Word Problems — Number, Age, Work, Motion, Mixture, Investment - Cheat sheet
Your last-minute revision companion for mastering word problems - formulas, strategies, and key patterns for UPCAT success
Problem-Solving Strategy (3 R's + ESP)
Important facts
- EQUATE: Form equations from problem statements
- SOLVE: Use algebraic methods to find variable values
- PROVE: Check answers by substituting back into original conditions
- Always define variables clearly before solving
- Look for key phrases that indicate mathematical operations
Key definitions
READ
Example
Identify given information and what needs to be found
Thoroughly understand what the problem is asking for
REPRESENT
Example
Let x = the number, Let t = time in hours
Assign variables to unknown quantities
RELATE
Example
Sum of two numbers is 50 → x + y = 50
Connect variables using problem relationships
Translation Phrases
Important facts
- For 'less than' and 'subtracted from' - switch the order of numbers
- 'Of' usually means multiplication in word problems
- 'Is', 'was', 'will be', 'equals' indicate the equals sign
- Always check if phrases require switching number order
Key definitions
Addition Phrases
Example
4 more than x → x + 4
plus, more than, increased by, sum of, when added
Subtraction Phrases
Example
4 less than x → x - 4
minus, less than, subtracted from, decreased by, difference
Multiplication Phrases
Example
twice x → 2x, half of x → x/2
times, product of, of, twice, thrice, half
Division Phrases
Example
x divided by 4 → x/4
divided by, quotient of, per
Number Problems
Formulas
Consecutive integers: x, x+1, x+2, x+3, ...
Meaning
x = first integer, each next integer is 1 more
Watch out
Don't confuse with consecutive odd/even integers
When to use
When problem mentions consecutive whole numbers
Consecutive odd/even: x, x+2, x+4, x+6, ...
Meaning
x = first odd/even number, each next is 2 more
Watch out
The difference is 2, not 1
When to use
When problem specifies consecutive odd or even numbers
Two-digit number: 10x + y
Meaning
x = tens digit, y = ones digit
Watch out
Reversed number is 10y + x, not xy
When to use
For digit problems involving place values
Important facts
- Sum of n consecutive integers starting from a: n·a + n(n-1)/2
- For consecutive odd/even integers, difference between terms is 2
- When digits are reversed: 10y + x becomes the new number
- Always check if your answer makes sense in context
Key definitions
Consecutive Numbers
Example
5, 6, 7, 8 are consecutive integers
Numbers that follow each other in order without gaps
Digit Problem
Example
In 56: tens digit = 5, ones digit = 6
Problems involving place value of digits in numbers
Age Problems
Formulas
Present Age ± Years = Future/Past Age
Meaning
Add years for future age, subtract for past age
Watch out
All people age at the same rate - add/subtract same amount
When to use
When comparing ages at different time periods
Important facts
- Create a table with people as rows and time periods as columns
- Age differences between people remain constant over time
- If A is 5 years older than B now, A will always be 5 years older than B
- Assign variables to present ages, then calculate other time periods
Key definitions
Age Table
Example
Present | Past (2 years ago) | Future (3 years from now)
Organized table showing ages of people at different times
Diagrams to know
- Age comparison table with present, past, and future columns
Work Problems
Formulas
Work = Rate × Time
Meaning
Amount of work done = rate of work × time spent
Watch out
Rate is fraction of job done per unit time
When to use
For all work-related problems
Rate = 1/Time to complete job alone
Meaning
If person completes job in n hours, rate = 1/n jobs per hour
Watch out
Rate is always a fraction, not a whole number
When to use
To find individual work rates
Combined Rate = Rate₁ + Rate₂ + ...
