🔢 Quant

Linear Equations & Word Translation

Translating English into equations, solving one and two unknowns, tiered rates, and the unique/none/infinite trichotomy.

~5h

to master

Translation is the real test

Most GMAT "algebra" questions are word problems: the algebra itself is one or two lines — the skill is converting English into equations without losing a constraint. Build a reflex table:

English	Math
increased by, more than, total of	$+$
decreased by, fewer than, less than	$-$
of, times, product, twice/double/triple	$\times$
per, ratio of, out of, quotient	$\div$
is, was, equals, costs	$=$
what number / how many	the variable

⚠️GMAT Trap

"5 less than $x$ " is $x - 5$ , not $5 - x$ . Subtraction phrases reverse the reading order. "Subtracted from" works the same way: " $3$ subtracted from $y$ " $= y - 3$ .

Solving one and two unknowns

📐Core Rule

Whatever you do to one side, do to the other. Add/subtract anything; multiply/divide by anything nonzero.

For two unknowns, two non-equivalent linear equations pin down a unique solution. Two methods:

Substitution: solve one equation for one variable, push it into the other.
Elimination: scale the equations so one variable's coefficients match, then add or subtract the equations.

Three endings are possible, and the GMAT tests all three:

A clean unique solution (most problems).
You derive something like $0 = 0$ → the equations were the same line: infinitely many solutions.
You derive a contradiction like $0 = 7$ → no solution (parallel lines).

✏️Worked Example

A club collected $1,300 from 50 members; regular dues are $20 and premium dues are $35. How many premium members? Let $p$ = premium count, so regulars are $50 - p$ : $35p + 20(50 - p) = 1300 \Rightarrow 15p = 300 \Rightarrow p = 20$ . One variable, one constraint baked into " $50 - p$ " — using two variables and two equations also works, but costs time.

The same pattern with money split unevenly: .

Tiered and piecewise setups

Many applied problems pay different rates over different ranges — commissions, taxes, billing. Handle each tier separately and add: a 15% commission on the first $500 plus 20% beyond $500 means total sales

S > 500

earns

0.15(500) + 0.20(S - 500)

. The classic mistake is applying the top rate to the whole amount. Drill it: .

💡Exam Tip

Pick one variable, not many. "Ben has 4 more than twice what Ana has" → Ana $= a$ , Ben $= 2a + 4$ . Each extra variable demands an extra equation; experienced solvers encode relationships directly into expressions.

⚡Shortcut

Answer choices are equations too. When options are numbers, test the middle option in the original sentence (not your possibly-wrong equation). If it overshoots, move down; undershoots, move up. Two tests usually finish it.

Checklist

Define variables in writing before forming equations (" $p$ = premium members")
Reversal phrases ("less than", "subtracted from") handled correctly
Tiered rates: split at the boundary, never blend
$0=0$ → infinite; contradiction → none
Verify by substituting back into the words, not the equation

Sample Questions

22 practice questions

Hard

A salesperson earns a commission of 10 percent on the first $800 of weekly sales and 25 percent on all sales beyond $800. In a week her total commission was $455. What were her total sales for the week, in dollars?

Medium

A profit of $70{,}000 is shared by a firm's 3 partners and 8 employees. If each partner receives twice as much as each employee, how much does each partner receive?

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