Word Problems — Number, Age, Work, Motion, Mixture, Investment - Slides
Word problems are mathematical scenarios that require translation of written statements into mathematical equations. This chapter covers six major types of word problems commonly found in the UPCAT and other Philippine college entrance exams. We'll learn the systematic approach using the 3 R's and ESP method: Read, Represent, Relate, Equate, Solve, and Prove. Each problem type has specific patterns and solving strategies that will help you tackle complex real-world mathematical situations.
Slides
Introduction to Word Problems
Notes
This overview slide introduces students to the comprehensive nature of word problems and sets expectations for what they'll learn.
Examples
- A number problem: 'Find two consecutive integers whose sum is 25'
- An age problem: 'Maria is 5 years older than Juan. In 3 years, Maria will be twice as old as Juan'
Word problems are mathematical puzzles that describe real-life situations using words instead of numbers and symbols. Success in solving them requires a structured approach and pattern recognition. The key is to identify what type of problem you're dealing with and apply the appropriate solving strategy.
Main points
- Word problems translate real-world situations into mathematical equations
- Six main types: Number, Age, Work, Motion, Mixture, and Investment problems
- Systematic approach using 3 R's and ESP method
- Essential skill for UPCAT and other entrance exams
Diagram
Mind map showing the six main types of word problems and their subcategories
The 3 R's and ESP Method
Notes
This method provides a reliable framework that students can apply to any word problem type.
Examples
- READ: Identify what is given and what is asked
- REPRESENT: Let x = the unknown number
- RELATE: Express other quantities in terms of x
This systematic approach ensures you don't miss important information and helps organize your solution process. Each step builds on the previous one, leading to a complete and verifiable solution.
Main points
- READ the problem thoroughly
- REPRESENT the unknown using variables
- RELATE variables to each other and given values
- EQUATE using facts in the problem
- SOLVE the equation
- PROVE and check the answers
Diagram
Flowchart showing the systematic 6-step approach to solving word problems
Translating Words to Mathematical Expressions
Notes
Emphasize that some phrases like 'less than' and 'subtracted from' require switching the order of terms.
Examples
- '4 more than x' becomes 'x + 4'
- '4 less than x' becomes 'x - 4'
- '4 subtracted from x' becomes 'x - 4'
Learning these keyword translations is crucial for converting word problems into mathematical equations. Pay special attention to phrases that require switching the order of numbers, such as 'less than' and 'subtracted from'.
Main points
- Addition: plus, more than, increased by, sum of
- Subtraction: minus, less than, subtracted from, decreased by
- Multiplication: times, product of, of, twice, thrice
- Division: divided by, quotient of, per
- Equality: is, was, will be, equal, results to
Diagram
Examples of translating common word phrases into mathematical expressions
Number Problems - Basic Concepts
Notes
Start with simple one-variable problems before moving to more complex relationships.
Examples
- Find two numbers whose sum is 50 and difference is 10
- One number is 3 times another, and their sum is 28
Number problems are the foundation of word problems. They teach the essential skill of translating verbal descriptions into mathematical equations. The key is identifying the relationship between numbers and expressing it algebraically.
Main points
- Simple number problems involve finding unknown numbers based on given relationships
- Use variables to represent the unknown numbers
- Translate word statements into algebraic equations
- Most common: finding numbers with specific sum, difference, or ratio
Diagram
Decision tree for approaching number problems
Number Problems - Worked Example
Notes
Always emphasize the importance of checking answers by substituting back into the original problem.
Examples
- Left side: 5x - 8 (product decreased by 8)
- Right side: 2x + 13 (sum of twice and 13)
- Equation: 5x - 8 = 2x + 13
Step-by-step solution: 5x - 8 = 2x + 13. Solving: 5x - 2x = 13 + 8, so 3x = 21, therefore x = 7. PROVE: 5(7) - 8 = 35 - 8 = 27, and 2(7) + 13 = 14 + 13 = 27. Both sides equal 27, so our answer is correct.
Main points
- Problem: The product of a number and 5, decreased by 8, is equal to the sum of twice the number and 13.
- READ: One unknown number with a specific relationship
- REPRESENT: Let x = the number
- RELATE: Product of number and 5, decreased by 8 = 5x - 8
- Sum of twice the number and 13 = 2x + 13
Diagram
Step-by-step solution process for the number problem example
Consecutive Integer Problems
Notes
Students often confuse consecutive odd/even integers. Emphasize that both types increase by 2, not 1.
Examples
- Three consecutive integers: x, x+1, x+2
- Three consecutive odd integers: x, x+2, x+4
- Four consecutive even integers: x, x+2, x+4, x+6
The key insight is that consecutive integers increase by 1, while consecutive odd or even integers increase by 2. Always let x represent the first (smallest) number in the sequence to avoid negative coefficients.
