Solved Exercises on Multi-Step Word Problems in Grade 7

Master multi-step word problems: algebraic reasoning, problem-solving strategies, and mathematical modeling through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Shopping Problem
Exercise 1
Sarah bought 3 notebooks and 2 pens for $15. Each pen costs $2 more than each notebook. Find the cost of each item.
Definition:

Multi-step word problem: A mathematical problem requiring multiple operations and logical steps to reach a solution.

Algebraic reasoning: Using variables and equations to represent and solve problems.

Problem-solving method:
  1. Read and understand the problem
  2. Identify unknowns and assign variables
  3. Set up equations based on given information
  4. Solve the equations step by step
  5. Check the solution
Step 1: Define variables

Let x = cost of one notebook (in dollars)

Then (x + 2) = cost of one pen (since each pen costs $2 more)

Step 2: Set up equation

3 notebooks + 2 pens = $15

3x + 2(x + 2) = 15

Step 3: Solve the equation

3x + 2x + 4 = 15

5x + 4 = 15

5x = 11

x = 2.2

Step 4: Find pen cost

Cost of one pen = x + 2 = 2.2 + 2 = $4.20

Step 5: Verify solution

Check: 3($2.20) + 2($4.20) = $6.60 + $8.40 = $15 ✓

Final answer:

Each notebook costs $2.20 and each pen costs $4.20.

Applied rules:

• Variable assignment: Assign variables to unknown quantities

• Equation setup: Translate words into mathematical expressions

• Algebraic manipulation: Combine like terms and solve systematically

Read Problem Define Variables Set Up Equation Solve Equation Verify Answer Done
2 Rate and Time Problem
Exercise 2
A car travels from City A to City B at 60 mph and returns at 40 mph. If the total trip takes 5 hours, how far apart are the cities?
Definition:

Rate problem: Uses the relationship Distance = Rate × Time.

Multi-stage journey: A trip with different speeds for different segments.

Step 1: Define variables

Let d = distance between cities (in miles)

Time going = d/60 hours, Time returning = d/40 hours

Step 2: Set up equation

Total time = Time going + Time returning = 5 hours

d/60 + d/40 = 5

Step 3: Find common denominator

LCM of 60 and 40 is 120

(2d)/120 + (3d)/120 = 5

5d/120 = 5

Step 4: Solve for d

5d/120 = 5

5d = 600

d = 120

Step 5: Verify solution

Going: 120/60 = 2 hours, Returning: 120/40 = 3 hours

Total: 2 + 3 = 5 hours ✓

Final answer:

The cities are 120 miles apart.

Applied rules:

• Distance formula: Distance = Rate × Time

• Time calculation: Time = Distance ÷ Rate

• Fraction addition: Find common denominators to combine fractions

Going: 60 mph Returning: 40 mph 120 miles 2 hours 3 hours City A City B Round Trip Journey
3 Percentage and Ratio Problem
Exercise 3
In a class of 30 students, 40% are boys and the rest are girls. If 25% of the boys and 30% of the girls participate in sports, how many students participate in sports?
Definition:

Percentage: A fraction expressed as a part of 100.

Ratio problem: A problem involving proportional relationships between quantities.

Step 1: Find number of boys

Boys = 40% of 30 = 0.40 × 30 = 12 boys

Step 2: Find number of girls

Girls = 30 - 12 = 18 girls

Step 3: Find boys in sports

Boys in sports = 25% of 12 = 0.25 × 12 = 3 boys

Step 4: Find girls in sports

Girls in sports = 30% of 18 = 0.30 × 18 = 5.4 ≈ 5 girls

Step 5: Find total participants

Total = 3 + 5 = 8 students participate in sports

Final answer:

8 students participate in sports.

Applied rules:

• Percentage calculation: Part = Percentage × Whole

• Sequential calculations: Solve step by step in order

• Real-world context: Round to nearest whole number when counting people

12 Boys (40%) 18 Girls (60%) 3 boys in sports 5 girls in sports 8 Total Sports Participation in Class
Problem-Solving Concepts, Rules and Methods
Distance = Rate × Time
Distance Formula
Percentage
Part/Whole × 100%
Expressing ratios as parts of 100
Rate
Distance/Time
How fast something changes
Ratio
a:b or a/b
Comparison of quantities
Key definitions:

Multi-Step Word Problem: A mathematical problem requiring multiple operations and logical steps to reach a solution.

