USTET Mathematics — Word Problems — Number, Age, Work, Motion, Mixture, InvestmentSlides
Word Problems — Number, Age, Work, Motion, Mixture, Investment slides, sized for screen and print. Flip through them for a five-minute pre-mock refresh, or print the deck for on-paper annotation. Either way, the slides cover Word Problems — Number, Age, Work, Motion, Mixture, Investment at the depth University of Santo Tomas tests for the USTET 2026.
Exam context
University of Santo Tomas runs the University of Santo Tomas Entrance Test on Early Q4 2026. Its Mathematics section sits under a "Core section" weighting, and Word Problems — Number, Age, Work, Motion, Mixture, Investment is the 4th chapter in the 9-chapter USTET Mathematics rotation. The USTET passing mark is Competitive overall score, and the most recent 2026 paper drew about a meaningful share of questions from Mathematics.
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
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.
Notes
This overview slide introduces students to the comprehensive nature of word problems and sets expectations for what they'll learn.
Topic
Introduction
Slide Id
S1
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mermaid
Image Prompt
Slide Number
1
Mermaid Diagram
Code
mindmap root((Word Problems)) Number Problems Consecutive Integers Digit Problems Basic Number Relations Age Problems Present Ages Past Ages Future Ages Work Problems Individual Rates Combined Work Completion Time Motion Problems Distance Rate Time Meeting Problems Overtaking Problems Mixture Problems Solutions Concentrations Dilutions Investment Problems Simple Interest Compound Interest Principal Rate Time
Type
mermaid_mindmap
Description
Mind map showing the six main types of word problems and their subcategories
The 3 R's and ESP Method
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.
Notes
This method provides a reliable framework that students can apply to any word problem type.
Topic
Problem-Solving Method
Slide Id
S2
Visual Type
mermaid
Image Prompt
Slide Number
2
Mermaid Diagram
Code
flowchart TD A[fa:fa-book READ Problem] --> B[fa:fa-calculator REPRESENT with Variables] B --> C[RELATE Variables and Values] C --> D[fa:fa-equals EQUATE Using Facts] D --> E[fa:fa-cogs SOLVE Equation] E --> F[fa:fa-check PROVE Answer] F --> G{Answer Correct?} G -->|Yes| H[fa:fa-star Final Answer] G -->|No| I[fa:fa-times Check Work] I --> B
Type
mermaid_flowchart
Description
Flowchart showing the systematic 6-step approach to solving word problems
Translating Words to Mathematical Expressions
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'.
Notes
Emphasize that some phrases like 'less than' and 'subtracted from' require switching the order of terms.
Topic
Translation Skills
Slide Id
S3
Visual Type
mermaid
Image Prompt
Slide Number
3
Mermaid Diagram
Code
flowchart LR A[Words] --> B[Mathematical Expression] C[4 more than x] --> D[x + 4] E[4 less than x] --> F[x - 4] G[twice x] --> H[2x] I[x divided by 4] --> J[x/4] K[4 subtracted from x] --> L[x - 4]
Type
mermaid_flowchart
Description
Examples of translating common word phrases into mathematical expressions
Number Problems - Basic Concepts
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.
Notes
Start with simple one-variable problems before moving to more complex relationships.
Topic
Number Problems
Slide Id
S4
Visual Type
mermaid
Image Prompt
Slide Number
4
Mermaid Diagram
Code
flowchart TD A[Number Problem] --> B{How many unknowns?} B -->|One| C[Let x = the number] B -->|Two| D[Express second in terms of first] C --> E[Translate relationship] D --> E E --> F[Set up equation] F --> G[Solve for x] G --> H[Find all numbers]
Type
mermaid_flowchart
Description
Decision tree for approaching number problems
Number Problems - Worked Example
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.
Notes
Always emphasize the importance of checking answers by substituting back into the original problem.
Topic
Number Problems Example
Slide Id
S5
Visual Type
mermaid
Image Prompt
Slide Number
5
Mermaid Diagram
Code
flowchart TD A[5x - 8 = 2x + 13] --> B[5x - 2x = 13 + 8] B --> C[3x = 21] C --> D[x = 7] D --> E[Check: 5times7 - 8 = 27] E --> F[Check: 2times7 + 13 = 27] F --> G[fa:fa-check Answer Verified]
Type
mermaid_flowchart
Description
Step-by-step solution process for the number problem example
Consecutive Integer Problems
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.
Notes
Students often confuse consecutive odd/even integers. Emphasize that both types increase by 2, not 1.
Topic
Consecutive Integers
Slide Id
S6
Visual Type
mermaid
Image Prompt
Slide Number
6
Mermaid Diagram
Code
flowchart TD A[Consecutive Integer Problem] --> B{What type?} B -->|Regular| C[x, x+1, x+2, ...] B -->|Odd| D[x, x+2, x+4, ...] B -->|Even| E[x, x+2, x+4, ...] C --> F[Difference of 1] D --> G[Difference of 2] E --> G F --> H[Set up equation] G --> H
Type
mermaid_flowchart
Description
Classification of consecutive integer problems and their representation patterns
Age Problems - Basic Approach
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.
Notes
The table method is crucial for preventing errors in age problems. Always organize information before writing equations.
Topic
Age Problems
Slide Id
S7
Visual Type
mermaid
Image Prompt
Slide Number
7
Mermaid Diagram
Code
flowchart TD A[Age Problem] --> B[Create Age Table] B --> C[Present Ages] B --> D[Past Ages] B --> E[Future Ages] C --> F[Assign Variables] D --> G[Subtract Years] E --> H[Add Years] F --> I[Set up equation using given relationship] G --> I H --> I
Type
mermaid_flowchart
Description
Systematic approach to organizing and solving age problems using tables
Age Problems - Worked Example
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.
