First Find the Information Structure
Suppose a problem says:
A rectangular garden is \(12\text{ m}\) long and has an area of \(84\text{ m}^2\). What is its width?
Before solving anything, sort the information:
\[ \ell=12\text{ m}, \] \[ A=84\text{ m}^2, \] \[ w=?. \]We also know the relationship
\[ A=\ell w. \]Now the problem is no longer a paragraph. It is a small network of known quantities, an unknown quantity, and a relationship connecting them.
\[ \boxed{ \text{problem statement} \to \text{knowns + unknowns + constraints + relationships} } \]What Is Known Information?
Known information is information the problem supplies or information that follows directly from accepted definitions and stated conditions.
Known information may include:
- numerical values,
- measurements,
- counts,
- equations,
- geometric relationships,
- logical conditions,
- units,
- definitions.
For example, if a right triangle has legs
\[ a=6, \qquad b=8, \]then both leg lengths are known information.
What Is Unknown Information?
Unknown information is what the problem asks us to determine or what must be found before the requested result can be obtained.
An unknown may be:
- a number,
- a length,
- a rate,
- an angle,
- a probability,
- a function,
- a set,
- a parameter,
- an entire mathematical object.
The unknown should be named clearly.
Instead of writing only
\[ x, \]say what \(x\) represents:
\[ x=\text{width of the garden}. \]Givens and Data
A given is information explicitly supplied in the problem.
For example:
“A car travels \(180\text{ miles}\) in \(3\text{ hours}\). Find its average speed.”
The givens are:
\[ d=180\text{ mi}, \] \[ t=3\text{ h}. \]The requested unknown is:
\[ v=?. \]The known relationship is:
\[ v=\frac{d}{t}. \]Known Relationships
A problem may provide values without explicitly stating the equation that connects them.
That relationship may come from prior mathematical knowledge.
Example
If the problem gives the radius of a circle and asks for area, the relevant relationship is
\[ A=\pi r^2. \]So the information inventory is not only:
\[ r=\text{known}, \qquad A=\text{unknown}. \]It also includes the bridge:
\[ \boxed{A=\pi r^2}. \]Without a relationship between the knowns and unknowns, there is no path to a solution.
Constraints and Conditions
A constraint restricts which values are possible.
Examples:
\[ x>0, \] \[ n\in\mathbb{Z}, \] \[ 0\leq p\leq1, \] \[ a+b>c \]for sides of a nondegenerate triangle.
Constraints are part of the problem's information structure even when they are not the quantity being solved for.
They help reject impossible answers.
Units Are Part of the Information
A value and its unit form one quantity.
The statements
\[ 5\text{ m} \]and
\[ 5\text{ s} \]contain the same numerical value but represent completely different quantities.
When organizing a problem, keep the unit attached:
\[ d=120\text{ km}, \qquad t=2\text{ h}. \]This makes the expected unit of the unknown visible:
\[ v=\frac{d}{t} \]must have units
\[ \frac{\text{km}}{\text{h}}. \]Hidden and Implied Information
Some information is not written as a separate sentence but follows from the meaning of the problem.
Example: Rectangle
If a figure is identified as a rectangle, then we know:
- opposite sides are equal,
- all interior angles are right angles,
- area satisfies \(A=\ell w\),
- perimeter satisfies \(P=2\ell+2w\).
These facts are not guesses. They follow from the definition and known properties of a rectangle.
But be careful: implied information must come from valid mathematical or contextual assumptions, not from what “usually happens.”
Relevant and Irrelevant Information
Not every number in a problem must be used.
Example
A rectangular field is \(50\text{ m}\) long and \(30\text{ m}\) wide. It was purchased in 2018 for \$80,000. Find its area.
Relevant:
\[ 50\text{ m}, \qquad 30\text{ m}. \]Irrelevant to the area calculation:
- 2018,
- \$80,000.
A common problem-solving mistake is assuming that every supplied number must appear in the calculation.
Assigning Symbols to Unknowns
Symbols make unknown information easier to manipulate.
