Sherpa Prep GRE Arithmetic Mastery Guide

Core Importance of Arithmetic in the Revised GRE

• Concepts of Arithmetic, Number Properties, and Algebra are the most critical topics tested on the revised GRE.
• These topics are directly tested in approximately one-third of all GRE questions.
• A majority of other question types—including Word problems, Geometry, and problems involving Charts and Graphs—require proficiency in arithmetic and solving for $x$.
• Arithmetic is considered the "heart" of GRE math; without these skills, it is impossible to solve more exotic mathematical concepts.
• Breakdown of questions testing Arithmetic, Algebra, and Number Properties:
  • Algebra: $35\%$ of questions.
  • Integer Properties: $25\%$ of questions.
  • Exponents and Roots: $20\%$ of questions.
  • Fractions or Decimals: $12.5\%$ of questions.
  • Functions: $7.5\%$ of questions.
• While few questions test basic arithmetic skills in isolation, almost all questions require the ability to work with numbers quickly and accurately.

PEMDAS: The Universal Order of Operations

• Arithmetic must be performed in a specific, agreed-upon order known as PEMDAS:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction
• The Equivalence Rule: Multiplication and Division are considered equal operations, as are Addition and Subtraction. Division is simply multiplication by a reciprocal, and subtraction is the addition of a negative number.
• Left-to-Right Processing: If a problem contains multiple instances of the same operation level (e.g., multiplication and division), they must be resolved from left to right.
  • Example: $4 \div 2 \times 4$ must be solved by dividing $4 \div 2 = 2$ first, then multiplying $2 \times 4 = 8$. Calculating $2 \times 4 = 8$ first would incorrectly yield $4 \div 8 = \frac{1}{2}$.
• Quantitative Comparison Example:
  • Quantity A: $1 + 4(5 - 3)^2 - 3 \times 6$
  • Quantity B: $-1$
  • Step 1 (Parentheses): $5 - 3 = 2$
  • Step 2 (Exponent): $2^2 = 4$
  • Step 3 (Rewrite): $1 + 4(4) - 3 \times 6$
  • Step 4 (Multiplication, left to right): $4 \times 4 = 16$ and $3 \times 6 = 18$
  • Step 5 (Final Calculation): $1 + 16 - 18 = 17 - 18 = -1$
  • Result: Answer C (Quantities are equal).

The Onscreen GRE Calculator

• General Features: The calculator is provided to reduce the focus on computation and increase focus on reasoning. It is available onscreen for computerized tests and as a handheld device for paper tests. Private calculators are prohibited.
• Basic Functions: Addition, subtraction, multiplication, division, and square root.
• Memory Functions:
  • M+ (Memory Sum): Inserts a number into the memory location.
  • MR (Memory Recall): Displays the stored number.
  • MC (Memory Clear): Erases the memory storage.
• Transfer Display: Allows the user to move a number from the calculator window directly into the answer box for Numeric Entry questions.
• Significant Limitations:
  • Inefficiency: Using the calculator for every problem can take $20\text{--}30$ seconds per question, totaling $4\text{--}6$ minutes per math section ($21$ or $26$ minutes long).
  • Display Limit: The screen can only display a maximum of eight digits. Calculations resulting in more than $99,999,999$ or less than $0.0000001$ will display an "Error" message.
  • Capabilities: It cannot handle fractions, exponents, variables, or stored formulas.
  • Input Risks: It is easy to mistype entries when rushing; pre-estimation is recommended to ensure answers are "in the ballpark."
• Example of Calculator Failure (Units Digit):
  • Question: What is the units digit of $123,456,789 \times 123$?
  • Calculation: $9 \times 3 = 27$. The units digit is $7$. The calculator cannot process this because the product exceeds eight digits.
• Example of Calculator Failure (Remainder Issues):
  • Question: If $a = 6 \times 7$ and $b = 3 \times 5$, what is the remainder of $a \div b$?
  • Calculator Result: $42 \div 15 = 2.8$.
  • True Result: The remainder is the whole number left over. $15 \times 2 = 30$; thus, $42 - 30 = 12$. The remainder is $12$, not $0.8$ or $8$.

