GMAT Question Patterns: Algebra (Focus Edition)

Question

Algebra Patterns (Focus Edition)

Table of Contents:

Pattern #1: Solve for Variable (23 questions)  
 
 1.1 Variable Isolation (5) 
 1.2 Simple Numeric Solutions (5) 
 1.3 Multiple Variables or Equations (5) 
 1.4 Algebraic Word Problems (3) 
 1.5 Powers and Exponents (covered in Powers/Exponents)

Pattern #2: Algebraic Expressions (8 questions)

2.1 Expression Equivalence (5) 
 2.2 Equation Equivalence (2) 
 2.3 Step-Based Simplification (1)

Pattern #3: Quadratic Equations (13 questions)  
 
 3.1 Finding Roots (5) 
 3.2 Factoring Expressions (5) 
 3.3 Identities & Symmetric Expressions (2) 
 3.4 Word Problems (1) 
 3.5 Polynomials & Higher powers (covered in Powers/Exponents)

Algebra Pattern #1 - Solve for a Variable

Sub-pattern #1: Solve in Terms of Another Variable

Example 1.1.1(Difficulty: Sub 505 Level)

If  b 
eq 1 , what is the value of  a  in terms of  b  if  \frac{ab}{(a-b)} = 1 ?

A.  \frac{b}{(1 - b)}  
B.  \frac{(b - 1)}{b}  
C.  \frac{b}{(b + 1)}  
D.  \frac{(b + 1)}{b}  
E.  \frac{(1 - b)}{b}

Question Discussion Link

Example 1.1.2 (Difficulty: Sub 505 Level)

If  s > 0  and  \sqrt{\frac{r}{s}} = s , what is  r  in terms of  s ?

A.  \frac{1}{s}  
B.  \sqrt{s}  
C.  s\sqrt{s}  
D.  s^3  
E.  s^2 - s

Question Discussion Link

Sub-pattern #2:     Linear Equations

Example 1.2.1 (Difficulty: Sub 505 Level)

If  -3  is  6  more than  x , what is the value of  \frac{x}{3} ?

A.  -9  
B.  -6  
C.  -3  
D.  -1  
E.  1

Question Discussion Link

Example 1.2.2 (Difficulty: Sub 505 Level)

If  \sqrt{(3x + 4)} = x , what is the value of  x ?

A.  -4  
B.  -2  
C.  1  
D.  2  
E.  4

Question Discussion Link

Example 1.2.3 (Difficulty: Sub 505 Level)

If the sum of  5, 8, 12, 15  is equal to the sum of  3, 4, x, x + 3 , what is the value of  x ?

A.  14  
B.  15  
C.  16  
D.  17  
E.  18

Question Discussion Link

Example 1.2.4 (Difficulty: Sub 505 Level)

If  \frac{1}{2}  the result obtained when 2 is subtracted from 5x is equal to 10 + 3x, what is the value of x?

A.  -22  
B.  -4  
C.  4  
D.  18  
E.  22

Question Discussion Link

Example 1.2.5 (Difficulty: Sub 505 Level)

If  3x + 2y = 7  and  2x - y = 7 , what is the value of  x ?

A.  0  
B.  1  
C.  \frac{7}{5}  
D.  \frac{21}{11}  
E.  3

Question Discussion Link

Sub-pattern #3: Multiple Variables & Equations

Example 1.3.1 (Difficulty: Sub 505 Level)

If  x^2 = 2y^3  and  2y = 4 , what is the value of  x^2 + y ?

A.  -14  
B.  -2  
C.  3  
D.  6  
E.  18

Question Discussion Link

Example 1.3.2 (Difficulty: Sub 505 Level)

What is the value of  x^2yz − xyz^2 , if  x = −2 ,  y = 1 , and  z = 3 ?

A.  20  
B.  24  
C.  30  
D.  32  
E.  48

Question Discussion Link

Example 1.3.3 (Difficulty: Sub 505 Level)

If  x = \frac{-5}{8}  and  y = \frac{-1}{2} , what is the value of  -2x - y^2 ?

