Algebra is the backbone of GMAT Focus Quant. Beyond the questions labelled as algebra, it drives word problems, rates, functions and most Data Sufficiency value questions. The core skills are turning words into equations, manipulating expressions, and knowing when you have enough information to find an answer.
Three ideas cover a lot of ground. First, count equations and unknowns, but do not trust the count blindly: two equations can be the same equation in disguise, and a single equation can be enough when the question asks for a combination such as a + b rather than each variable. Second, look for structure before you expand. Expressions like x + 1/x, or a function applied to itself, usually unlock through a known identity or a substitution rather than brute force. Third, use the constraints. A condition such as “integer” or “positive” can turn one equation with two unknowns into a single possible solution.
Common traps include answering for the wrong expression (computing f(f(5)) when the question asks for f(5)), losing a negative root, and assuming every variable must be found. In Data Sufficiency, decide what the question really needs, then check whether each statement delivers exactly that.
4 practice questions
Data SufficiencyWord ProblemsAlgebraEasy
Question 1
A café sells only two drinks: lattes at 4 dollars each and teas at 3 dollars each. On Monday the café's total revenue from drinks was 90 dollars. How many lattes did the café sell on Monday?
(1) The café sold a total of 25 drinks on Monday.
(2) The café sold more lattes than teas on Monday.
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: A
Key Concept or Approach
Let $L$ = lattes and $T$ = teas, non-negative integers with $4L + 3T = 90$. A second independent linear equation fixes $L$; an inequality usually leaves several integer solutions.
Step 2: Statement (2): $L > T$. Integer solutions of $4L + 3T = 90$ include $(L, T) = (15, 10), (18, 6), (21, 2)$, all with $L > T$. Two different values of $L$, so not sufficient.
Step 3: Statement (1) alone answers the question; statement (2) alone does not.
Final Answer
Answer: A
Common trap: C: students assume a single equation plus an inequality is needed together, not noticing (1) already gives two linear equations.
Problem SolvingFunctions and Custom CharactersAlgebraHard
Question 2
The function $f$ is defined by $f(x) = ax + b$, where $a$ and $b$ are constants and $a > 0$. If $f(f(x)) = 9x - 8$ for all $x$, what is $f(5)$?
A$13$
B$17$
C$23$
D$29$
E$37$
Show answer & explanation
Correct answer: A
Key Concept or Approach
Compose the linear function and match coefficients.
Common trap: 37 - that is f(f(5)) = 9(5) - 8, not f(5).
Data SufficiencyFunctions and Custom CharactersAlgebraHard
Question 3
The function $f$ is defined by $f(x) = ax + b$, where $a$ and $b$ are constants. What is the value of $f(4)$?
(1) $f(1) = 3$
(2) $f(2) + f(6) = 14$
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: B
Key Concept or Approach
For a linear function, the value at the midpoint of two inputs equals the average of the values: $f(4) = \frac{f(2) + f(6)}{2}$. You need $4a + b$, not $a$ and $b$ separately.
Step-by-Step Solution
Step 1: Statement (1): $a + b = 3$. $a = 1, b = 2$ gives $f(4) = 6$; $a = 0, b = 3$ gives $f(4) = 3$. Not sufficient.
Step 2: Statement (2): $(2a + b) + (6a + b) = 8a + 2b = 14$, so $4a + b = 7$, i.e. $f(4) = 7$. Sufficient.
Step 3: Only statement (2) is sufficient.
Final Answer
Answer: B
Common trap: C: students think they must solve for $a$ and $b$ separately and so need two equations.
Problem SolvingAlgebraRootsHard
Question 4
If $x + \frac{1}{x} = 3$ and $x > 1$, what is the value of $x^2 - \frac{1}{x^2}$?
A$\sqrt{5}$
B$3\sqrt{5}$
C$7$
D$4\sqrt{5}$
E$9$
Show answer & explanation
Correct answer: B
Key Concept or Approach
Factor as a difference of squares: $x^2 - \frac{1}{x^2} = (x+\frac{1}{x})(x-\frac{1}{x})$.
