Algebra and Quantitative Problem Solving

Build Math From the Situation

The GED Mathematical Reasoning test is 115 minutes long and is not a pure calculation contest. It is delivered in two parts: the first 5 questions are answered with the onscreen calculator locked (a short no-calculator block), after which the TI-30XS MultiView onscreen calculator unlocks for the rest. About 45% of the items are quantitative problem solving (number operations, ratios, percents, measurement) and about 55% are algebraic problem solving (expressions, equations, inequalities, functions). You need a score of at least 145 out of 200 to pass.

For this section, focus on the bridge between arithmetic and algebra: figure out what the problem is asking, assign a variable when something is unknown, and use the units to confirm the answer makes sense before you commit.

The Core Translation Table

Rational-number work appears everywhere: decimals, fractions, signed numbers, square roots, exponents, and scientific notation inside real settings. The wording tells you the operation. Per means divide (cost per ounce = cost / ounces). Of usually means multiply (20% of 350 = 0.20 x 350 = 70). For percent decrease, the base is the original larger amount, never the smaller amount after the drop. A common trap divides by the new value and reports a wrong percent.

Algebra That Actually Scores

The most reliable algebra routine is short and strict. First, name the unknown with a variable. Second, write the relationship in symbols. Third, solve using inverse operations while keeping both sides balanced. Fourth, substitute the answer back into the sentence. This last step kills answer choices that are numerically close but contextually wrong.

Worked example. A repair company charges a 28 dollar visit fee plus 16 dollars per quarter hour. The total bill is 92 dollars. Let q be the number of quarter-hour units: 28 + 16q = 92. Subtract 28 to get 16q = 64, then divide by 16 to get q = 4. Because each unit is one quarter hour, the repair lasted 4 quarter hours = 1 full hour. A distractor will offer "4 hours" to catch readers who skip the unit.

Inequality example. A tutoring budget is at most 75 dollars. There is a 9 dollar registration fee and each session costs 18 dollars: 9 + 18s <= 75. Subtract 9, then divide by 18, giving s <= 3.67. The real answer is at most 3 full sessions, because partial sessions are not purchasable unless the prompt allows them.

Process for Quantitative Word Problems

1. Underline the actual task: final cost, original amount, unit rate, number of items, or a comparison.
2. List the known values with their units (dollars, feet, hours, percent).
3. Decide whether the relationship is additive, multiplicative, proportional, or variable-based.
4. Estimate the answer first, then use the calculator only to confirm.
5. Reject any choice with impossible units, the wrong direction, or an unreasonable scale.

Graphs, Slope, and Functions

GED algebra often appears as a graph, table, or rule rather than a tidy equation. Slope is the rate of change: vertical change divided by horizontal change. In a table, subtract two output values and divide by the matching change in input. In the form y = mx + b, m is the slope and b is the y-intercept (the output when x = 0). A positive slope rises left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has undefined slope.

A function adds one rule: each input maps to exactly one output. A table where x = 2 gives both 5 and 9 is not a function; a graph that a vertical line can cross twice is not a function. To evaluate f(4), replace every input variable with 4 and simplify carefully, respecting order of operations.

Final Check

A passing Math answer survives three questions: Did I model the situation correctly, did I calculate accurately, and does the result fit the units and restrictions? If any of the three fails, recompute before choosing. Spend the no-calculator first 5 items showing your arithmetic on the scratch board so a small slip does not cost a point you cannot recover with the calculator later.