Word Problems — Speed/Distance/Age, Discount & Interest - Revision Notes
This chapter focuses on solving real-world mathematical problems involving speed, distance, time, age relationships, discounts, and interest calculations. These problem-solving skills are essential for Civil Service examinations and practical applications in business, finance, and everyday life. Understanding these concepts will help you tackle word problems systematically and accurately.
Speed, Distance, and Time Problems
Formulas
Speed = Distance ÷ Time
Example
A car travels 210 km in 3 hours. Speed = 210 ÷ 3 = 70 km/hr
Variables
Speed (km/hr, m/s), Distance (km, m), Time (hr, min, sec)
Application
Finding how fast someone or something is moving
Distance = Speed × Time
Example
A train travels at 60 km/hr for 4 hours. Distance = 60 × 4 = 240 km
Variables
Distance (km, m), Speed (km/hr, m/s), Time (hr, min, sec)
Application
Finding how far someone traveled
Average Speed = Total Distance ÷ Total Time
Example
180 km at 60 mph, return at 90 mph. Times: 3hr + 2hr = 5hr. Average = 360 ÷ 5 = 72 mph
Variables
Average Speed (km/hr), Total Distance (km), Total Time (hr)
Application
Finding average speed for trips with different speeds
Exam tips
- Always identify what is given and what needs to be found
- Create a table for complex problems with multiple speeds or times
- Remember: Distance has units like km, m; Speed has units like km/hr, m/s; Time has units like hr, min
- For average speed problems, never add the speeds - always use total distance ÷ total time
Key points
- Speed is the distance traveled per unit time (Speed = Distance ÷ Time)
- Distance is the total path covered (Distance = Speed × Time)
- Time is the duration of travel (Time = Distance ÷ Speed)
- Average speed = Total distance ÷ Total time (for round trips or multiple segments)
- Unit conversions: 1 km/hr = 5/18 m/s, 1 m/s = 18/5 km/hr
- For problems with two different speeds, create a table with distance, speed, and time for each segment
Definitions
Speed
The rate at which distance is covered over time
Importance
Essential for calculating travel time and distance in transportation problems
Average Speed
Total distance traveled divided by total time taken
Importance
Used when speed varies during the journey or for round trips
Common mistakes
- Adding speeds instead of using the average speed formula for round trips
- Forgetting to convert units (hours to minutes, km/hr to m/s)
- Not creating a systematic table for multi-segment journeys
- Confusing distance with displacement in physics contexts
Age Word Problems
Formulas
Present Age ± n = Past/Future Age
Example
If Karylle is now 15, then 4 years ago she was 15 - 4 = 11 years old
Variables
Present age (years), n (number of years), Past/Future age (years)
Application
Converting between different time periods in age problems
Age Ratio: If a:b, then ages are ax and bx
Example
If ages are in ratio 3:2, and x = 5, then ages are 15 and 10 years
Variables
a, b (ratio numbers), x (common multiplier)
Application
When ages are given in ratio form
Exam tips
- Create a table with columns for each person and rows for different time periods
- Clearly define your variable (usually the present age of one person)
- Check your final answer by substituting back into the original problem
- Age problems often have two equations that need to be solved simultaneously
Key points
- Age problems involve relationships between people's ages at different times
- Present age, past age (n years ago), and future age (n years later) are key concepts
- If present age is x, then: Past age = x - n, Future age = x + n
- Age ratios: If ages are in ratio a:b, then ages are ax and bx
- Set up equations based on the given relationships between ages
- Use variables to represent unknown ages and solve systematically
Definitions
Age Relationship
Mathematical connection between people's ages at different time periods
Importance
Forms the basis for setting up equations in age problems
Time Period Shift
Moving between past, present, and future ages by adding or subtracting years
Importance
Essential for translating word problems into mathematical equations
Common mistakes
- Confusing past and future age calculations (adding instead of subtracting)
- Not maintaining consistency in time periods across all people in the problem
- Forgetting to check if the solution makes logical sense (negative ages, etc.)
