Solved Exercises on Multi-step Word Problems in Grade 8

Master multi-step word problems: systematic problem-solving, mathematical reasoning, and strategic thinking through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Shopping Problem
Exercise 1
Sarah buys 3 notebooks for $2.50 each, 2 pens for $1.75 each, and a backpack for $15.00. She pays with a $50 bill. How much change does she receive?
Definition:

Multi-step word problem: A problem requiring multiple calculations in sequence to find the solution

Problem-solving method:
  1. Read carefully: Identify all given information and what is being asked
  2. Plan the steps: Break the problem into smaller parts
  3. Calculate each part: Perform operations in order
  4. Combine results: Put together the partial answers
  5. Check the answer: Verify it makes sense in context
Notebooks cost
3 × $2.50
Pens cost
2 × $1.75
Total cost
$7.50 + $3.50 + $15.00
Step 1: Calculate notebook cost

3 notebooks × $2.50 each = $7.50

Step 2: Calculate pen cost

2 pens × $1.75 each = $3.50

Step 3: Calculate total cost

$7.50 (notebooks) + $3.50 (pens) + $15.00 (backpack) = $26.00

Step 4: Calculate change

$50.00 (paid) - $26.00 (total) = $24.00

Sarah receives $24.00 in change
Final answer:

Sarah receives $24.00 in change

Applied rules:

• Multiplication: Calculate individual item costs

• Addition: Sum all costs to get total

• Subtraction: Find change by subtracting total from payment

• Order of operations: Perform calculations in logical sequence

2 Distance-Speed-Time Problem
Exercise 2
A car travels for 2 hours at 60 mph, then for another 1.5 hours at 40 mph. What is the total distance traveled?
Definition:

Distance formula: Distance = Speed × Time

First leg
60 × 2
Second leg
40 × 1.5
Total distance
120 + 60
Step 1: Calculate first leg distance

Distance = Speed × Time = 60 mph × 2 hours = 120 miles

Step 2: Calculate second leg distance

Distance = Speed × Time = 40 mph × 1.5 hours = 60 miles

Step 3: Calculate total distance

Total distance = 120 miles + 60 miles = 180 miles

Total distance = 180 miles
Final answer:

The car traveled a total distance of 180 miles

Applied rules:

• Distance formula: Distance = Speed × Time

• Addition: Combine distances from different segments

• Unit consistency: Ensure units match in calculation

3 Percentage Problem
Exercise 3
A store increases the price of a $40 shirt by 25%, then offers a 20% discount on the new price. What is the final price of the shirt?
Definition:

Percentage increase: New value = Original × (1 + percentage)

Percentage decrease: New value = Original × (1 - percentage)

Original price
$40
After 25% increase
$40 × 1.25
After 20% discount
$50 × 0.80
Step 1: Calculate 25% increase

New price = $40 × (1 + 0.25) = $40 × 1.25 = $50

Step 2: Calculate 20% discount on new price

Discounted price = $50 × (1 - 0.20) = $50 × 0.80 = $40

Step 3: Final result

The final price is $40

Final price = $40
Final answer:

The final price of the shirt is $40

Applied rules:

• Percentage conversion: Convert percentage to decimal

• Sequential operations: Perform operations in the given order

• Compound changes: Apply changes sequentially, not simultaneously

Problem-Solving Strategies and Methods
\(\text{Distance} = \text{Speed} \times \text{Time}\)
Distance Formula
Percentage Increase
\(\text{New Value} = \text{Original} \times (1 + \frac{\%}{100})\)
Adding a percentage to a value
Percentage Decrease
\(\text{New Value} = \text{Original} \times (1 - \frac{\%}{100})\)
Subtracting a percentage from a value
Profit/Loss
\(\text{Profit} = \text{Selling Price} - \text{Cost Price}\)
Calculating profit or loss
Key definitions:

Multi-step problem: Requires multiple calculations performed in sequence

Systematic approach: Organized method to solve complex problems

Verification: Checking if the answer makes sense in context

Complete methodology:
  1. Understand: Read the problem carefully, identify given information and unknowns
  2. Plan: Decide the order of operations needed
  3. Solve: Execute calculations step by step
  4. Check: Verify the answer is reasonable and correct
Tip 1: Always write down what you know and what you need to find.
Tip 2: Draw diagrams or tables to visualize the problem.
Tip 3: Keep track of units throughout calculations.
Tip 4: Check if your final answer has the correct unit.
Common errors: Misreading the question, incorrect order of operations, unit mismatches.
Exam preparation: Practice different problem types, master basic formulas, work systematically.
Solution: Exercises 4 to 5
4 Ratio and Proportion Problem
Exercise 4
In a school, the ratio of boys to girls is 3:4. If there are 210 boys, how many students are there in total?
Definition:

Ratio: Comparison of two quantities by division

Proportion: Two ratios that are equal

Given ratio
Boys:Girls = 3:4
Actual boys
210
Scale factor
210 ÷ 3
Step 1: Find the scale factor

If the ratio is 3:4 and actual boys = 210, then scale factor = 210 ÷ 3 = 70

Step 2: Calculate number of girls

Girls = 4 × scale factor = 4 × 70 = 280

Step 3: Calculate total students

Total students = Boys + Girls = 210 + 280 = 490

Step 4: Verify the ratio

Boys:Girls = 210:280 = 3:4 ✓

Total students = 490
Final answer:

There are 490 students in total

Applied rules:

• Ratio interpretation: Understand what the ratio represents

• Scale factor: Find multiplier to convert ratio to actual numbers

• Proportional reasoning: Maintain ratio relationships

5 Algebraic Word Problem
Exercise 5
The sum of three consecutive even integers is 78. What are the three integers?
Definition:

Consecutive even integers: Even numbers that follow each other (e.g., 2, 4, 6)

Algebraic representation: Let the middle integer be x

Let middle integer
x
Three integers
(x-2), x, (x+2)
Equation
(x-2) + x + (x+2) = 78
Step 1: Define variables

Let the three consecutive even integers be (x-2), x, and (x+2)

Step 2: Set up equation

(x-2) + x + (x+2) = 78

Step 3: Solve for x

x - 2 + x + x + 2 = 78

3x = 78

x = 26

Step 4: Find the three integers

First integer: x - 2 = 26 - 2 = 24

Second integer: x = 26

Third integer: x + 2 = 26 + 2 = 28

Step 5: Verify the solution

24 + 26 + 28 = 78 ✓

The three integers are 24, 26, and 28
Final answer:

The three consecutive even integers are 24, 26, and 28

Applied rules:

• Variable representation: Express unknowns algebraically

• Equation formation: Translate words to mathematical expressions

• Algebraic solving: Solve the equation systematically

• Verification: Check solution in original problem

Problem-Solving Laws, Methods, and Strategies
\(\text{Distance} = \text{Speed} \times \text{Time}\)
Basic Formula
Key definitions:

Multi-step word problem: A problem that requires multiple calculations in sequence to reach the solution

Problem-solving strategy: Systematic approach to analyze and solve complex problems

Mathematical modeling: Translating real-world situations into mathematical expressions

Complete methodology:
  1. Comprehension: Read the problem thoroughly, identify what is given and what is asked
  2. Organization: List known information and unknowns, draw diagrams if helpful
  3. Planning: Determine the sequence of operations needed
  4. Execution: Carry out calculations step by step
  5. Verification: Check if the answer is reasonable and correct
Tip 1: Always read the problem twice before starting to solve.
Tip 2: Underline or highlight key numbers and operations.
Tip 3: Draw a picture or make a table for complex problems.
Tip 4: Estimate the answer first to check reasonableness later.
Tip 5: Write down each step clearly to avoid calculation errors.
Common errors: Skipping steps, misreading the question, calculation mistakes, unit errors.
Key formulas: Distance = Speed × Time, Percentage = Part/Whole × 100, Profit = Selling Price - Cost.
Problem types: Shopping problems, rate problems, mixture problems, age problems, geometric applications.
Essential problem-solving laws:

• Identify the question: What exactly are you trying to find?

