7.4 Algebra Word-Problem Translation
Key Takeaways
- GED algebra word problems test modeling first and calculation second, so define the variable before you compute anything.
- Define the variable WITH units so the equation answers the exact question the item asks.
- Comparison words each control a specific operation: 'less than' reverses subtraction order, 'at least' means >=, 'twice' means multiply by 2, and 'total' means add.
- Let the structure pick the model: one unknown becomes an equation, a limit becomes an inequality, two unknowns with two facts become a system, and area or product situations become quadratics.
- After solving, check whether the answer can physically exist; negative time, negative length, or a fractional person is usually rejected.
GED Focus: Modeling Before Arithmetic
The GED Mathematical Reasoning test wraps algebra inside everyday situations: paychecks, discounts, phone plans, ticket sales, mileage, and construction materials. The targets include writing expressions, equations, inequalities, systems, and quadratics that represent a context. Your first job is not calculation; it is translation. Most word-problem points are won or lost in setting up the correct model, not in the final arithmetic that the TI-30XS MultiView calculator can handle for you.
Expect mixed answer formats: some items are multiple choice, others are fill-in-the-blank where you type a number, and a few are drag-and-drop where you build the equation itself. In every case, a correct model is the gateway, so resist the urge to compute before you have written the relationship in symbols.
The Four-Step Translation Process
Use the same routine every time so timed pressure does not break your method.
- Define the variable with units (let m = number of months).
- Identify the total, comparison, rate, or starting value in the sentence.
- Write the expression, equation, inequality, or system.
- Solve, then check the answer against the original sentence.
| Words in problem | Algebra meaning | Example |
|---|---|---|
| Total, sum, combined | Add the parts | adult + child tickets |
| Difference, less than | Subtract in 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 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 and the monthly cost is 18m, so write 30 + 18m = 174. Subtract 30: 18m = 144. Divide by 18: m = 8. The member paid for 8 months. Check: 30 + 18(8) = 30 + 144 = 174. The biggest trap here is treating the $30 fee as a per-month rate; it is a one-time starting value, so it is added once, not multiplied by m.
Worked Examples: Inequality and System
Budget inequality. A student has no more than $65 for notebooks. Each costs $4.50 and shipping is $6 once. Let n = number of notebooks. The model is 4.50n + 6 <= 65. Subtract 6: 4.50n <= 59. Divide by 4.50: n <= 13.11. Because notebooks are whole, the student can buy at most 13. You must round DOWN here even though 13.11 rounds up normally, since 14 notebooks would exceed the budget.
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 and s = student tickets. The count equation is a + s = 40 and the money equation is 9a + 5s = 280. Substitute s = 40 - a: 9a + 5(40 - a) = 280, so 9a + 200 - 5a = 280, giving 4a = 80 and a = 20. Then s = 20.
Choosing the Model and Checking Reasonableness
Let the situation pick the form:
- One unknown total becomes a one-variable equation.
- A limit, budget, or requirement becomes an inequality.
- Two unknown quantities with two separate facts become a system.
- A rectangular area, a falling object, or a product where x multiplies another expression containing x becomes a quadratic.
Before solving, estimate the answer size. If a bill is $82 after a $10 fee at $12 per month, the answer should be near 6 months, not 60; estimation catches misplaced decimals and reversed operations. After solving, always ask: what does this number mean, and can it exist? A negative count, a fractional person, or a negative length is rejected even when it is algebraically valid. A clear model also prevents the classic reversal of "5 less than a number" into 5 - x instead of the correct x - 5.
Percent and Consecutive-Number Models
Two word-problem types recur on the GED. For percent of change, write the increase or decrease as a multiplier: a 20 percent raise on salary s is 1.20s, and a 15 percent discount on price p is 0.85p. For consecutive integers, let the first be x, the next x + 1, and the next x + 2; consecutive even or odd integers step by 2, so x, x + 2, x + 4. If three consecutive integers sum to 72, then x + (x + 1) + (x + 2) = 72, giving 3x + 3 = 72 and x = 23, so the integers are 23, 24, and 25.
Another recurring type is the mixture or weighted total, such as combining a $9 item and a $5 item to a known total, which leads to the ticket-style system shown earlier. Recognizing these patterns turns a wordy prompt into a single clean equation quickly.
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?
A worker earns $15 per hour and wants to earn at least $480 in a week. Which inequality finds the hours h needed?