- Setting up incorrect equations from complex word relationships
Discount Problems
Formulas
Discount = List Price - Selling Price
Example
List price ₱1500, selling price ₱1200. Discount = 1500 - 1200 = ₱300
Variables
Discount (₱), List Price (₱), Selling Price (₱)
Application
Finding the amount of discount given
Discount % = (Discount ÷ List Price) × 100
Example
Discount ₱30 on list price ₱150. Discount % = (30 ÷ 150) × 100 = 20%
Variables
Discount % (percentage), Discount (₱), List Price (₱)
Application
Finding the percentage rate of discount
Selling Price = List Price × (1 - Discount Rate)
Example
List price ₱1500, 20% discount. Selling price = 1500 × (1 - 0.20) = ₱1200
Variables
Selling Price (₱), List Price (₱), Discount Rate (decimal form)
Application
Quick calculation of final price after discount
Exam tips
- Always identify which values are given: list price, selling price, discount amount, or discount %
- Remember: Discount % is always calculated based on the list price, not selling price
- Check if your discount percentage makes sense (should be less than 100%)
- Practice both methods: direct calculation and using (100% - discount%) approach
Key points
- Discount is the reduction in the original price of an item
- List price (marked price) is the original tagged price
- Selling price is the actual price paid after discount
- Discount can be expressed as amount (₱) or percentage (%)
- Types: Trade discount, quantity discount, promotional discount
- Alternative calculation: Selling price = List price × (100% - Discount%)
Definitions
List Price
The original marked price of an item before any discounts
Importance
Starting point for all discount calculations
Discount Rate
The percentage reduction applied to the list price
Importance
Determines how much customers save on their purchase
Selling Price
The final price customers pay after discount is applied
Importance
The actual amount received by the seller
Common mistakes
- Confusing discount amount with discount percentage
- Using selling price instead of list price in percentage calculations
- Forgetting to convert percentage to decimal form in calculations
- Not distinguishing between different types of discounts in complex problems
Interest Problems (Simple and Compound)
Formulas
Simple Interest: I = Prt
Example
₱60,000 at 10% for 9 months. I = 60,000 × 0.10 × 0.75 = ₱4,500
Variables
I (interest), P (principal), r (rate as decimal), t (time in years)
Application
Calculating interest earned or paid on loans and investments
Future Value (Simple): F = P(1 + rt)
Example
₱2M at 0.5% for 5 years. F = 2,000,000(1 + 0.005 × 5) = ₱2,050,000
Variables
F (future value), P (principal), r (rate), t (time)
Application
Finding total amount after simple interest
Future Value (Compound): F = P(1 + r)ᵗ
Example
₱15,000 at 3% for 3 years. F = 15,000(1.03)³ = ₱16,390.91
Variables
F (future value), P (principal), r (rate), t (time)
Application
Calculating compound interest growth
Exam tips
- Always convert percentage to decimal and time to years before calculating
- For simple interest: I = Prt, then F = P + I
- For compound interest: F = P(1 + r)ᵗ, then I = F - P
- Remember: Simple interest grows linearly, compound interest grows exponentially
- Check units: P and F in pesos, r as decimal, t in years, I in pesos
Key points
- Simple interest is calculated only on the principal amount
- Compound interest is calculated on principal plus accumulated interest
- Principal (P) is the original amount borrowed or invested
- Rate (r) is the annual interest rate (convert % to decimal)
- Time (t) must be expressed in years
- Maturity/Future value is the total amount after interest
- Simple interest: I = Prt, Future value: F = P + I or F = P(1 + rt)
- Compound interest: F = P(1 + r)ᵗ, Interest: I = F - P
Definitions
Principal
The original amount of money borrowed or invested
Importance
Base amount on which all interest calculations are made
Simple Interest
Interest calculated only on the principal amount throughout the investment period
Importance
Common in basic loans and simple investment scenarios
Compound Interest
Interest calculated on principal plus previously earned interest
Importance
Reflects real-world investment growth and compound growth scenarios
Maturity Value
Total amount (principal + interest) at the end of the investment period
Importance
Final amount received by investor or paid by borrower
Common mistakes
- Forgetting to convert percentage rates to decimal form (divide by 100)
- Not converting months to years (divide months by 12)
- Using simple interest formula when compound interest is required
- Confusing interest amount with total maturity value
- Incorrect calculation of compound interest exponents
Connections
- Speed-distance-time problems connect to physics concepts of motion and velocity
- Interest calculations are fundamental to banking, finance, and economics
- Discount problems relate to business mathematics and consumer economics
- Age problems develop logical reasoning and equation-solving skills used in algebra
- All problem types require unit conversion skills essential in science and engineering
- These concepts appear in various forms across Civil Service, UPCAT, and professional examinations
Quick review questions
A car travels 150 km in 2.5 hours. What is its speed in km/hr?
Answer
60 km/hr
Speed = Distance ÷ Time = 150 ÷ 2.5 = 60 km/hr
If a shirt originally costs ₱800 and is sold for ₱640, what is the discount percentage?
Answer
20%
Discount = 800 - 640 = ₱160. Discount % = (160 ÷ 800) × 100 = 20%
John is twice as old as Mary. In 5 years, John will be 35. How old is Mary now?
Answer
15 years old
John's present age = 35 - 5 = 30. Since John is twice Mary's age: Mary = 30 ÷ 2 = 15
What simple interest is earned on ₱10,000 at 6% per year for 8 months?
Answer
₱400
I = Prt = 10,000 × 0.06 × (8/12) = 10,000 × 0.06 × 0.667 = ₱400
A person travels 60 km at 40 km/hr, then 80 km at 50 km/hr. What is the average speed?
Answer
44.44 km/hr
Time1 = 60/40 = 1.5 hr, Time2 = 80/50 = 1.6 hr. Average = (60+80)/(1.5+1.6) = 140/3.1 = 45.16 km/hr
Diagrams
Word Problems Overview - Main Concepts
Comprehensive mind map showing all major concepts in word problems, their relationships, and key formulas
Problem-Solving Strategy Flowchart
Step-by-step flowchart for systematically approaching and solving different types of word problems
Speed-Distance-Time Problem Solving Process
Detailed flowchart showing decision process and calculations for speed, distance, and time problems including average speed scenarios
Interest Calculation Decision Tree
Decision tree for determining which interest formulas to use and how to approach different types of interest calculations
Exam strategy
For word problems, always follow the systematic approach: (1) Read carefully and identify what's given and what's asked, (2) Choose appropriate formulas, (3) Set up equations or calculations step by step, (4) Solve systematically, (5) Check your answer for reasonableness. Create tables for complex problems involving multiple variables or time periods. Practice converting units and working with percentages in decimal form. Remember that most exam questions test your ability to translate word problems into mathematical expressions, so focus on understanding the relationships described in the problems rather than just memorizing formulas.