PSAT Algebra — writing the equation before solving it
Almost nothing in the Algebra domain of the PSAT is hard as algebra. The equations are linear, the numbers are small, and a calculator is allowed on the whole section. What the questions test is the step before any of that: turning a described situation into an equation, and knowing which number in it is charged once and which is charged per unit.
This material is twelve of those, in the five shapes the domain uses. Three ask for a single unknown from one linear equation — a booking fee and a rate per guest, a call-out fee and an hourly rate, a fixed late fee and a charge per day. In each one the distractors are the results of the two mistakes that actually happen: forgetting the fixed part altogether, and adding it where it should be subtracted. Three are systems of two equations, one of which asks not for either unknown but for their sum, which falls out of adding the two equations without solving for anything.
Two are inequalities, and both turn on the same small question the wording settles: whether the number at the limit is itself allowed. "No more than two thousand pounds" includes two thousand, so forty packages fit exactly; "at least eighty-five points" includes eighty-five, so seventeen assignments are enough. The wrong options in these are not miscalculations but misreadings of that boundary, or perfectly valid answers that are simply not the greatest or the least.
Two ask what a part of a linear model stands for in the situation it describes — the constant in a rental cost, the coefficient in a leaking tank — and two are about equivalent forms: rearranging the perimeter formula for one of its letters, and identifying which expression equals another once the parentheses are opened.
Every explanation works the arithmetic through in prose rather than asserting the answer, and names the mistake behind each wrong option, so a student who chose one can see exactly where their number came from. There are no graphs, and the only drawing is a labeled rectangle beside the perimeter and area formulas; otherwise every quantity is stated in words or in symbols inside the question itself, which is a real limitation of a text-only bank and is why the category page recommends the College Board's own practice for questions that depend on reading a drawn graph. All sums of money are written in words, and every situation here was invented for the purpose.
- Turn a described situation into a linear equation before doing any arithmetic
- Tell the part of a charge that is paid once from the part that is paid per unit
- Solve a system of two linear equations by substitution or by elimination, and choose whichever is quicker
- Find a combination such as the sum of two unknowns by adding the equations, without solving for either
- Read a stated limit as an inequality, and decide whether the number at the limit is itself allowed
- Say what a constant term and a coefficient mean in the situation a model describes
- Rearrange a formula for a named letter by undoing its operations in reverse order
- Recognize the mistakes that produce each wrong answer: the fee ignored, the fee added, the two numbers swapped
Every situation here was written for this exercise, and no two questions share one. Among them: a caterer charging a booking fee of one hundred twenty dollars and thirty dollars a guest against a budget of seven hundred twenty; a technician's bill of two hundred twenty dollars made of a call-out fee and an hourly rate; ten items bought for twenty-four dollars where notebooks cost three and pens two; thirty heads and eighty legs in one farmyard; the pair of equations $3x + 2y = 18$ and $2x + 3y = 12$, where the question asks for $x + y$; a van limited to two thousand pounds carrying packages of fifty; a course awarding five points an assignment where eighty-five are needed to pass; and the perimeter formula $P = 2l + 2w$ to be rearranged for the length. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not a test center.
Sample question
The perimeter of a rectangle is given by $P = 2l + 2w$, where $l$ is its length and $w$ its width. Which of these gives $l$ in terms of $P$ and $w$?
- $l = \frac{P - w}{2}$
- $l = \frac{P - 2w}{2}$
- $l = \frac{P + 2w}{2}$
- $l = P - 2w$
See the answer
$l = \frac{P - 2w}{2}$
Start from $P = 2l + 2w$ and take the term without $l$ to the other side: $P - 2w = 2l$. Now divide both sides by $2$, which gives $l = \frac{P - 2w}{2}$. The option $\frac{P + 2w}{2}$ loses the sign when moving $2w$ across, adding where it should subtract. The option $P - 2w$ never divides at all, so it leaves $2l$ where $l$ should be and comes out twice too large. The option $\frac{P - w}{2}$ subtracts one width instead of two, forgetting that a rectangle has two of them; try it on a square of side one, where $P$ is four, and it gives a length of one and a half.
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