Refresh Your Memory: Crucial GRE Quant Formulas for Last-Minute Prep

As your GRE Quantitative Reasoning test date approaches, every minute counts. While deep conceptual understanding is paramount, a quick refresh of key formulas can be the difference between confidently solving a problem and struggling for valuable seconds. This guide is designed to help you quickly review the most crucial GRE Quant formulas, ensuring they're top-of-mind for test day.

Think of this as your essential checklist for that final formula sweep. We'll categorize them to make your review systematic and efficient.

Why a Last-Minute Formula Review Matters

The GRE Quant section isn't just about knowing formulas; it's about applying them strategically. However, if you have to spend time recalling a formula, you're losing precious seconds and mental energy. A solid grasp of these foundations allows you to focus on the problem-solving logic, which is where the GRE truly tests you. This review aims to solidify that instant recall.

I. Arithmetic and Number Properties

These are the building blocks. Ensure these fundamental concepts are second nature.

Percentages

• Percent Change: $\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$
  • Example: If a price goes from \$80 to \$100, the percent increase is $\frac{100-80}{80} \times 100\% = \frac{20}{80} \times 100\% = 25\%$.
• Finding Original Value after Percent Change: $\text{Original Value} = \frac{\text{New Value}}{1 \pm \text{percent change (as a decimal)}}$
  • Example: If an item costs \$120 after a 20% increase, its original price was $\frac{120}{1+0.20} = \frac{120}{1.2} = \$100$.

Averages

• Arithmetic Mean (Average): $\text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}}$
• Sum of Terms: $\text{Sum} = \text{Average} \times \text{Number of terms}$
  • Tip: This rearrangement is often more useful for problem-solving. If you know the average and the number of terms, you can find the sum.

Ratios & Proportions

• A ratio $a:b$ can be written as a fraction $\frac{a}{b}$.
• Proportions: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Sequences & Series

• Sum of an Arithmetic Series: $\text{Sum} = \frac{\text{Number of terms}}{2} \times (\text{First term} + \text{Last term})$

II. Algebra

Algebraic manipulation is critical for nearly every GRE Quant problem type.

Exponents and Roots

• $x^a \cdot x^b = x^{a+b}$
• $\frac{x^a}{x^b} = x^{a-b}$
• $(x^a)^b = x^{ab}$
• $(xy)^a = x^a y^a$
• $(\frac{x}{y})^a = \frac{x^a}{y^a}$
• $x^0 = 1$ (for $x \neq 0$)
• $x^{-a} = \frac{1}{x^a}$
• $\sqrt{x} = x^{\frac{1}{2}}$
• $\sqrt[n]{x} = x^{\frac{1}{n}}$
• $\sqrt{xy} = \sqrt{x}\sqrt{y}$

Quadratic Equations

• Quadratic Formula: For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Factoring: $(x+p)(x+q) = x^2 + (p+q)x + pq$
  • Look for two numbers that multiply to $c$ and add to $b$.

Special Products

• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$
  • This is extremely common on the GRE.
• Perfect Squares:
  • $(a+b)^2 = a^2 + 2ab + b^2$
  • $(a-b)^2 = a^2 - 2ab + b^2$

Inequalities

• When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
  • Example: If $-2x < 6$, then $x > -3$.

III. Geometry

Geometry questions often rely on direct application of formulas.

Triangles

• Area of a Triangle: $\frac{1}{2} \times \text{base} \times \text{height}$
• Pythagorean Theorem: $a^2 + b^2 = c^2$ (for right triangles, where $c$ is the hypotenuse)
• Special Right Triangles:
  • 45-45-90 Triangle: Sides are in ratio $x : x : x\sqrt{2}$
  • 30-60-90 Triangle: Sides are in ratio $x : x\sqrt{3} : 2x$ (short leg : long leg : hypotenuse)
• Sum of Angles: $180^\circ$ for any triangle.
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Quadrilaterals

• Square:
  • Area: $s^2$
  • Perimeter: $4s$
  • Diagonal: $s\sqrt{2}$
• Rectangle:
  • Area: $\text{length} \times \text{width}$
  • Perimeter: $2(\text{length} + \text{width})$
  • Diagonal: $\sqrt{\text{length}^2 + \text{width}^2}$ (Pythagorean theorem)
• Parallelogram: Area: $\text{base} \times \text{height}$
• Trapezoid: Area: $\frac{1}{2}(\text{base}1 + \text{base}2) \times \text{height}$