A.  \frac{-3}{2}  
B.  -1  
C.  1  
D.  \frac{3}{2}  
E.  \frac{7}{4}

Question Discussion Link

Example 1.3.4 (Difficulty: Sub 505 Level)

If  x + y = 2  and  x^2 + y^2 = 2 , what is the value of  xy ?

A.  -2  
B.  -1  
C.  0  
D.  1  
E.  2

Question Discussion Link

Example 1.3.5 (Difficulty: Sub 505 Level)

If  x = -5  and  y = -2 , what is the value of  3(x-y)^2 - xy ?

A.  -157  
B.  -37  
C.  17  
D.  37  
E.  137

Question Discussion Link

Sub-pattern #4: Algebra Word Problems   
 Word problems that rely mostly on Algebra to be solved. Word Problems will be covered in a topic of their own.  
 Plugging numbers often does wonders to simplify these questions.

Example 1.4.1 (Difficulty: 505-555 Level)

In an electric circuit, two resistors with resistances  x  and  y  are connected in parallel. In this case, if  r  is the combined resistance, then the reciprocal of  r  is equal to the sum of reciprocals of  x  and  y . What is  r  in terms of  x  and  y ?

A.  xy  
B.  x + y  
C.  \frac{1}{(x + y)}  
D.  \frac{xy}{(x + y)}  
E.  \frac{(x + y)}{xy}

Question Discussion Link

Example 1.4.2 (Difficulty: Sub 505 Level)

x = \frac{1}{4}sv^2  
In the formula, if  x/s = 25  and  v > 0 , what's the value of  v ?

A.  9  
B.  10  
C.  11  
D.  29  
E.  100

Question Discussion Link

Example 1.4.3 (Difficulty: Sub 505 Level)

x = \frac{1}{6}gt^2  
In the formula, if  g  is constant and  x = -6  when  t = 2 , what's the value of  x  when  t = 4 ?

A.  -24  
B.  -20  
C.  -15  
D.  20  
E.  24

Question Discussion Link

Pattern #2: Simplifying Algebraic Expressions :

Sub-pattern #1: "Equivalent to"  
 These questions ask which expression is  equivalent  to a given algebraic expression. 
 Often tests simplification, factoring, or fraction manipulation.

Example 2.1.1 (Difficulty: Sub 505 Level)  
Which of the following is equivalent to  \frac{(a + b)}{2} + \frac{(a - b)}{3} ?

A.  \frac{2a}{5}  
B.  \frac{5a}{6}  
C.  \frac{(5a + b)}{6}  
D.  \frac{(5a + 2b)}{6}  
E.  \frac{(5a - b)}{6}

Question Discussion Link

Example 2.1.2. (Difficulty: 505-555 Level)  
If  k  and  n  are positive integers such that  n > k , then  k! + (n-k)(k-1)!  is equivalent to:

A.  k \cdot n!  
B.  k! \cdot n  
C.  (n - k)!  
D.  n \cdot (k + 1)!  
E.  n \cdot (k - 1)!

Question Discussion Link

Example 2.1.3 (Difficulty: 505-555 Level)  
If  n  is a positive integer, which of the following is equivalent to  n! + (n + 1)! + (n + 2)! ?

A.  (n!)^3  
B.  3(n + 1)!  
C.  n!(n + 1)^3  
D.  n!(n + 2)^2  
E.  n!(n + 3)

Question Discussion Link

Example 2.1.4. (Difficulty: Sub 505 Level)  
If  (x + 1)(y + 1) 
eq 0 ,  y = \frac{1}{(x + 1)} , and  z = \frac{1}{(y + 1)} , which of the following is equivalent to z in terms of x?

A.  1  
B.  x + 1  
C.  x + 2  
D.  \frac{(x + 1)}{(x + 2)}  
E.  \frac{(x + 2)}{(x + 1)}

Question Discussion Link

Example 2.1.5. (Difficulty: 505-555 Level)  
If  xyz ≠ 0 , which of the following is equivalent to  \frac{(\frac{x}{y})}{z} ÷ \frac{x}{(\frac{y}{z})} ?

