match decimal places:
   3.20
+ 2.85
------

Step 2: Add the decimals

Add each column starting from the right:
   3.20
+ 2.85
------
   6.05

Step 3: Verify the calculation

Separate calculation: 3 + 2 = 5, 0.20 + 0.85 = 1.05, Total = 5 + 1.05 = 6.05 ✓

Emma's total distance was 6.05 km
Final answer:

Emma's total distance was 6.05 km

Applied rules:

• Decimal alignment: Match decimal places before adding

• Carrying: Handle carrying over in decimal addition

• Units: Include appropriate unit in the final answer

6 Gift Purchase Problem
Exercise 6
Lisa wants to buy a gift that costs £18.95. She has £15.30. How much more money does she need?
Definition:

Difference finding problems: Calculating how much more is needed or how much is left.

Cost of Gift
£18.95
Amount Available
£15.30
Amount Needed
£18.95 - £15.30
Step 1: Set up the subtraction

£18.95 - £15.30

Align decimal points:
   18.95
- 15.30
------

Step 2: Perform the subtraction

Subtract each column starting from the right:
   18.95
- 15.30
------
    3.65

Step 3: Verify the answer

Amount available (£15.30) + Amount needed (£3.65) = Cost (£18.95) ✓

Lisa needs £3.65 more
Final answer:

Lisa needs £3.65 more

Applied rules:

• Subtraction setup: Larger number minus smaller number

• Decimal alignment: Align decimal points for accuracy

• Verification: Check that the sum of parts equals the whole

Solution: Exercises 7 to 9
7 Temperature Problem
Exercise 7
The temperature was 23.6°C in the morning. It dropped by 4.85°C in the evening. What was the evening temperature?
Definition:

Temperature change problems: Calculating final temperatures after increases or decreases.

Morning Temp
23.6°C
Drop
4.85°C
Evening Temp
23.6 - 4.85
Step 1: Set up the subtraction

23.6 - 4.85

Rewrite 23.6 as 23.60 to align decimal places:
   23.60
- 4.85
------

Step 2: Perform the subtraction

Borrowing may be needed:
   23.60
- 4.85
------
   18.75

Step 3: Verify the result

Evening temp (18.75°C) + Drop (4.85°C) = Morning temp (23.60°C) ✓

Evening temperature was 18.75°C
Final answer:

Evening temperature was 18.75°C

Applied rules:

• Decimal alignment: Align decimal points when subtracting

• Temperature context: Understand the meaning of positive/negative changes

• Verification: Confirm the relationship between values

8 Paint Problem
Exercise 8
A paint tin contained 5.2 liters of paint. After painting, 1.75 liters remained. How much paint was used?
Definition:

Volume subtraction problems: Calculating how much liquid was consumed or removed.

Initial Volume
5.2 L
Remaining Volume
1.75 L
Used Volume
5.2 - 1.75
Step 1: Align decimal points

5.2 - 1.75

Rewrite 5.2 as 5.20 to match decimal places:
   5.20
- 1.75
------

Step 2: Perform the subtraction

Borrowing may be needed:
   5.20
- 1.75
------
   3.45

Step 3: Verify the calculation

Used paint (3.45L) + Remaining (1.75L) = Initial (5.20L) ✓

3.45 liters of paint was used
Final answer:

3.45 liters of paint was used

Applied rules:

• Decimal alignment: Match decimal places before subtracting

• Volume context: Understand the relationship between initial, remaining, and used amounts

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

9 Running Problem
Exercise 9
David ran 4.5 km on Monday and 3.75 km on Tuesday. On Wednesday, he ran 2.8 km less than his combined Monday and Tuesday distance. How far did he run on Wednesday?
Definition:

Multi-step distance problems: Problems requiring multiple operations to find the final answer.

Mon + Tue
4.5 + 3.75
Wednesday
(Mon + Tue) - 2.8
Step 1: Calculate Monday + Tuesday

4.5 + 3.75

Rewrite 4.5 as 4.50 to align decimals:
   4.50
+ 3.75
------
   8.25

Step 2: Calculate Wednesday's distance

8.25 - 2.8

Rewrite 2.8 as 2.80 to align decimals:
   8.25
- 2.80
------
   5.45

Step 3: Verify the result

Wednesday (5.45km) + Difference (2.8km) = Mon+Tue (8.25km) ✓

David ran 5.45 km on Wednesday
Final answer:

David ran 5.45 km on Wednesday

Applied rules:

• Multi-step approach: Break complex problems into smaller parts

• Sequential operations: Complete one calculation before starting the next

• Verification: Check each step of the multi-step process

Solution: Exercise 10
10 Fruit Shopping Problem
Exercise 10
A fruit vendor had 12.5 kg of apples. He sold 3.75 kg in the morning and 4.6 kg in the afternoon. At the end of the day, he found 0.85 kg were spoiled. How many kilograms of good apples remained?
Definition:

Multi-operation word problems: Complex problems involving multiple additions and subtractions.