Main points
- Consecutive integers: numbers that follow each other in order (1, 2, 3, 4...)
- If x is the first integer, then x+1, x+2, x+3 are the next consecutive integers
- Consecutive odd integers differ by 2: x, x+2, x+4, x+6...
- Consecutive even integers also differ by 2: x, x+2, x+4, x+6...
Diagram
Classification of consecutive integer problems and their representation patterns
Age Problems - Basic Approach
Notes
The table method is crucial for preventing errors in age problems. Always organize information before writing equations.
Examples
- Present ages: Maria = x, Juan = y
- 5 years ago: Maria = x-5, Juan = y-5
- 3 years from now: Maria = x+3, Juan = y+3
Age problems become manageable when you organize the information systematically. Create a table with columns for each person and rows for different time periods. This visual organization helps prevent confusion and ensures all relationships are properly captured.
Main points
- Age problems involve relationships between people's ages at different times
- Key insight: Everyone ages at the same rate
- Use a table to organize ages: Present, Past, Future
- Assign variables to represent ages in the time frame mentioned in the question
Diagram
Systematic approach to organizing and solving age problems using tables
Age Problems - Worked Example
Notes
Use the table format consistently to avoid confusion with different time periods in age problems.
Examples
- Age table helps visualize the problem clearly
- Always check if the relationship holds true
- x-2 = 2(x-7) simplifies to x = 12
Solving: x-2 = 2x-14, so 14-2 = 2x-x, giving us 12 = x. Therefore Alice is 12 and Ben is 7. PROVE: Two years ago, Alice was 10 and Ben was 5. Indeed, 10 = 2×5, confirming our answer.
Main points
- Problem: Alice is 5 years older than Ben today. Two years earlier, Alice was twice as old as Ben.
- Present ages: Alice = x, Ben = x-5
- Two years ago: Alice = x-2, Ben = x-7
- Relationship: Alice's past age = 2 × Ben's past age
- Equation: x-2 = 2(x-7)
Diagram
Type
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Work Problems - Understanding Rates
Notes
The rate concept is fundamental. If someone takes 4 hours for a job, they do 1/4 of the job each hour.
Examples
- Rate = 1/time (if time is to complete 1 job)
- Combined rate = Rate₁ + Rate₂ + Rate₃ + ...
- Time together = 1 ÷ (Combined rate)
Work problems use the concept of rates. If Charlie can paint a room in 6 hours, his rate is 1/6 room per hour. If Donna can paint the same room in 3 hours, her rate is 1/3 room per hour. Working together, their combined rate is 1/6 + 1/3 = 1/2 room per hour.
Main points
- Work formula: Work = Rate × Time
- If a person can complete a job in n hours, their rate is 1/n jobs per hour
- When people work together, their rates add up
- Total work completed = Sum of individual work contributions
Diagram
Step-by-step process for solving work problems involving multiple workers
Motion Problems - Distance Formula
Notes
Drawing diagrams is essential for motion problems. Visualize the movement to understand the relationships.
Examples
- Meeting: Combined distances = Total separation
- Overtaking: Both objects travel same distance
- Round trip: Distance going = Distance returning
Motion problems involve objects moving at constant speeds. The key relationships depend on whether objects move toward each other, away from each other, or in the same direction. Always draw a diagram to visualize the situation and use tables to organize the information systematically.
Main points
- Basic formula: Distance = Rate × Time (D = RT)
- Three types: Meeting problems, Overtaking problems, Round trip problems
- Meeting problems: Objects move toward each other
- Overtaking problems: One object catches up to another
- Use tables to organize distance, rate, and time information
Diagram
Classification and solution approach for different types of motion problems
Motion Problems - Worked Example
Notes
Meeting problems are easier when you visualize objects moving toward each other until they meet at one point.
Examples
- Both travel for the same time t
- Their combined distances equal the separation
- Always verify by checking the total distance
Solving: 65t = 117, so t = 1.8 hours (1 hour 48 minutes). PROVE: Eddie travels 40×1.8 = 72 km, Fiona travels 25×1.8 = 45 km. Total: 72+45 = 117 km ✓
Main points
- Problem: Eddie and Fiona live 117 km apart. They leave simultaneously toward each other.
- Eddie travels at 40 kph, Fiona at 25 kph
- Let t = time until they meet
- Eddie's distance: 40t, Fiona's distance: 25t
- Equation: 40t + 25t = 117
Diagram
Visual representation of the meeting problem showing how distances combine
Mixture Problems - Solution Concepts
Notes
The table method is essential for mixture problems. Always track the pure solute amounts carefully.