If the unknown is a width, write:
\[ w=\text{width}. \]If two unknowns are needed, distinguish them:
\[ x=\text{adult tickets}, \] \[ y=\text{student tickets}. \]Good symbol choices reduce memory demands and make equations easier to read.
Organizing Information Visually
A simple organizer can be written as:
| Known | Unknown | Constraint | Relationship |
|---|---|---|---|
| \(\ell=12\text{ m}\) | \(w=?\) | \(w>0\) | \(A=\ell w\) |
| \(A=84\text{ m}^2\) |
Problems with Multiple Unknowns
Some problems require finding more than one quantity.
For example:
\[ x+y=20, \] \[ x-y=4. \]There are two unknowns:
\[ x, \qquad y. \]There are also two independent relationships.
The number of unknowns alone does not determine whether a problem can be solved. We also need enough independent information to determine them.
Derived Information
Sometimes we can calculate new information from the original givens before solving the final question.
Example
A right triangle has legs \(6\) and \(8\).
The hypotenuse is not explicitly given, but it can be derived:
\[ c^2=6^2+8^2, \] \[ c^2=100, \] \[ c=10. \]So \(c=10\) becomes newly known information.
\[ \boxed{ \text{givens} \to \text{derived knowns} \to \text{final unknown} } \]Insufficient Information
Some problems cannot be solved uniquely from the information supplied.
Example
“A rectangle has area \(24\text{ m}^2\). Find its length.”
We know:
\[ \ell w=24. \]But many pairs satisfy this:
\[ (1,24), \quad (2,12), \quad (3,8), \quad (4,6), \]and infinitely many positive real pairs as well.
The problem needs another independent condition to determine one unique length.
Redundant Information
Sometimes the problem provides more information than is strictly necessary.
Redundant information can be useful for:
- verification,
- cross-checking,
- detecting inconsistent data.
Example
A rectangle is said to have
\[ \ell=5, \qquad w=4, \qquad A=20. \]The area is not needed to find the perimeter, but it confirms that the dimensions are mutually consistent because
\[ 5(4)=20. \]Reading Word Problems Structurally
When reading a word problem, use this sequence:
- Read once for the situation.
- Read again for the question.
- Mark the quantities and units.
- Name the unknown.
- Identify constraints.
- Find or recall the relationship connecting the quantities.
- Ignore information that does not affect the requested result.
This converts reading into mathematical modeling.
Worked Examples
Example 1: Speed
A cyclist travels \(45\text{ km}\) in \(1.5\text{ h}\). Find average speed.
Known:
\[ d=45\text{ km}, \] \[ t=1.5\text{ h}. \]Unknown:
\[ v=?. \]Relationship:
\[ v=\frac{d}{t}. \]Solution:
\[ v=\frac{45}{1.5}=30. \] \[ \boxed{v=30\text{ km/h}}. \]Example 2: Rectangle Width
A rectangle has area \(96\text{ cm}^2\) and length \(12\text{ cm}\). Find the width.
Known:
\[ A=96\text{ cm}^2, \qquad \ell=12\text{ cm}. \]Unknown:
\[ w=?. \]Relationship:
\[ A=\ell w. \]Solution:
\[ 96=12w, \] \[ \boxed{w=8\text{ cm}}. \]Example 3: Irrelevant Information
A tank contains \(300\text{ L}\) of water and drains at \(15\text{ L/min}\). The tank was installed in 2024. How long until it is empty?
Relevant:
\[ 300\text{ L}, \qquad 15\text{ L/min}. \]Irrelevant: installation year 2024.
Unknown:
\[ t=?. \]Relationship:
\[ t=\frac{\text{volume}}{\text{rate}}. \] \[ t=\frac{300}{15} = \boxed{20\text{ min}}. \]Example 4: Insufficient Information
A triangle has area \(30\text{ cm}^2\). Find its base.
Using
\[ A=\frac12bh, \]we have
\[ 30=\frac12bh. \]Both \(b\) and \(h\) are unknown. Without another independent condition, the base is not uniquely determined.
\[ \boxed{\text{insufficient information}} \]Example 5: Hidden Information from a Definition
A square has side length \(7\text{ m}\).
