Word Problems — Speed/Distance/Age, Discount & Interest - Cheat Sheet
Your last-minute revision companion for mastering word problems in speed, distance, time, age, discount, and interest calculations — essential for CSE, UPCAT, and other entrance exams.
Speed, Distance, and Time
Formulas
Speed = Distance ÷ Time
Meaning
Speed (km/h or m/s), Distance (km or m), Time (hours or seconds)
Watch out
Check units! Convert km/h to m/s by multiplying by 5/18
When to use
When finding how fast something moves
Distance = Speed × Time
Meaning
Distance (km or m), Speed (km/h or m/s), Time (hours or seconds)
Watch out
Ensure speed and time units match (hours with hours, seconds with seconds)
When to use
When finding how far something traveled
Time = Distance ÷ Speed
Meaning
Time (hours or seconds), Distance (km or m), Speed (km/h or m/s)
Watch out
Convert minutes to hours by dividing by 60, not multiplying
When to use
When finding how long a journey takes
Average Speed = Total Distance ÷ Total Time
Meaning
For round trips or multiple segments
Watch out
Never just average the two speeds — must use total distance over total time
When to use
When different speeds are used for different parts of journey
Common values
Value
5 km/h
Symbol
v
Quantity
Walking speed
Value
10-15 km/h
Symbol
v
Quantity
Running speed
Value
40-60 km/h
Symbol
v
Quantity
Car speed (city)
Important facts
- 1 km/h = 5/18 m/s (multiply by 5/18)
- 1 m/s = 18/5 km/h (multiply by 18/5)
- 1 hour = 60 minutes = 3600 seconds
- When solving round trips, calculate time for each segment separately
- Speed is always positive in physics problems
Key definitions
Speed
Example
60 km/h means 60 km in 1 hour
Distance covered per unit time
Average Speed
Example
100 km in 2 hours = 50 km/h average
Total distance divided by total time for entire journey
Diagrams to know
- Speed-Distance-Time triangle
- Unit conversion charts
Age Problems
Formulas
Present Age = x, Age after n years = x + n
Meaning
x = current age, n = number of years in future
Watch out
Add for future ages, subtract for past ages
When to use
When dealing with future ages
Present Age = x, Age n years ago = x - n
Meaning
x = current age, n = number of years in past
Watch out
Don't mix up past and future — ago means subtract
When to use
When dealing with past ages
Ratio Ages: a:b = ax:bx
Meaning
Ages in ratio a:b are ax and bx where x is common factor
Watch out
Both ages must use same multiplier x
When to use
When ages are given in ratio form
Important facts
- Set up equations for each time period mentioned
- Use same variable for same person across all time periods
- Age difference between two people remains constant over time
- If A is n years older than B now, A will always be n years older than B
- Check your answer by substituting back into original conditions
Key definitions
Present Age
Example
If John is 20 now, his present age is 20
Current age at the time of solving
Age Ratio
Example
3:2 means first person is 1.5 times older
Proportion of one age to another
Diagrams to know
- Age timeline showing past, present, and future
- Age relationship tables
Discount Problems
Formulas
Discount = List Price - Selling Price
Meaning
Discount (amount saved), List Price (original price), Selling Price (final price)
Watch out
List price is always higher than selling price
When to use
When finding the peso amount of discount
Discount % = (Discount ÷ List Price) × 100
Meaning
Discount % (percentage), Discount (peso amount), List Price (original price)
Watch out
Divide by list price, not selling price
When to use
When finding percentage discount
Selling Price = List Price × (100% - Discount%)
Meaning
Alternative way to find selling price directly
Watch out
Convert percentage to decimal (20% = 0.20)
When to use
When discount percentage is given
Discount Amount = List Price × Discount Rate
Meaning
Direct calculation of discount in pesos
Watch out
Convert percentage to decimal before multiplying
When to use
When discount rate is given as percentage
Common values
Value
10%, 15%, 20%, 25%, 50%
Symbol
d%
Quantity
Common discount rates
Important facts
- Discount is always calculated from the original list price
- Selling Price = List Price - Discount Amount
- A 20% discount means you pay 80% of the original price
- Trade discount is for retailers, not end customers
- Quantity discount applies when buying in bulk
Key definitions
List Price
Example
₱1,500 marked on price tag
Original marked price before any discount
Selling Price
Example
₱1,200 after 20% discount
Final price after discount is applied
Discount Rate
Example
20% off means 0.20 discount rate
Percentage reduction from list price
Diagrams to know
- Price breakdown showing list price, discount, and selling price
- Percentage circle showing discount portion
Simple Interest
Formulas
I = PRT
Meaning
Interest (₱), Principal (₱), Rate (decimal), Time (years)
Watch out
Convert percentage rate to decimal (5% = 0.05)
When to use
Basic simple interest calculation
P = I ÷ (RT)
Meaning
Finding principal when interest, rate, and time are known