Examples
- 10% saline solution: 10% salt, 90% water
- Mixing 30% and 10% solutions to get 25%
- Amount of pure solute = Percentage × Total volume
Mixture problems involve combining solutions of different concentrations. The critical insight is that when you mix solutions, the total amount of pure solute equals the sum of solute from each individual solution. Use tables to track percentages, volumes, and amounts of pure solute.
Main points
- Solution = Solute + Solvent (salt + water = saline solution)
- Percentage refers to the concentration of solute
- Pure substance = 100% solution, Pure solvent = 0% solution
- Key equation: (% solution) × (Volume of solution) = Volume of solute
- When mixing: Total solute = Sum of individual solutes
Diagram
Systematic approach to organizing and solving mixture problems
Investment Problems - Interest Formulas
Notes
Common errors include forgetting to convert percentages to decimals and mismatching time units with interest rate periods.
Examples
- ₱5,000 at 6% for 2 years: I = 5000 × 0.06 × 2 = ₱600
- Total amount after 2 years: ₱5,000 + ₱600 = ₱5,600
- Monthly rate for 1 year: Use T = 12 months
Investment problems use interest calculations to find unknown amounts, rates, or time periods. Simple interest is calculated only on the principal amount. Always convert percentages to decimals (5% = 0.05) and ensure time units match the interest rate period (annual rate requires time in years).
Main points
- Simple Interest: I = P × R × T
- P = Principal (amount invested)
- R = Rate (annual interest rate as decimal)
- T = Time (in years)
- Total Amount = Principal + Interest
Diagram
Step-by-step process for solving simple interest investment problems
Investment Problems - Worked Example
Notes
Investment problems often involve splitting a total amount between different interest rates based on income relationships.
Examples
- Two investments totaling ₱20,000
- Use relationship between the incomes
- Always verify the income relationship
Solving: 1400 - 0.07x = 0.42x - 70, so 1470 = 0.49x, giving x = ₱3,000. Therefore, ₱3,000 was invested at 6% and ₱17,000 at 7%. PROVE: 6% income = ₱180, 7% income = ₱1,190. Indeed, 1,190 = 7(180) - 70 ✓
Main points
- Problem: A man invested ₱20,000 in two parts at 6% and 7%
- The 7% investment earned ₱70 less than 7 times the 6% investment income
- Let x = amount invested at 6%
- Amount at 7% = ₱20,000 - x
- Income equation: 0.07(20000-x) = 7(0.06x) - 70
Diagram
Type
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Problem-Solving Strategies and Common Mistakes
Notes
Emphasize that consistent practice and systematic approaches lead to success in word problems.
Examples
- Wrong: Let x = the larger number (when finding the smaller)
- Right: Let x = the smaller number, then larger = x + difference
- Always check: Does the answer make sense?
Success in word problems comes from systematic organization and careful checking. Use the 3 R's and ESP method consistently. Draw diagrams for motion problems, create tables for age and mixture problems, and always verify your final answer makes sense in the context of the original problem.
Main points
- Always organize information using tables or diagrams
- Check units throughout your solution
- Verify answers by substituting back into the original problem
- Common mistakes: Wrong variable assignments, unit mismatches, algebra errors
- Practice different problem types to recognize patterns
Diagram
Comprehensive guide to successful word problem solving strategies and avoiding common pitfalls
Summary and Key Takeaways
Notes
This summary reinforces the importance of mastering all six problem types for exam success.
Examples
- Number problems: Focus on relationships between unknowns
- Age problems: Use tables to track different time periods
- Work problems: Remember rates add when working together
- Motion problems: Visualize the movement with diagrams
- Mixture problems: Track pure solute amounts
- Investment problems: Match time units with interest periods
Word problems are fundamental to mathematical problem-solving and appear frequently in entrance exams. The six types covered - number, age, work, motion, mixture, and investment - form the foundation for more complex mathematical modeling. Consistent practice with the systematic approach will build confidence and accuracy.
Main points
- Master the 3 R's and ESP method for all word problem types
- Each problem type has specific patterns and solution strategies
- Organization through tables and diagrams prevents errors
- Always verify answers by checking the original conditions
- Regular practice builds pattern recognition and speed
Diagram
Distribution of word problem types commonly found in UPCAT and other entrance exams
In summary
Word problems are an essential component of mathematical reasoning that bridge abstract mathematics with real-world applications. Mastering the six types covered in this chapter - number, age, work, motion, mixture, and investment problems - provides a solid foundation for success in the UPCAT and other Philippine college entrance exams. The systematic 3 R's and ESP approach (Read, Represent, Relate, Equate, Solve, Prove) ensures consistent success across all problem types. Remember that organization through tables and diagrams, careful attention to units, and verification of answers are key to avoiding common mistakes. Regular practice will develop pattern recognition skills that make these problems more manageable and build confidence for exam day.