Because it is a square, all four sides have the same length.
So the perimeter is
\[ P=4(7) = \boxed{28\text{ m}}. \]The equality of all four sides is implied by the definition of a square.
Example 6: Derived Information
A car travels \(150\text{ miles}\) in \(3\text{ hours}\), then continues for another \(2\text{ hours}\) at the same average rate. How far does it travel in the second part?
First derive the rate:
\[ v=\frac{150}{3}=50\text{ mph}. \]Then use the derived known:
\[ d=vt=50(2)=\boxed{100\text{ miles}}. \]Common Errors
1. Solving before Identifying the Unknown
You may compute a quantity the problem never asked for.
2. Assuming Every Number Must Be Used
Some information is intentionally irrelevant or useful only for checking.
3. Dropping Units
Units often reveal which operations make sense.
4. Treating an Assumption as a Given
Use only stated information, valid definitions, and justified implications.
5. Ignoring Constraints
A numerical solution may violate a required condition such as \(x>0\).
6. Failing to Notice Insufficient Information
Not every problem has enough independent information for a unique answer.
7. Failing to Use Derived Information
Sometimes a useful quantity must be computed before the final unknown can be found.
Practice
Identify Known and Unknown
A circle has radius \(4\text{ cm}\). Find its circumference.
Check Answer
Known:
\[ r=4\text{ cm}. \]Unknown:
\[ C=?. \]Relationship:
\[ C=2\pi r. \]Find the Irrelevant Information
A train travels \(240\text{ miles}\) in \(4\text{ hours}\). It has 8 cars. Find its average speed.
Check Answer
The number of cars is irrelevant to the average-speed calculation.
Check Sufficiency
A rectangle has perimeter \(20\text{ m}\). Can its length be determined uniquely?
Check Answer
No. The equation
\[ 2\ell+2w=20 \]contains two unknowns and allows many positive pairs.
Derived Information
A vehicle travels \(90\text{ km}\) in \(2\text{ h}\). What useful derived quantity can be found before predicting how far it travels in \(5\text{ h}\)?
Check Answer
The rate:
\[ v=\frac{90}{2}=\boxed{45\text{ km/h}}. \]Constraint
If \(x\) represents the length of a physical object, what common constraint is reasonable?
Check Answer
\[ \boxed{x>0} \]for a positive nonzero length, unless the context explicitly allows a degenerate zero-length case.
The Deeper Idea
A mathematical problem is an information system.
Some information is already available:
\[ K=\{\text{known quantities and conditions}\}. \]Some information is sought:
\[ U=\{\text{unknown quantities}\}. \]Relationships connect the two:
\[ R(K,U)=0 \]or in some other mathematical form.
Problem solving then becomes the task of using the relationships and constraints to transform enough known information into the unknown.
\[ \boxed{ \text{knowns} + \text{relationships} + \text{constraints} \longrightarrow \text{unknowns} } \]That is why organizing information is not merely preliminary bookkeeping. It is the beginning of the mathematical model itself.
Summary
- Known information includes givens, definitions, valid implications, units, and stated conditions.
- Unknown information is what must be determined.
- Known relationships connect the givens to the unknowns.
- Constraints restrict which answers are allowed.
- Units are part of the information and should remain attached to quantities.
- Some information is implied by mathematical definitions or valid contextual assumptions.
- Not every supplied number is relevant to the requested result.
- Symbols should be assigned clear meanings when representing unknowns.
- Tables and diagrams can separate knowns, unknowns, constraints, and relationships.
- Problems may contain multiple unknowns and require multiple independent relationships.
- New known information can be derived from earlier givens.
- Some problems are underdetermined and do not contain enough information for a unique solution.
- Redundant information can help verify consistency even when it is not needed to compute the answer.
Before asking “What formula should I use?” ask “What do I actually know, what am I trying to find, and what connects the two?” Once those roles are clear, the strategy is usually much easier to see.
Problem Solving and Verification
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- Problem Solving and Verification — return to the lesson directory.