A.  1  
B.  \frac{1}{y^2}  
C.  \frac{1}{z^2}  
D.  \frac{1}{(yz)}  
E.  \frac{1}{(y^2z^2)}

Question Discussion Link

Sub-pattern #2: "Equation Equivalent to"  
 Questions that ask: “Which equation is NOT equivalent to...” or “Which form is equivalent to...” 
 Focus is on transforming equations, not just expressions.

Example 2.2.1. (Difficulty: Sub 505 Level)  
Which of the following equations is NOT equivalent to  4x^2 = y^2 - 9 ?

A.  4x^2 + 9 = y^2  
B.  4x^2 - y^2 = -9  
C.  4x^2 = (y + 3)(y - 3)  
D.  2x = y - 3  
E.  x^2 = \frac{(y^2 - 9)}{4}

Question Discussion Link

Example 2.2.2. (Difficulty: 505-555 Level)  
 R = \frac{24F}{N}/(P + \frac{A}{12})

A bank uses this formula to approximate the annual percentage rate R of a loan. Which of the following is an equivalent form?

A.  \frac{24F(12P + A)}{(12N)}  
B.  \frac{24F}{(12NP + AN)}  
C.  \frac{2F}{(N(P + A))}  
D.  \frac{288F}{(PN + AN)}  
E.  \frac{288F}{(12NP + AN)}

Question Discussion Link

Sub-pattern #3: Simplification  
 These questions break algebra into steps, often referencing real-world or process-based formats.

Example 2.3.1. (Difficulty: 505-555 Level)

Step 1: Multiply x by 3. 
Step 2: Take the square root of the result. 
Step 3: Take the reciprocal of the result. 
Step 4: Add 1. 
Step 5: Square the result.

What is the resulting expression?

A.  \frac{(1 + 3x)}{(3x)}  
B.  \frac{(1 + 9x)}{(9x)}  
C.  \frac{1}{(3x + 3\sqrt{3x} + 1)}  
D.  \frac{1 + 2\sqrt{3x} + 3x)}{(3x)}  
E.  \frac{1 + 6\sqrt{x} + 9x)}{(9x)}

Question Discussion Link

Pattern #3: Quadratic Equations and Factoring

Sub-pattern 3.1 Finding Roots  
  Solve for the values of  x  when a standard quadratic is given.  
  Often asks for sum or product of roots, or the difference between them.

Example 3.1.1 (Difficulty: 555-605 Level)  
By how much does the larger root of the equation  2x^2 + 5x = 12  exceed the smaller root?

A.  \frac{5}{2}  
B.  \frac{10}{3}  
C.  \frac{7}{2}  
D.  \frac{14}{3}  
E.  \frac{11}{2}

Question Discussion Link

Example 3.1.2 (Difficulty: 655-705 Level)  
Let  a, b, c,  and  d  be nonzero real numbers. If the quadratic equation  ax(cx + d) = -b(cx + d)  is solved for  x , which of the following is a possible ratio of the 2 solutions?

A.  -\frac{ab}{cd}  
B.  -\frac{ac}{bd}  
C.  -\frac{ad}{bc}  
D.  \frac{ab}{cd}  
E.  \frac{ad}{bc}

Question Discussion Link

Example 3.1.3 (Difficulty: 555-605 Level)  
Which of the following equations has  1 + \sqrt{2}  as one of its roots?

A.  x^2 + 2x – 1 = 0  
B.  x^2 – 2x + 1 = 0  
C.  x^2 + 2x + 1 = 0  
D.  x^2 – 2x – 1 = 0  
E.  x^2 – x – 1 = 0

Question Discussion Link

Example 3.1.4 (Difficulty: Sub 505 Level)  
What is the larger of the 2 solutions of the equation  x^2 − 4x = 96 ?

A.  8  
B.  12  
C.  16  
D.  32  
E.  100

Question Discussion Link

Example 3.1.5 (Difficulty: 805+ Level)  
The set of solutions for the equation  (x^2 – 25)^2 = x^2 – 10x + 25  contains how many real numbers?

