Math · Algebra & word problems
Translating Multi-Step Word Problems
Estimated study time. Take the time you need.
Start here: key ideas
- Define one variable with units.
- Translate relationships rather than isolated keywords.
- Separate fixed amounts from rates.
- Check units, scale, and the original relationship.
What you’ll be able to do
- Define unknowns with units.
- Translate relationships before calculating.
- Validate the result against the question.
Name the unknown, model relationships, solve, and return to the question.
Complete Solving Linear Equations first.
Define before calculating
A variable represents the unknown; define it with a unit, the label describing its measure. Read the entire relationship. A fixed amount occurs once; a rate is per unit. Thus a $10 fee plus $3 per visit is 10+3v.
Build and solve
An equation says two modeled quantities are equal. Write it before arithmetic. For three equal packages plus a $4 fee costing $25, 3p+4=25, then p=7.
Return to the question
A reasonableness check tests whether a result fits the story. Substitute it, inspect its unit, and answer the requested quantity.
| Question | Variable/unit | Equation | Check |
|---|---|---|---|
| Classes | c classes | 18+7c=60 | 18+7(6)=60 |
Worked examples
Worked example 1: A rental is $25 plus $0.40 per mile and totals $61. Then 25+0.40m=61, so m=90 miles.
Worked example 2: An 80-L tank drains 6 L/min until 32 L remains. Write 80−6t=32; therefore t=8 minutes. The answer is time, not liters.
A reliable way to reason
Translation works best by identifying actors and relationships, not isolated clue words. “Three fewer than twice the number” describes a result produced by doubling first and subtracting three: 2x−3. By contrast, “three minus twice the number” is 3−2x. Read the sentence aloud after writing the expression to see whether its order matches.
Units reveal the operation. A rate of $4 per ticket multiplied by t tickets produces dollars; adding a $6 fee is valid because both terms then measure dollars. Dividing dollars by dollars per ticket produces tickets. Write units beside a difficult step as a built-in check. After solving, return to all conditions. A computed 4.6 buses means five buses are required if everyone must ride; 4 buses would not satisfy the condition. A negative time usually signals a modeling or algebra error. Finally, distinguish intermediate quantities. If a question asks for money remaining, finding money spent is necessary but incomplete. Subtract it from the starting amount, state the requested unit, and substitute the result into the original relationship.
Guided reasoning and error checks
Worked comparison: A plan charges $22 monthly plus $0.08 per text; another charges $34 monthly plus $0.04 per text. To find when costs match, let t be texts and write 22+0.08t=34+0.04t. Subtract 0.04t and 22: 0.04t=12, so t=300 texts. Check both plans: each costs $46. Below 300 texts the lower fixed fee is cheaper; above 300 the lower per-text rate is cheaper.
Optional additional practice: consecutive numbers
Worked count: Three consecutive even integers total 72. Let n be the smallest; the next are n+2 and n+4. Then n+(n+2)+(n+4)=72, so 3n+6=72, 3n=66, and n=22. The integers are 22, 24, and 26; their sum is 72. Defining the sequence correctly is what makes the equation match the stated relationship.
Final self-check
Estimation should happen before detailed calculation. If five items cost about $8 each plus a small fee, a total near $40 is plausible; $4 or $400 is not. After the exact equation is solved, compare the result with that estimate. Agreement supports the model, while a large mismatch signals that the rate, fixed amount, or requested quantity may have been misidentified.
Key terms
- variable: A symbol for an unknown quantity.
- unit: A label describing what a number measures.
- equation: A statement that two quantities are equal.
- fixed amount: A quantity independent of the variable.
- rate: A quantity measured per unit.
- reasonableness check: A test that an answer fits facts and scale.
Summary
- Define the unknown before operations.
- Separate fixed and per-unit amounts.
- Answer the requested quantity, not an intermediate result.
- Carry units through the equation and test the result against every condition in the original situation.
Question → variable with units → relationship equation → solve → check.
Sources
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