USTET Mathematics — Word Problems — Number, Age, Work, Motion, Mixture, InvestmentCheat Sheet
Word Problems — Number, Age, Work, Motion, Mixture, Investment cheat sheet for USTET aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. University of Santo Tomas's most-tested concepts, all in one place.
Exam context
The University of Santo Tomas Entrance Test is conducted by University of Santo Tomas and is scheduled for Early Q4 2026. The Mathematics subtest is marked as "Core section" in the official pattern, and Word Problems — Number, Age, Work, Motion, Mixture, Investment appears in position 4th of 9 in the USTET Mathematics review rotation. Passing mark: Competitive overall score. Recent USTET 2026 papers have drawn roughly a meaningful share of questions from this subject.
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
Sections
Section Title
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
Term
READ
Example
Identify given information and what needs to be found
Definition
Thoroughly understand what the problem is asking for
Term
REPRESENT
Example
Let x = the number, Let t = time in hours
Definition
Assign variables to unknown quantities
Term
RELATE
Example
Sum of two numbers is 50 → x + y = 50
Definition
Connect variables using problem relationships
Section Title
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
Term
Addition Phrases
Example
4 more than x → x + 4
Definition
plus, more than, increased by, sum of, when added
Term
Subtraction Phrases
Example
4 less than x → x - 4
Definition
minus, less than, subtracted from, decreased by, difference
Term
Multiplication Phrases
Example
twice x → 2x, half of x → x/2
Definition
times, product of, of, twice, thrice, half
Term
Division Phrases
Example
x divided by 4 → x/4
Definition
divided by, quotient of, per
Formulas
Formula
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
Formula
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
Formula
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
Section Title
Number Problems
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
Term
Consecutive Numbers
Example
5, 6, 7, 8 are consecutive integers
Definition
Numbers that follow each other in order without gaps
Term
Digit Problem
Example
In 56: tens digit = 5, ones digit = 6
Definition
Problems involving place value of digits in numbers
Formulas
Formula
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
Section Title
Age Problems
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
Term
Age Table
Example
Present | Past (2 years ago) | Future (3 years from now)
Definition
Organized table showing ages of people at different times
Diagrams To Know
- Age comparison table with present, past, and future columns
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Section Title
Work Problems
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
Term
Work Rate
Example
If job takes 6 hours, rate = 1/6 job per hour
Definition
Fraction of total job completed per unit time
Formulas
Formula
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
Formula
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
Formula
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
Section Title
Motion Problems
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
Term
Relative Speed
Example
Car at 60 kph meets car at 40 kph → relative speed = 100 kph
Definition
Combined speed when objects move in opposite directions
Formulas
Formula
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
Formula
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
Section Title
Mixture Problems
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
Term
Solution
Example
Saline solution = salt (solute) + water (solvent)
Definition
Homogeneous mixture of solute (dissolved substance) and solvent
Term
Concentration
Example
20% salt solution has 20% salt, 80% water
Definition
Percentage of solute in the total solution
Term
Pure substance
Example
Pure alcohol = 100% alcohol solution
Definition
100% concentration of the desired substance
Formulas
Formula
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
Formula
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
Section Title
Investment Problems
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
Term
Principal
Example
Invest ₱10,000 → Principal = ₱10,000
Definition
Original amount of money invested
Term
Interest Rate
Example
6% annual rate → 6% return per year
Definition
Percentage return per time period
Term
Simple Interest
Example
₱1000 at 5% for 2 years → Interest = ₱1000 × 0.05 × 2 = ₱100
Definition
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
Rows
Values
- Work = Rate × Time
- Distance = Speed × Time
Property
Main Formula
Values
- Fraction of job per unit time
- Speed in distance per unit time
Property
Rate Definition
Values
- Add individual rates
- Add speeds (opposite direction)
Property
Combined Rates
Values
- Always 1 (complete job)
- Given total distance
Property
Total Quantity
Values
- Two people work together
- Two objects meet or chase
Property
Common Setup
Columns
- Aspect
- Work Problems
- Motion Problems
Table Title
Work vs Motion Problems
Rows
Values
- x, x+1, x+2, ...
- 1
- 7, 8, 9, 10
Property
Consecutive integers
Values
- x, x+2, x+4, ...
- 2
- 7, 9, 11, 13
Property
Consecutive odd
Values
- x, x+2, x+4, ...
- 2
- 8, 10, 12, 14
Property
Consecutive even
Columns
- Type
- Pattern
- Difference
- Example
Table Title
Consecutive Number Types
Previous chapter
Algebra — Sets, Exponents, Radicals, Polynomials & Equations
Next chapter
Geometry — Lines, Angles, Polygons, Triangles & Circles
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