Time: The Hidden Topic and Optimization Strategies

• Time management is implicitly tested in every GRE question. Improving efficiency is vital for a high score.
• Speed Strategies:
  1. Improve the speed of routine calculations via shortcuts.
  2. Use the best framework or setup for problems (e.g., choosing a $30$-second method over a $3$-minute method).
• Breaking Numbers Down: This is the "smart, simple way" to handle complex calculations. Making numbers smaller makes them easier to work with.
  • Example for Division ($81 \div 27$): Represent as a fraction and cancel common factors: $\frac{81}{27} = \frac{9(9)}{9(3)} = \frac{9}{3} = 3$.
  • Example for Quantitative Comparison ($x = \frac{350}{14}$, $y = \frac{900}{18}$):
    • $x = \frac{7(50)}{7(2)} = \frac{50}{2} = 25$
    • $y = \frac{9(100)}{9(2)} = \frac{100}{2} = 50$
    • $x + y = 75$, which is greater than $70$ (Quantity A is larger).

Mandatory Multiplication and Power Tables

• Memorization through $12 \times 12$ is essential for breaking down numbers like $72, 84,$ and $96$.
• Perfect Squares Trick: To multiply two numbers separated by $2$ (e.g., $n - 1$ and $n + 1$), subtract $1$ from the square of the number between them ($n^2 - 1$).
  • $5 \times 7 = 6^2 - 1 = 35$
  • $11 \times 13 = 12^2 - 1 = 143$
  • $14 \times 16 = 15^2 - 1 = 224$
  • $19 \times 21 = 20^2 - 1 = 399$
• Perfect Squares and Cubes List:
  • $10^2 = 100$, $11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $15^2 = 225$, $20^2 = 400$, $25^2 = 625$, $30^2 = 900$.
  • $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$, $6^3 = 216$, $10^3 = 1,000$.
• Special Powers:
  • Powers of 2: $2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64$.
  • Powers of 3: $3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243$.
  • Powers of 4: $4^1=4, 4^2=16, 4^3=64, 4^4=256$.
  • Powers of 5: $5^1=5, 5^2=25, 5^3=125, 5^4=625$.

Square Roots and Approximations

• Memorizing roots from $1$ to $10$ is critical for Geometry (specifically Right Isosceles triangles $x:x:x\sqrt{2}$ and $30\text{--}60\text{--}90$ triangles $x:x\sqrt{3}:2x$).
• Root Approximations:
  • $\sqrt{1} = 1$
  • $\sqrt{2} \approx 1.4$
  • $\sqrt{3} \approx 1.7$
  • $\sqrt{4} = 2$
  • $\sqrt{5} \approx 2.2$
  • $\sqrt{6} \approx 2.4$
  • $\sqrt{7} \approx 2.6$
  • $\sqrt{8} \approx 2.8$
  • $\sqrt{9} = 3$
  • $\sqrt{10} \approx 3.2$
• The Adjustment Rule: For roots greater than $\sqrt{4}$, add $0.2$ per digit increase. For roots smaller than $\sqrt{4}$, subtract $0.3$ per digit decrease.
• Note: Roots cannot be added directly (e.g., $\sqrt{2} + \sqrt{6} + \sqrt{8} \neq \sqrt{16}$).

The Fraction to Decimal "Conversion List"

• $\frac{1}{2} = 0.5$
• $\frac{1}{3} = 0.\overline{3}$
• $\frac{1}{4} = 0.25$
• $\frac{1}{5} = 0.2$
• $\frac{1}{7} \approx 0.14$
• $\frac{1}{8} = 0.125$
• $\frac{1}{9} = 0.\overline{1}$
• $\frac{1}{10} = 0.1$
• $\frac{1}{11} = 0.\overline{09}$
• $\frac{1}{99} = 0.\overline{01}$
• $\frac{1}{100} = 0.01$
• Application Example: To find the decimal for $\frac{3}{11}$, multiply $0.0909... \times 3 = 0.2727...$.