Meaning
When working together, add individual rates
Watch out
Only add rates, not times
When to use
When multiple people/machines work together
1/n + 1/m = 1/h
Meaning
n, m = individual times, h = combined time
Watch out
This only works for two workers
When to use
When two people work together
Important facts
- Total work is always considered as 1 (one complete job)
- Work rates add when people work together
- Individual times do NOT add when working together
- Use work table: Work = Rate × Time for each person
Key definitions
Work Rate
Example
If job takes 6 hours, rate = 1/6 job per hour
Fraction of total job completed per unit time
Motion Problems
Formulas
Distance = Velocity × Time (D = VT)
Meaning
D = distance traveled, V = speed, T = time
Watch out
Keep units consistent (km/h with hours, m/s with seconds)
When to use
For all motion and distance problems
Meeting Problems: D₁ + D₂ = Total Distance
Meaning
When objects move toward each other, add distances
Watch out
Both objects travel for the same amount of time
When to use
When two objects start from different points and meet
Overtaking Problems: D₁ = D₂
Meaning
When one object overtakes another, distances are equal
Watch out
Objects may start at different times
When to use
When faster object catches up to slower object
Common values
Value
4-6 kph
Symbol
v
Quantity
Walking speed
Value
15-25 kph
Symbol
v
Quantity
Cycling speed
Value
60-100 kph
Symbol
v
Quantity
Car speed
Important facts
- Use motion table: Distance = Velocity × Time for each object
- Objects moving toward each other: add speeds
- Objects moving in same direction: subtract speeds
- Time is usually the same for all objects in meeting problems
Key definitions
Relative Speed
Example
Car at 60 kph meets car at 40 kph → relative speed = 100 kph
Combined speed when objects move in opposite directions
Mixture Problems
Formulas
Amount of solute = Percentage × Total volume
Meaning
Pure substance amount = concentration × solution volume
Watch out
Convert percentage to decimal (30% = 0.30)
When to use
To find amount of pure substance in solution
Final concentration = Total solute / Total volume
Meaning
New percentage after mixing solutions
Watch out
Add solute amounts and volume amounts separately
When to use
To verify final mixture concentration
Common values
Value
0% solute
Symbol
H₂O
Quantity
Pure water
Value
100% concentration
Symbol
varies
Quantity
Pure substance
Important facts
- Use mixture table: % solution × Volume = Amount of solute
- Total solute = sum of solute from each solution mixed
- Total volume = sum of volumes of all solutions mixed
- Diluting means adding 0% solution (pure solvent)
Key definitions
Solution
Example
Saline solution = salt (solute) + water (solvent)
Homogeneous mixture of solute (dissolved substance) and solvent
Concentration
Example
20% salt solution has 20% salt, 80% water
Percentage of solute in the total solution
Pure substance
Example
Pure alcohol = 100% alcohol solution
100% concentration of the desired substance
Investment Problems
Formulas
Interest = Principal × Rate × Time (I = PRT)
Meaning
I = interest earned, P = amount invested, R = interest rate, T = time
Watch out
Rate must be in decimal form (6% = 0.06)
When to use
For simple interest calculations
Total Amount = Principal + Interest (A = P + I)
Meaning
Final amount after earning interest
Watch out
Don't forget to add principal to interest
When to use
To find total value after investment period
Common values
Value
2-4% annually
Symbol
r
Quantity
Typical savings rate
Value
5-8% annually
Symbol
r
Quantity
Investment rate
Important facts
- Use investment table: Interest = Principal × Rate × Time
- Annual rate for 1 year: Time = 1
- Monthly rate for 1 year: Time = 12
- Convert percentages to decimals in calculations
Key definitions
Principal
Example
Invest ₱10,000 → Principal = ₱10,000
Original amount of money invested
Interest Rate
Example
6% annual rate → 6% return per year
Percentage return per time period
Simple Interest
Example
₱1000 at 5% for 2 years → Interest = ₱1000 × 0.05 × 2 = ₱100
Interest calculated only on the principal amount
Must remember
- ALWAYS define your variables clearly before setting up equations
- READ the problem twice - identify what's given and what's being asked
- For work problems: Rate = 1/(time to complete job alone)
- In motion problems: Distance = Speed × Time, keep units consistent
- Age problems: Create a table with people and time periods
- Mixture problems: Amount of solute = Percentage × Total volume
- Investment problems: I = PRT, convert percentages to decimals
- Consecutive integers differ by 1, consecutive odd/even differ by 2
- ALWAYS check your answer by substituting back into original problem
- Word problems test translation skills - master key phrases and their math meanings
Last minute tips
- Draw tables for work, motion, age, mixture, and investment problems - they organize information clearly
- For 'less than' and 'subtracted from' phrases, always switch the order of the numbers
- In work problems, when people work together, ADD their rates, don't add their individual completion times
- Motion problems: if objects move toward each other, their relative speed is the sum of individual speeds
- Always verify your answer makes sense in the real-world context of the problem
Comparison tables
Work vs Motion Problems
| Aspect | Work Problems | Motion Problems |
|---|---|---|
| Main Formula | Work = Rate × Time | Distance = Speed × Time |
| Rate Definition | Fraction of job per unit time | Speed in distance per unit time |
| Combined Rates | Add individual rates | Add speeds (opposite direction) |
| Total Quantity | Always 1 (complete job) | Given total distance |
| Common Setup | Two people work together | Two objects meet or chase |
Consecutive Number Types
| Type | Pattern | Difference | Example |
|---|---|---|---|
| Consecutive integers | x, x+1, x+2, ... | 1 | 7, 8, 9, 10 |
| Consecutive odd | x, x+2, x+4, ... | 2 | 7, 9, 11, 13 |
| Consecutive even | x, x+2, x+4, ... | 2 | 8, 10, 12, 14 |
Diagrams
Word Problems Strategy Map
Comprehensive overview of word problem types and solution approach using 3 R's + ESP method
Word Problem Solution Flowchart
Step-by-step decision tree for solving different types of word problems systematically
UPCAT Word Problem Distribution
Typical distribution of word problem types in UPCAT mathematics sections