Notes
Use the table format consistently to avoid confusion with different time periods in age problems.
Topic
Age Problems Example
Slide Id
S8
Visual Type
none
Image Prompt
Slide Number
8
Mermaid Diagram
Type
none
Work Problems - Understanding Rates
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.
Notes
The rate concept is fundamental. If someone takes 4 hours for a job, they do 1/4 of the job each hour.
Topic
Work Problems
Slide Id
S9
Visual Type
mermaid
Image Prompt
Slide Number
9
Mermaid Diagram
Code
flowchart TD A[Work Problem] --> B[Identify individual completion times] B --> C[Calculate individual rates] C --> D[Rate = 1/time for each person] D --> E[Add rates for combined work] E --> F[Combined rate = Rate1 + Rate2] F --> G[Time together = 1/Combined rate]
Type
mermaid_flowchart
Description
Step-by-step process for solving work problems involving multiple workers
Motion Problems - Distance Formula
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.
Notes
Drawing diagrams is essential for motion problems. Visualize the movement to understand the relationships.
Topic
Motion Problems
Slide Id
S10
Visual Type
mermaid
Image Prompt
Slide Number
10
Mermaid Diagram
Code
flowchart TD A[Motion Problem] --> B{Type of motion?} B -->|Meeting| C[Distance1 + Distance2 = Total Distance] B -->|Overtaking| D[Distance1 = Distance2] B -->|Round Trip| E[Distance going = Distance returning] C --> F[Use D = RT for each object] D --> F E --> F F --> G[Set up equation and solve]
Type
mermaid_flowchart
Description
Classification and solution approach for different types of motion problems
Motion Problems - Worked Example
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 ✓
Notes
Meeting problems are easier when you visualize objects moving toward each other until they meet at one point.
Topic
Motion Problems Example
Slide Id
S11
Visual Type
mermaid
Image Prompt
Slide Number
11
Mermaid Diagram
Code
flowchart LR A[Eddie 40 kph] -->|travels 40t km| B[Meeting Point] B <--travels 25t km| C[Fiona 25 kph] D[Total Distance: 40t + 25t = 117] --> E[65t = 117] E --> F[t = 1.8 hours]
Type
mermaid_flowchart
Description
Visual representation of the meeting problem showing how distances combine
Mixture Problems - Solution Concepts
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.
Notes
The table method is essential for mixture problems. Always track the pure solute amounts carefully.
Topic
Mixture Problems
Slide Id
S12
Visual Type
mermaid
Image Prompt
Slide Number
12
Mermaid Diagram
Code
flowchart TD A[Mixture Problem] --> B[Identify given solutions] B --> C[Create mixture table] C --> D[Columns: Percent, Volume, Pure Solute] D --> E[Pure Solute = Percent × Volume] E --> F[Total pure solute = Sum of individual solutes] F --> G[Set up equation and solve]
Type
mermaid_flowchart
Description
Systematic approach to organizing and solving mixture problems
Investment Problems - Interest Formulas
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).
Notes
Common errors include forgetting to convert percentages to decimals and mismatching time units with interest rate periods.
Topic
Investment Problems
Slide Id
S13
Visual Type
mermaid
Image Prompt
Slide Number
13
Mermaid Diagram
Code
flowchart TD A[Investment Problem] --> B[Identify P, R, T] B --> C[Convert % to decimal] C --> D[Match time units to rate period] D --> E[Calculate Interest: I = P × R × T] E --> F[Total Amount = P + I] F --> G[Check units and reasonableness]
Type
mermaid_flowchart
Description
Step-by-step process for solving simple interest investment problems
Investment Problems - Worked Example
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 ✓
Notes
Investment problems often involve splitting a total amount between different interest rates based on income relationships.
Topic
Investment Problems Example
Slide Id
S14
Visual Type
none
Image Prompt
Slide Number
14
Mermaid Diagram
Type
none
Problem-Solving Strategies and Common Mistakes
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.
Notes
Emphasize that consistent practice and systematic approaches lead to success in word problems.
Topic
Problem-Solving Strategies
Slide Id
S15
Visual Type
mermaid
Image Prompt
Slide Number
15
Mermaid Diagram
Code
mindmap root((Problem Solving Tips)) Organization Use tables Draw diagrams Label variables clearly Common Mistakes Wrong variable assignment Unit mismatches Algebra errors Not checking answers Verification Substitute back Check reasonableness Verify all conditions Practice Pattern recognition Multiple problem types Timed practice
Type
mermaid_mindmap
Description
Comprehensive guide to successful word problem solving strategies and avoiding common pitfalls
Summary and Key Takeaways
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.
Notes
This summary reinforces the importance of mastering all six problem types for exam success.
Topic
Summary
Slide Id
S16
Visual Type
mermaid
Image Prompt
Slide Number
16
Mermaid Diagram
Code
pie title Word Problem Types in UPCAT "Number Problems" : 20 "Age Problems" : 15 "Work Problems" : 15 "Motion Problems" : 20 "Mixture Problems" : 15 "Investment Problems" : 15
Type
mermaid_pie
Description
Distribution of word problem types commonly found in UPCAT and other entrance exams
References
- BRAINBOX UPCAT AND OTHER COLLAGE ENTRANCE — Mathematics Proficiency.pdf
- THE UPCAT CHAMPION CET — Quantitative Reasoning.pdf
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.
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Geometry — Lines, Angles, Polygons, Triangles & Circles
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