Solved Exercises on Problem-Solving Strategies in Grade 8

Master problem-solving strategies: systematic approaches, working backwards, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Guess and Check Strategy
Exercise 1
Two consecutive integers have a product of 132. Find the integers.
Definition:

Guess and Check: A problem-solving strategy where you make educated guesses, check if they work, and refine based on results

Guess and check process:
  1. Understand the problem requirements
  2. Make an educated first guess
  3. Check if your guess satisfies all conditions
  4. Refine your guess based on results
  5. Repeat until solution is found
Problem
n(n+1) = 132
First Guess
11×12 = 132
Solution
11 and 12
Step 1: Understand the problem

We need two consecutive integers whose product is 132

Let n = first integer, then (n+1) = second integer

So n(n+1) = 132

Step 2: Estimate reasonable values

√132 ≈ 11.5, so try values around 11 and 12

Step 3: Try first guess

Try n = 11: 11 × 12 = 132 ✓

This works!

Step 4: Verify the solution

Check: 11 and 12 are consecutive integers

Product: 11 × 12 = 132 ✓

n = 11, n+1 = 12
Final answer:

The two consecutive integers are 11 and 12.

Applied rules:

• Estimation: Use √product to estimate values

• Verification: Always check your answer

• Strategy: Guess and check works well for integer problems

2 Working Backwards
Exercise 2
Sarah spent half her money on a book, then spent $5 on candy, then spent half of what was left on a gift. She ended with $10. How much did she start with?
Definition:

Working Backwards: A strategy that starts with the final result and reverses operations to find the initial value

Final Amount
$10
Before Gift
$20
Before Candy
$25
Start Amount
$50
Step 1: Start with final amount

Sarah ended with $10

Step 2: Reverse the last operation

She spent half of what was left on a gift to end with $10

Before gift: $10 × 2 = $20

Step 3: Reverse the candy purchase

She spent $5 on candy to have $20 left

Before candy: $20 + $5 = $25

Step 4: Reverse the book purchase

She spent half her money on a book to have $25 left

If $25 is half of what she had before buying book, then before book: $25 × 2 = $50

Step 5: Verify the solution

Start with $50

Book: $50 ÷ 2 = $25 spent, $25 left

Candy: $25 - $5 = $20 left

Gift: $20 ÷ 2 = $10 spent, $10 left ✓

Started with $50
Final answer:

Sarah started with $50.

Applied rules:

• Reverse operations: Undo each step in reverse order

• Division becomes multiplication: If x/2 = y, then x = 2y

• Addition becomes subtraction: If x + a = y, then x = y - a

3 Draw a Diagram
Exercise 3
A rectangular garden is 3 feet longer than it is wide. If the perimeter is 30 feet, find the dimensions.
Definition:

Draw a Diagram: A strategy that creates a visual representation of the problem to better understand relationships

Width
w
Length
w + 3
Perimeter
2w + 2(w+3) = 30
Step 1: Draw the rectangle and label dimensions

Width = w, Length = w + 3

Step 2: Write the perimeter equation

Perimeter = 2(length) + 2(width)

30 = 2(w + 3) + 2w

Step 3: Solve the equation

30 = 2w + 6 + 2w

30 = 4w + 6

24 = 4w

w = 6 feet

Step 4: Find the length

Length = w + 3 = 6 + 3 = 9 feet

Step 5: Verify the solution

Perimeter = 2(6) + 2(9) = 12 + 18 = 30 feet ✓

Width = 6 ft, Length = 9 ft
Final answer:

The garden is 6 feet wide and 9 feet long.

Applied rules:

• Diagram helps: Visualize relationships between dimensions

• Perimeter formula: P = 2l + 2w

• Algebraic translation: Convert visual into equation

Problem-Solving Strategies Fundamentals
P = 2l + 2w
Perimeter Formula
Guess and Check
Iterative approach
Make educated guesses
Work Backwards
Reverse operations
Start with result
Draw Diagram
Visual representation
Picture the problem
Key definitions:

Problem-Solving Strategy: A systematic approach to tackle mathematical problems

Heuristic: A problem-solving technique that may not guarantee a solution but often leads to one

Consecutive Integers: Integers that follow each other in order (n, n+1, n+2, ...)

Perimeter: The distance around the outside of a shape

Verification: Checking that your solution is correct

Systematic Approach: Following a consistent, organized method

Logical Reasoning: Using facts and rules to reach conclusions

Algebraic Translation: Converting word problems into mathematical equations

General Problem-Solving Process:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Choose an appropriate strategy based on the problem type
  3. Solve: Carry out the chosen strategy step-by-step
  4. Check: Verify that your answer makes sense and is correct
Tip 1: Read the problem multiple times to fully understand it.
Tip 2: Identify key information and what you need to find.
Tip 3: Try multiple strategies if one doesn't work.
Tip 4: Always verify your solution in the context of the problem.
Common errors: Misreading the problem, skipping verification steps, using wrong strategy for problem type.
Exam preparation: Practice multiple strategies, work on word problems, develop systematic approach.
Solution: Exercises 4 to 5
4 Make a Table
Exercise 4
A bacteria culture doubles every hour. If there are initially 50 bacteria, how many will there be after 6 hours?
Definition:

Make a Table: A strategy that organizes information in rows and columns to identify patterns and relationships

Initial
50
Rate
Doubles each hour
After 6h
50×2^6 = 3200
Step 1: Create a table showing growth pattern
HourBacteria Count
050
1100
2200
3400
4800
51600
63200
Step 2: Identify the pattern

Each hour, the count is multiplied by 2

This is exponential growth: P(t) = P₀ × 2^t

Step 3: Apply the formula

P(6) = 50 × 2^6 = 50 × 64 = 3,200

Step 4: Verify with table

Checking the table: after 6 hours, count is 3,200 ✓

After 6 hours: 3,200 bacteria
Final answer:

After 6 hours, there will be 3,200 bacteria.