Initial
12.5 kg
Sold Total
3.75 + 4.6
Good Apples
12.5 - Sold - Spoiled
Step 1: Calculate total sold

3.75 + 4.6

Rewrite 4.6 as 4.60 to align decimals:
   3.75
+ 4.60
------
   8.35

Step 2: Calculate remaining after sales

12.5 - 8.35

Rewrite 12.5 as 12.50 to align decimals:
  12.50
- 8.35
------
   4.15

Step 3: Calculate good apples

Remaining after sales - spoiled
4.15 - 0.85 = 3.30

Step 4: Verify the calculation

Sold (8.35kg) + Spoiled (0.85kg) + Good (3.30kg) = Initial (12.50kg) ✓

3.3 kg of good apples remained
Final answer:

3.3 kg of good apples remained

Applied rules:

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

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

• Verification: Confirm the sum of all parts equals the original amount

Rules and methods, laws,...
\( \text{Change} = \text{Given} - \text{Total Cost} \)
Money Calculation
Addition
\(a + b = c\)
Combine quantities
Subtraction
\(a - b = c\)
Find difference
Multi-step
\(a - b - c = d\)
Sequential operations
Key definitions:

Word problems: Real-life scenarios requiring mathematical operations

Decimals: Numbers with fractional parts separated by a decimal point

Problem-solving: Systematic approach to analyze and solve mathematical challenges

Complete methodology:
  1. Read carefully: Identify what the problem is asking
  2. Find key information: Extract numbers and operations needed
  3. Choose the operation: Addition, subtraction, or multiple operations
  4. Align decimals: Line up decimal points for accurate calculations
  5. Calculate: Perform the mathematical operations
  6. Verify: Check if the answer makes sense in the context
Tip 1: Always align decimal points when adding or subtracting.
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 check if your answer makes sense in the context of the problem.
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 adding/subtracting

• Money calculations: Express results with appropriate currency symbols

• 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

Questions & Answers

Question: I often get confused about whether to add or subtract in word problems. How can I tell which operation to use?

Answer: Great question! Here are some clues to help determine the operation:

  • Use addition when: combining amounts, finding totals, calculating "altogether," "in all," or "sum"
  • Use subtraction when: finding differences, calculating change, determining "left over," "remaining," "more than," or "less than"
  • Look for key phrases: "spent," "used," "sold," "cut off" usually indicate subtraction
  • Think about the context: If the answer should be larger than individual amounts, consider addition; if smaller, consider subtraction

For example, if someone "has £20 and spends £5," you subtract because the amount decreases. If they "buy items costing £5 and £3," you add to find the total cost.

Question: When I add or subtract decimals, I sometimes get the decimal point in the wrong place. How can I make sure I align them correctly?

Answer: Decimal alignment is crucial for accurate calculations! Here's how to do it properly:

  • Line up the decimal points vertically - not the digits themselves
  • Rewrite whole numbers as decimals (e.g., 5 becomes 5.00) to make alignment clearer
  • Fill empty spaces with zeros if needed to maintain place value
  • Place the decimal point in your answer directly below the aligned decimal points

Example: Adding 12.5 and 3.75:
   12.50 (rewritten as hundredths)
+ 3.75
------
   16.25

Notice how the decimal points are aligned vertically in the calculation.

Question: Some word problems seem to have too much information. How do I know what to use and what to ignore?

Answer: This is a common challenge! Here's how to identify relevant information:

  • Read the question carefully - what exactly is being asked?
  • Circle or underline the question part of the problem
  • Identify the numbers that relate directly to the question
  • Ignore extra details that don't impact the calculation
  • Check your work - does your answer address the specific question?

For example, in "Sarah bought a book for £12.45, a pen for £2.50, and had £20. How much change did she receive?" The pen price is irrelevant if the question only asks about the book. But in this case, both prices are needed since we want the total spent.

Always ask yourself: "Does this number help me answer the specific question?"