Circles

• Area of a Circle: $\pi r^2$
• Circumference: $2\pi r$ or $\pi d$
• Arc Length: $\frac{\text{central angle}}{360^\circ} \times 2\pi r$
• Area of a Sector: $\frac{\text{central angle}}{360^\circ} \times \pi r^2$

3D Shapes

• Rectangular Solid (Box):
  • Volume: $\text{length} \times \text{width} \times \text{height}$
  • Surface Area: $2(\text{lw} + \text{lh} + \text{wh})$
• Cube:
  • Volume: $s^3$
  • Surface Area: $6s^2$
• Cylinder:
  • Volume: $\pi r^2 h$
  • Surface Area: $2\pi r^2 + 2\pi rh$ (two bases + lateral surface)

Coordinate Geometry

• Distance Formula: $d = \sqrt{(x2-x1)^2 + (y2-y1)^2}$
• Midpoint Formula: $(\frac{x1+x2}{2}, \frac{y1+y2}{2})$
• Slope of a Line: $m = \frac{y2-y1}{x2-x1}$
• Equation of a Line: $y = mx+b$ (slope-intercept form)

IV. Data Analysis and Probability

These areas test your ability to interpret and calculate from given data.

Statistics

• Mean (Average): See Arithmetic section.
• Median: The middle value in a sorted set of numbers. If an even number of terms, it's the average of the two middle terms.
• Mode: The value that appears most frequently in a set.
• Range: Highest value - Lowest value.
• Standard Deviation: Measures the spread of data around the mean. While the GRE usually doesn't require calculating it, understand that a smaller standard deviation means data points are closer to the mean.
• Weighted Average: $\frac{\text{Sum of (value} \times \text{weight)}}{\text{Sum of weights}}$
  • Example: If a class has 30 students with an average score of 80 and 20 students with an average score of 90, the combined average is $\frac{(30 \times 80) + (20 \times 90)}{30+20} = \frac{2400+1800}{50} = \frac{4200}{50} = 84$.

Probability

• Basic Probability: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
• Probability of A AND B (independent events): $P(A \text{ and } B) = P(A) \times P(B)$
• Probability of A OR B (mutually exclusive events): $P(A \text{ or } B) = P(A) + P(B)$
• Probability of A OR B (not mutually exclusive): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
• Complement Rule: $P(\text{event occurs}) = 1 - P(\text{event does not occur})$

Combinations and Permutations

• Permutations (Order matters): $P(n, k) = \frac{n!}{(n-k)!}$
  • Example: How many ways to arrange 3 books on a shelf from a set of 5? $P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$.
• Combinations (Order does NOT matter): $C(n, k) = \frac{n!}{k!(n-k)!}$
  • Example: How many ways to choose 3 books from a set of 5? $C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10$.

V. Last-Minute Formula Review Tips

1. Don't Just Memorize, Understand: Simply rattling off formulas isn't enough. Understand what each variable represents and when to apply the formula.
2. Practice Application: The best way to refresh is to solve a few problems using each formula. Even quick mental checks can help.
3. Focus on Weak Areas: If you consistently miss geometry questions, spend more time on those formulas.
4. Create a "Formula Cheat Sheet" (for practice): Writing down the formulas yourself helps embed them in your memory. You won't have one on test day, but the act of creating one is a powerful learning tool.
5. Utilize Practice Questions: Many online resources and practice apps, like the GRE Quantitative Reasoning prep app, offer numerous practice problems where you can apply these formulas under timed conditions, simulating the actual test environment. This helps you identify which formulas you're quick to recall and which need a bit more review.
6. Review Common Pitfalls: Understand why certain wrong answers are tempting, especially in areas like percentages or geometric figures.

Conclusion

A thorough review of these crucial GRE Quant formulas can significantly boost your confidence and performance on test day. This isn't about rote memorization, but about ensuring that when a problem calls for a specific tool, that tool is readily available in your mental toolkit. Take a deep breath, review these essentials, and walk into your exam prepared to apply your knowledge effectively. Good luck!