Problem-Solving Strategy: A systematic approach to analyze and solve mathematical problems.

Mathematical Modeling: Translating real-world situations into mathematical expressions.

Algebraic Reasoning: Using variables and equations to represent and solve problems.

Problem-solving methodology:
  1. Understand: Read the problem carefully and identify what's given and what's needed
  2. Plan: Decide on a strategy and set up equations or steps
  3. Solve: Carry out the plan step by step
  4. Check: Verify that the answer makes sense in the context
Tip 1: Underline or circle key information in the problem.
Tip 2: Draw diagrams to visualize the problem.
Tip 3: Check if your answer is reasonable in the real world.
Tip 4: Work backwards to verify your solution.
Key characteristics: Requires organizing information, identifying relationships, and performing multiple calculations.
Common applications: Shopping, travel, finance, science, engineering, and everyday life decisions.
Solution: Exercises 4 to 5
4 Mixture Problem
Exercise 4
A chemist needs to mix a 15% acid solution with a 30% acid solution to get 12 liters of a 20% acid solution. How many liters of each solution should be used?
Definition:

Mixture problem: A problem involving combining different concentrations to achieve a desired mixture.

Step 1: Define variables

Let x = liters of 15% solution

Then (12 - x) = liters of 30% solution

Step 2: Set up equation

Amount of pure acid in 15% solution + Amount in 30% solution = Amount in 20% solution

0.15x + 0.30(12 - x) = 0.20(12)

Step 3: Solve the equation

0.15x + 3.6 - 0.30x = 2.4

-0.15x + 3.6 = 2.4

-0.15x = -1.2

x = 8

Step 4: Find amounts

15% solution: 8 liters

30% solution: 12 - 8 = 4 liters

Step 5: Verify solution

Acid in 15% solution: 0.15 × 8 = 1.2 liters

Acid in 30% solution: 0.30 × 4 = 1.2 liters

Total acid: 1.2 + 1.2 = 2.4 liters

Required acid: 0.20 × 12 = 2.4 liters ✓

Final answer:

8 liters of 15% solution and 4 liters of 30% solution should be used.

Applied rules:

• Mixture equation: Concentration × Volume = Amount of substance

• Systematic substitution: Express one variable in terms of another

• Verification: Check that the final mixture has the desired concentration

15% 8L 30% 4L 20% 12L 1.2L acid 1.2L acid 2.4L acid Mixture Problem Solution
5 Work Rate Problem
Exercise 5
Working alone, Alice can paint a room in 6 hours. Bob can paint the same room in 4 hours. If they work together, how long will it take to paint the room?
Definition:

Work rate problem: A problem involving how fast individuals or machines can complete a task.

Combined rate: When working together, rates add up.

Step 1: Find individual work rates

Alice's rate = 1 room per 6 hours = 1/6 room per hour

Bob's rate = 1 room per 4 hours = 1/4 room per hour

Step 2: Find combined rate

Combined rate = Alice's rate + Bob's rate

Combined rate = 1/6 + 1/4

Step 3: Add fractions

LCM of 6 and 4 is 12

1/6 = 2/12, 1/4 = 3/12

Combined rate = 2/12 + 3/12 = 5/12 rooms per hour

Step 4: Find time to complete 1 room

If they paint 5/12 of a room in 1 hour, then:

Time = 1 room ÷ (5/12 room per hour) = 1 × 12/5 = 12/5 = 2.4 hours

Step 5: Convert to minutes

2.4 hours = 2 hours and 0.4 × 60 = 24 minutes

Final answer:

Working together, they can paint the room in 2.4 hours (2 hours and 24 minutes).

Applied rules:

• Work rate: Rate = Work done ÷ Time taken

• Combined rates: When working together, add individual rates

• Time calculation: Time = Total work ÷ Combined rate

Alice (40%) Bob (60%) Together (100%) 1/6 room/hr 1/4 room/hr 5/12 room/hr 2.4 hours A B A+B Work Rate Collaboration
Problem-Solving Theory: Laws, Methods, Definitions, and Formulas
Work = Rate × Time
Work Formula
Key definitions:

Multi-Step Word Problem: A mathematical problem requiring multiple operations and logical steps to reach a solution, often involving real-world scenarios.

