7.4 Algebra Word-Problem Translation
Key Takeaways
- GED algebra word problems are usually tests of modeling before they are tests of calculation.
- Define the variable first so the equation matches the question being asked.
- Translate comparison words carefully: "less than," "at least," "twice," and "total" each control a different operation.
- Use units to decide whether an expression, equation, inequality, table, or graph is the best model.
- After solving, test whether the answer is reasonable in the original context.
GED Focus
The GED Mathematical Reasoning test often asks algebra through ordinary situations: paychecks, discounts, phone plans, tickets, mileage, and construction materials. The official targets include writing expressions, equations, inequalities, systems, and quadratics to represent context. Your first job is not calculation. Your first job is modeling.
The Four-Step Translation Process
Use the same process every time.
- Define the variable with units.
- Identify the total, comparison, rate, or starting value.
- Write the expression, equation, inequality, or system.
- Solve and check the answer in the original sentence.
| Words In Problem | Algebra Meaning | Example |
|---|---|---|
| Total, sum, combined | Add parts | adult + child tickets |
| Difference, less than | Subtract in the correct order | 7 less than x is x - 7 |
| Per, each, hourly | Multiply by a rate | 18h dollars |
| At least | >= | score >= 145 |
| No more than | <= | cost <= budget |
| Two quantities unknown | Often a system | adult tickets and student tickets |
Worked Example: One-Variable Equation
A gym charges a $30 joining fee and $18 per month. A member has paid $174 total. How many months has the member paid for?
Let m = number of months. The starting fee is 30. The monthly cost is 18m. Write 30 + 18m = 174. Subtract 30: 18m = 144. Divide by 18: m = 8. The answer is 8 months. Check: 30 + 18(8) = 174.
Worked Example: Inequality From A Budget
A student has no more than $65 for notebooks. Each notebook costs $4.50, and shipping is $6. Let n = number of notebooks.
The model is 4.50n + 6 <= 65. Subtract 6: 4.50n <= 59. Divide by 4.50: n <= 13.11. Since notebooks must be whole, the student can buy at most 13 notebooks.
Worked Example: System From Two Facts
A school sold 40 tickets. Adult tickets cost $9, student tickets cost $5, and the total collected was $280. Let a = adult tickets and s = student tickets.
The count equation is a + s = 40. The money equation is 9a + 5s = 280. Use substitution: s = 40 - a. Then 9a + 5(40 - a) = 280. Simplify: 9a + 200 - 5a = 280, so 4a = 80 and a = 20. Then s = 20. The school sold 20 adult and 20 student tickets.
Choosing The Right Model
A single unknown total usually becomes a one-variable equation. A limit or requirement usually becomes an inequality. Two unknown quantities with two separate facts usually become a system. A rectangular area, falling object, or product involving a changing length can become a quadratic, especially when x is multiplied by x or by another expression containing x.
Practical Check
Before solving, estimate the size of the answer. If a monthly bill is $82 after a $10 fee and the rate is $12 per month, the answer should be around 6 months, not 60. Estimation helps catch misplaced decimals and reversed operations.
On the GED, answer choices can sometimes be checked faster than building every step from scratch, but do not skip the model. A clear model prevents common traps, such as treating a fixed fee as a rate or reversing "5 less than a number." Always finish by asking: What does this number mean, and can it exist in the situation?
A taxi charges $4 to start and $2.75 per mile. Which equation gives the cost c for m miles?
A rectangle is 3 feet longer than it is wide. If w is the width, which expression represents the area?