Applied rules:

• Pattern recognition: Tables help identify growth patterns

• Exponential growth: P(t) = P₀ × b^t

• Organization: Tables organize complex information

5 Multiple Strategies Combined
Exercise 5
Three friends have ages that are consecutive even integers. Their combined age is 54. Find their ages using multiple strategies.
Definition:

Strategy Combination: Using multiple problem-solving approaches to verify and solve complex problems

Variables
n, n+2, n+4
Equation
n + (n+2) + (n+4) = 54
Solution
16, 18, 20
Step 1: Define the variables

Let n = age of youngest friend

Then ages are: n, n+2, n+4 (consecutive even integers)

Step 2: Write the equation

n + (n+2) + (n+4) = 54

3n + 6 = 54

Step 3: Solve algebraically

3n = 48

n = 16

So ages are 16, 18, and 20

Step 4: Verify with guess and check

Try: 14, 16, 18 → 14+16+18 = 48 (too low)

Try: 16, 18, 20 → 16+18+20 = 54 ✓

Step 5: Check with average approach

Average age = 54 ÷ 3 = 18

Three consecutive even integers around 18: 16, 18, 20 ✓

Ages: 16, 18, 20
Final answer:

The three friends are 16, 18, and 20 years old.

Applied rules:

• Verification: Multiple approaches confirm correctness

• Consecutive even: n, n+2, n+4

• Flexibility: Different strategies can lead to same solution

Problem-Solving Strategies Analysis Summary
n + (n+1) + (n+2) = \text{sum}
Consecutive Integer Formula
Key definitions:

Problem-Solving Strategy: A systematic method for approaching and solving mathematical problems

Heuristic: A technique that helps solve problems but doesn't guarantee a solution

Systematic Approach: Following a consistent, organized method to solve problems

Verification: Checking that your solution is correct and makes sense

Consecutive Integers: Integers that follow each other in order (n, n+1, n+2, ...)

Consecutive Even/Odd: Even or odd integers that follow each other (n, n+2, n+4, ...)

Algebraic Translation: Converting word problems into mathematical equations

Logical Reasoning: Using facts and rules to reach valid conclusions

Complete Problem-Solving Process:
  1. Understand: Carefully read the problem, identify what is asked
  2. Plan: Choose the most appropriate strategy
  3. Solve: Execute the strategy step-by-step
  4. Check: Verify the solution and ensure it makes sense
  5. Reflect: Consider alternative strategies or extensions
Tip 1: Don't rely on one strategy; try multiple approaches if needed.
Tip 2: Draw diagrams for geometry or spatial problems.
Tip 3: Use tables to organize information for complex problems.
Tip 4: Working backwards is effective for problems with sequential operations.
Applications: Used in real-world problem solving, standardized tests, and mathematical modeling.
Limitations: Some strategies may be inefficient for certain problem types; context matters.
Essential Rules:

• Understand first: Spend time understanding the problem before attempting to solve

• Choose wisely: Match strategy to problem type

• Organize: Keep work neat and organized

• Verify: Always check your solution

• Be flexible: Switch strategies if one isn't working

Questions & Answers

Question: How do I know which problem-solving strategy to use for a particular problem?

Answer: Match the strategy to the problem type:

  • Guess and Check: Problems with integer solutions or when you can estimate
  • Work Backwards: Problems with sequential operations ending in a known result
  • Draw a Diagram: Geometry problems or problems with spatial relationships
  • Make a Table: Problems with patterns or sequences
  • Algebra: Problems with unknowns that can be represented with variables

Start with the strategy that seems most natural for the problem, but be prepared to switch if you get stuck.

Question: What should I do if my first strategy doesn't work?

Answer: This is completely normal! Here's what to do:

  1. Don't panic: It's common for strategies not to work immediately
  2. Re-read the problem: Make sure you understood it correctly
  3. Try a different strategy: Use an alternative approach
  4. Take a break: Sometimes stepping away helps
  5. Ask for hints: Get a small clue without seeing the full solution

Remember: mathematicians often try multiple approaches before finding the right one.

Question: How do I know if my answer is reasonable?

Answer: Use these verification techniques:

  • Check units: Make sure your answer has the correct units
  • Estimate: Does your answer seem reasonable based on mental math?
  • Substitute: Plug your answer back into the original problem
  • Context: Does the answer make sense in the real-world scenario?
  • Alternative method: Try solving with a different strategy

For example, if a problem asks for the number of people and you get 12.7, reconsider since you can't have a fraction of a person.

Question: Is it okay to combine multiple strategies to solve one problem?

Answer: Absolutely! Combining strategies is often the most effective approach:

For example, you might:
- Draw a diagram to understand the problem
- Make a table to identify patterns
- Use algebra to solve precisely
- Verify with guess and check

Each strategy can provide different insights. The combination often leads to deeper understanding and more reliable solutions.

The goal is to solve the problem effectively, not to use just one approach.

Question: How can I improve my problem-solving skills?

Answer: Practice these habits:

  1. Read carefully: Take time to understand what's being asked
  2. Practice regularly: Work on problems daily
  3. Try multiple approaches: Don't settle for the first method
  4. Learn from mistakes: Understand why wrong answers failed
  5. Discuss with others: Share strategies and learn from peers
  6. Stay curious: Ask "why" and "what if" questions

Problem-solving is a skill that improves with practice and experience.