Problem-Solving Strategy: A systematic approach that includes understanding, planning, solving, and checking phases.

Mathematical Modeling: The process of translating real-world situations into mathematical expressions and equations.

Algebraic Reasoning: Using variables, equations, and logical thinking to represent and solve problems.

Problem-solving methodology:
  1. Understanding: Read carefully, identify knowns and unknowns, clarify the question
  2. Planning: Choose appropriate strategy, set up equations, organize information
  3. Solving: Execute the plan, perform calculations, keep track of steps
  4. Checking: Verify reasonableness, substitute back, ensure answer addresses the question
Tip 1: Draw diagrams or tables to organize complex information visually.
Tip 2: Write down all given information before attempting to solve.
Tip 3: Use estimation to check if your answer is reasonable.
Tip 4: Practice different problem types to build confidence.

Key characteristics: Requires careful reading, organization, and systematic approach.
Common applications: Real-world scenarios involving rates, percentages, ratios, and measurements.
Essential formulas and rules:

• Distance-Rate-Time: Distance = Rate × Time

• Work-Rate-Time: Work = Rate × Time

• Percentage: Part = Percentage × Whole

• Mixture Problems: Concentration × Volume = Amount of substance

• Combined Rates: When working together, add individual rates

Questions & Answers

Question: I often get confused about which variable to assign when setting up equations for word problems. Any tips?

Answer: Here are some effective strategies for assigning variables:

  • Assign to the unknown: Identify what you're being asked to find and assign a variable to that quantity
  • Use descriptive letters: If finding the cost of notebooks, use 'n' for notebook cost
  • Express others in terms of the first: If one quantity relates to another, express it in terms of your primary variable
  • Keep it simple: Don't create more variables than necessary

Example: If you need to find the cost of pens and notebooks where pens cost $2 more than notebooks, let n = cost of notebook, then (n+2) = cost of pen.

Always write your variable definitions clearly before setting up equations.

Question: How do I know when to add rates versus multiply them in word problems?

Answer: The operation depends on the situation:

  • Add rates when: Two or more entities work together on the same task
  • Multiply rates when: Calculating total work done or converting units

Adding rates: If Alice paints 1/6 of a room per hour and Bob paints 1/4 of a room per hour, together they paint 1/6 + 1/4 = 5/12 of the room per hour.

Multiplying rates: If driving at 60 mph for 3 hours, total distance is 60 × 3 = 180 miles.

Remember: when entities work together toward the same goal, add their rates.

Question: What should I do if I get a decimal answer when counting people or things?

Answer: This usually indicates one of two things:

  • Round to nearest whole number: If the decimal is close to a whole number and makes sense in context
  • Check your work: If the decimal is far from a whole number, you likely made an error

In our sports participation example, we got 5.4 girls participating. Since we can't have 0.4 of a person, we rounded to 5 girls.

However, if you get an answer like 3.7 people, recheck your calculations as this might indicate an error in your setup or arithmetic.

Always consider if your answer makes sense in the real-world context of the problem.

Question: How can I check if my answer is reasonable without doing the entire problem again?

Answer: Use these quick verification strategies:

  • Estimate: Round numbers to check if your answer is in the right ballpark
  • Plug back in: Substitute your answer into the original problem to see if it works
  • Check units: Make sure your answer has the correct units
  • Reality check: Does the answer make sense in the real-world context?

Example: If calculating time and getting 200 hours for a simple task, reconsider your answer.

In the car problem, we found 120 miles between cities. We verified: 120/60 + 120/40 = 2 + 3 = 5 hours ✓

Question: I struggle with word problems that involve multiple operations. How do I keep track of everything?

Answer: Use these organizational strategies:

  • Write everything down: Don't try to keep all information in your head
  • Make a table: Organize known and unknown values in a structured format
  • Label each step: Clearly mark what you're calculating at each stage
  • Check progress: After each step, verify that your result makes sense

For complex problems, break them into smaller sub-problems and solve each one separately.

Create a "road map" of what you need to find and the steps required to get there.