🔢 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

If Marco loses 9 kilograms, he will weigh three times as much as his brother. Together they now weigh 277 kilograms. What is Marco's present weight, in kilograms?

Medium

Last year a town council consisting of only Green and Liberal members had 18 more Green members than Liberal members. This year the council has the same number of members as last year, but it has 5 fewer Green members than last year. If this year the number of Green members is $\frac{5}{8}$ of the number of council members, how many members does the council have this year?

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