A.  0  
B.  1  
C.  2  
D.  3  
E.  4

Question Discussion Link

3.2 Factoring Expressions  
 Simplify or identify equivalent factored forms of algebraic expressions.

Example 3.2.1 (Difficulty: 555-605 Level)

99,999^2 - 1^2 =

A.  10^{10} - 2 
B.  (10^5 – 2)^2 
C.  10^4(10^5 – 2) 
D.  10^5(10^4 – 2) 
E.  10^5(10^5 – 2)

Question Discussion Link

Example 3.2.2 (Difficulty: 505-555 Level)

If  x
eq{-1} , then  \frac{(1 - x^{16}) }{ ((1+x)(1+x^2)(1+x^4)(1+x^8))}  is equal to:

A.  -1 
B.  1 
C.  x 
D.  1 - x 
E.  x - 1

Question Discussion Link

Example 3.2.3 (Difficulty: 555-605 Level)

If  x^2 + \frac{1}{x^2} = 4 , what is the value of  x^4 + \frac{1}{x^4} ?

A.  2 
B.  4 
C.  6 
D.  14 
E.  16

Question Discussion Link

Example 3.2.4 (Difficulty: Sub 505 Level)

What is the value of  x^2yz − xyz^2 , if  x = −2 ,  y = 1 , and  z = 3 ?

A.  20 
B.  24 
C.  30 
D.  32 
E.  48

Question Discussion Link

Example 3.2.5 (Difficulty: 555-605 Level)

If  x^2 > y^2 , then  \frac{\sqrt{(x^4 - 2x^2y^2 + y^4)} }{ (x + y)}  = ?

A.  x - y 
B.  x + y 
C.  \frac{(x - y)}{(x + y)} 
D.  \frac{(x^2 + y^2)}{(x + y)} 
E.  \frac{(x^2 - \sqrt{2}xy + y^2)}{(x + y)}

Question Discussion Link

3.3 Algebraic Identities and Symmetric Expressions  
 Use of known identities such as  (x + y)^2 = x^2 + 2xy + y^2  to evaluate or derive expressions.

Example 3.3.1 (Difficulty: Sub 505 Level) 
If  (x + 3)^2 = y , then in terms of  y ,  x^2 + 6x + 17 =

A.  y - 14 
B.  y - 8 
C.  y + 3 
D.  y + 8 
E.  y + 14

Question Discussion Link

Example 3.3.2 (Difficulty: 555-605 Level) 
The cost C, in dollars, of producing x units per day of a certain commodity is given by:
 
 C = 3x^2 + 5x + 20 
 
If the number of units produced per day is increased from 50 to 60, what is the corresponding increase in the production cost?

A.  370 
B.  380 
C.  1,150 
D.  3,310 
E.  3,350

Question Discussion Link

3.4 Word Problems (Algebra-heavy)  
 Just an example of an algebra-heavy word problem. Word problems will be covered in a dedicated topic. These can often be solved quite efficiently by plugging numbers.

Example 3.4.1 (Difficulty: 555-605 Level)

For a car that is traveling on a dry road at a speed of  r  miles per hour, the stopping distance  d , in feet, is given by the formula  d = 0.045r^2 + 1.1r . In terms of  r , how many feet greater is the stopping distance for a car traveling on a dry road at a speed of  2r  miles per hour than at a speed of  r  miles per hour?

A.  0.045r^2 + 1.1r 
B.  0.09r^2 + 2.2r 
C.  0.135r^2 + 2.2r 
D.  0.135r^2 + 1.1r 
E.  3.3r^2 + 0.225r

Question Discussion Link

That's it! for now. If you know if another new pattern, that I failed to spot, please feel free to share.

Bunuel · Answer

GMAT Question Patterns:

GMAT Question Patterns: Work and Rate Problems  
  GMAT Question Patterns: Arithmetic  
  GMAT Question Patterns: Absolute Value  
  GMAT Question Patterns: Distance and Speed  
  GMAT Question Patterns: Remainders  
  GMAT Question Patterns: Algebra  
  GMAT Question Patterns: Overlapping Sets

Bunuel · Answer

New one:  GMAT Question Patterns: Sequences

Bunuel · Answer

The new pattern topic:  GMAT Question Patterns: Functions and Custom Characters .

GMAT Question Patterns: Algebra (Focus Edition)

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