Advanced Arithmetic Shortcuts

• The Multiplication Trick: Split complicated numbers into two parts. Example: $9(21) = 9(20 + 1) = 180 + 9 = 189$.
• Chunking (Division): Break numbers into chunks divisible by the divisor. Example: $168 \div 14 \rightarrow (140 + 28) \div 14 = 10 + 2 = 12$.
• Multiplying/Dividing by 4 and 8:
  • To multiply/divide by $4$, double or halve the number twice.
  • To multiply/divide by $8$, double or halve the number three times.
• Tricks for 9, 11, and 99:
  • To multiply by $9$: Multiply by $10$ and subtract the original number ($14 \times 9 = 140 - 14 = 126$).
  • To multiply by $11$: Multiply by $10$ and add the original number ($14 \times 11 = 140 + 14 = 154$).
  • To multiply by $99$: Multiply by $100$ and subtract the original number ($8 \times 99 = 800 - 8 = 792$).
• Multiplying/Dividing by 5:
  • Multiply by $5$: Multiply by $10$ and divide by $2$.
  • Divide by $5$: Divide by $10$ and multiply by $2$. ($92 \div 5 = 9.2 \times 2 = 18.4$).
• The Addition Shortcut: Add tens digits and units digits separately ($47 + 38 \rightarrow (40 + 30) + (7 + 8) = 70 + 15 = 85$).
• Subtraction by Addition: Calculate the distance between numbers on a number line using a simple bridge number (like $100$ or $50$).
  • Example $123 - 87$: From $87$ to $100 = 13$; from $100$ to $123 = 23$. $13 + 23 = 36$.
• Quick Percents (10% Shortcut):
  • $10\%$: Slide decimal point one space left.
  • $5\%$: Take half of $10\%$.
  • $1\%$: Slide decimal point two spaces left.
  • Example: $6\%$ of $120 = 5\% + 1\% = 6 + 1.2 = 7.2$.

The Divisibility Rules (Integers 2-10)

• 2: Last digit is even (can be cut in half once).
• 3: Sum of digits is divisible by $3$.
• 4: Last two digits are divisible by $4$ (can be cut in half twice).
• 5: Ends in $0$ or $5$.
• 6: Divisible by both $2$ and $3$.
• 7: No simple digit rule; "Chunking" is faster ($224 = 210 + 14$; both are divisible by $7$, so $224$ is divisible by $7$).
• 8: Last three digits are divisible by $8$ (can be cut in half thrice).
• 9: Sum of digits is divisible by $9$.
• 10: Ends in $0$.

Problem Sets and Practical Drills

• Drill Highlights:
  • $35 \times 15 \times 12 = 6,300$ (True).
  • $14 \times 16 = 224$ (True).
  • $108$ divisibility check: It is even (divisible by $2$), sum of digits is $9$ (divisible by $3$ and $9$), and last two digits ($08$) are divisible by $4$. Since it is divisible by $2$ and $3$, it is divisible by $6$.
• Quant Question Logic:
  • $10^{15}$ is divisible by all exponents of $2$ and $5$ within its range but NOT divisible by $3$ or $6$ because the sum of its digits ($1 + 0...$) is $1$.
  • Factors are small numbers that divide cleanly into larger ones (e.g., $9$ is a factor of $5,427$ because $5+4+2+7=18$).

Practice Questions & Solutions

• Videos for all questions are found at www.sherpaprep.com/masterkey.
• Strategic Tip: Re-visit any problem that took more than two minutes or was answered incorrectly after a few days to ensure internalization.
• If errors persist, add the problem to a "LOG of ERRORS" and repeat every few weeks.