• Extract information: What facts are given in the problem?

• Choose operations: Which mathematical operations are needed?

• Check units: Ensure units are consistent throughout calculations

• Verify reasonableness: Does the answer make sense in context?

Questions & Answers

Question: I often get confused about the order of operations in multi-step problems. How do I know which calculation to do first?

Answer: Great question! The order of operations follows the acronym PEMDAS:

  • Parentheses: Do operations inside parentheses first
  • Exponents: Calculate powers and roots next
  • Multiplication and Division: From left to right
  • Addition and Subtraction: From left to right

However, in word problems, you also need to consider the logical sequence based on the situation:

  • Calculate individual costs before adding them together
  • Find per-unit rates before calculating totals
  • Apply discounts after calculating base prices

Always read the problem carefully to understand the logical sequence of events described.

Question: How do I know when to use algebra versus just arithmetic in word problems?

Answer: Here's how to decide:

Use arithmetic when:

  • All values are known and you need to find a specific result
  • The problem involves straightforward calculations (shopping, distance, percentages)
  • You're following a direct sequence of operations

Use algebra when:

  • You need to find unknown values
  • The problem describes relationships between quantities
  • There are multiple conditions that must be satisfied

As a rule of thumb, if the problem asks "what number" or "find the value," consider using algebra. If it's "calculate the total" or "how much," arithmetic might suffice.

Question: I sometimes get the wrong answer even though my calculations seem correct. What am I missing?

Answer: This commonly happens due to several reasons:

  • Unit mismatches: Mixing different units (miles vs. kilometers, hours vs. minutes)
  • Reading errors: Missing key details like "decreased by" vs. "increased by"
  • Sign errors: Confusing positive and negative values
  • Calculation order: Not following the correct sequence of operations

To prevent this:

  1. Read the problem twice before solving
  2. Write down all given information clearly
  3. Check units throughout your calculations
  4. Estimate the answer first to see if your result is reasonable
  5. Verify your answer by plugging it back into the original problem

Always take time to review your work, especially for multi-step problems!

Question: How can I organize my work better when solving multi-step problems?

Answer: Organization is crucial for multi-step problems! Here's a systematic approach:

  1. Label everything: Write "Given:", "Find:", and "Steps:"
  2. Number your steps: Step 1, Step 2, etc.
  3. Write complete sentences: Don't just write numbers
  4. Box your final answer: Make it stand out
  5. Check your work: Verify each calculation

Example format:

Given: 3 notebooks at $2.50 each, 2 pens at $1.75 each, backpack $15.00
Paid: $50.00
Find: Change received

Step 1: Cost of notebooks = 3 × $2.50 = $7.50
Step 2: Cost of pens = 2 × $1.75 = $3.50
Step 3: Total cost = $7.50 + $3.50 + $15.00 = $26.00
Step 4: Change = $50.00 - $26.00 = $24.00

Final Answer: $24.00

This organization helps you track your progress and catch errors!

Question: What should I do if I'm completely stuck on a multi-step word problem?

Answer: Being stuck is normal! Try these strategies:

  1. Take a break: Step away for a few minutes, then reread
  2. Draw a diagram: Visualize the problem situation
  3. Make up numbers: Use easier numbers to understand the process
  4. Work backwards: Start from what you want to find
  5. Look for patterns: See if it resembles a problem you've solved

Specific techniques for getting unstuck:

  • Break the problem into smaller parts
  • Identify what you do know (even if it seems small)
  • Try to express the problem in your own words
  • Consider what formulas might apply
  • Think about similar problems you've seen before

Remember, struggling with problems builds problem-solving skills!