Watch out
Time must be in years, convert months by dividing by 12
When to use
When principal is unknown
R = I ÷ (PT)
Meaning
Finding rate when interest, principal, and time are known
Watch out
Result will be in decimal form, multiply by 100 for percentage
When to use
When interest rate is unknown
T = I ÷ (PR)
Meaning
Finding time when interest, principal, and rate are known
Watch out
Result will be in years, multiply by 12 for months
When to use
When time period is unknown
F = P + I or F = P(1 + RT)
Meaning
Future Value (maturity value), Principal, Interest or combined formula
Watch out
Don't forget to add principal to interest for total amount
When to use
When finding total amount after interest
Common values
Value
0.5% - 2% annually
Symbol
r
Quantity
Typical savings rates
Value
5% - 15% annually
Symbol
r
Quantity
Typical loan rates
Important facts
- Simple interest is calculated only on the original principal
- Time must be converted to years (months ÷ 12)
- Rate must be in decimal form for calculations
- Interest earned each year is the same in simple interest
- Maturity value = Principal + Total Interest
Key definitions
Principal
Example
₱10,000 initial deposit
Original amount invested or borrowed
Simple Interest
Example
₱500 interest on ₱10,000 at 5% for 1 year
Interest calculated only on principal amount
Maturity Value
Example
₱10,500 after 1 year (₱10,000 + ₱500)
Principal plus accumulated interest
Interest Rate
Example
5% per year means 0.05 in calculations
Annual percentage charged or earned
Diagrams to know
- Simple interest timeline showing constant interest per year
- Principal-Interest-Maturity value breakdown
Compound Interest
Formulas
F = P(1 + r)^t
Meaning
Future Value, Principal, Rate (decimal), Time (years)
Watch out
Use parentheses correctly, calculate (1 + r) first, then raise to power t
When to use
Finding maturity value with compound interest
P = F ÷ (1 + r)^t
Meaning
Finding principal when future value is known
Watch out
This is the reverse of compound interest formula
When to use
Present value calculations
Ic = F - P
Meaning
Compound Interest = Future Value - Principal
Watch out
Don't confuse with simple interest calculation
When to use
Finding the interest earned
Important facts
- Compound interest grows faster than simple interest
- Interest is calculated on increasing principal each year
- Future value includes both principal and all accumulated interest
- Rate must be in decimal form (5% = 0.05)
- Time must be in years for annual compounding
Key definitions
Compound Interest
Example
Interest earns interest each year
Interest calculated on principal plus accumulated interest
Compounding
Example
Year 1: ₱100 becomes ₱105, Year 2: ₱105 becomes ₱110.25
Process where interest is added to principal for next calculation
Diagrams to know
- Compound vs simple interest growth comparison
- Year-by-year compound interest breakdown
Must remember
- Speed = Distance ÷ Time (remember the triangle: cover what you're solving for)
- Average speed = Total distance ÷ Total time (never average the speeds directly)
- Age difference between people stays constant over time
- Discount % = (Discount amount ÷ List price) × 100
- Simple Interest: I = PRT (P = principal, R = rate as decimal, T = time in years)
- Compound Interest: F = P(1 + r)^t (don't forget the exponent)
- Convert months to years by dividing by 12
- Convert percentage to decimal by dividing by 100
- Always check units match before calculating
- Selling price = List price - Discount amount
Last minute tips
- Draw timelines for age problems to visualize relationships clearly
- For speed problems, write D-S-T in a triangle and cover what you're solving for
- In discount problems, identify which price is the 'base' for percentage calculations
- For compound interest, use calculator for (1 + r)^t — don't try to do large exponents mentally
- Always substitute your answer back into the original problem to verify it makes sense
Comparison tables
Simple vs Compound Interest
| Feature | Simple Interest | Compound Interest |
|---|---|---|
| Interest calculated on | Principal only | Principal + accumulated interest |
| Formula | I = PRT | F = P(1 + r)^t |
| Growth pattern | Linear (constant) | Exponential (accelerating) |
| Interest per year | Same each year | Increases each year |
Time Conversions
| Unit | Convert to Years | Convert to Months | Convert to Days |
|---|---|---|---|
| Months | ÷ 12 | × 1 | × 30 |
| Days | ÷ 365 | ÷ 30 | × 1 |
| Years | × 1 | × 12 | × 365 |
Diagrams
Word Problems Overview
Complete overview of all word problem types covered in this chapter, showing the relationship between different concepts
Problem-Solving Strategy
Step-by-step flowchart for approaching any word problem, helping students identify the problem type and choose the correct formula
Interest Calculation Decision Tree
Decision tree for solving interest problems, helping students choose between simple and compound interest formulas and identify what they need to calculate