Sherpa Prep GRE Arithmetic Master Study Guide

Fundamental Concepts of GRE Arithmetic

• Introduction to Core Concepts: Of all concepts tested by the revised GRE, none are more important than those of Arithmetic, Number Properties, and Algebra.

• Direct and Indirect Impact: Not only do a third of GRE questions test these concepts directly, but a majority of other questions involve them indirectly. Word problems, Geometry, and problems involving Charts and Graphs all demand the ability to work with numbers and solve for $x$.

• The "Heart" of GRE Math: These concepts are considered the core; exotic concepts cannot be solved without them.

• Statistical Breakdown of Arithmetic, Algebra, and Number Properties:

  • Algebra: $35\%$ of questions.

  • Integer Properties: $25\%$ of questions.

  • Exponents & Roots: $20\%$ of questions.

  • Fractions or Decimals: $12.5\%$ of questions.

  • Functions: $7.5\%$ of questions.

• The Role of Arithmetic: While few questions solely test basic math skills (hence Arithmetic's absence from the pie chart), strong arithmetic skills are vital for working with numbers quickly.

PEMDAS: The Order of Operations

• Definition: PEMDAS is the universally agreed-upon sequence for performing arithmetic operations:

  • $P$ = Parentheses

  • $E$ = Exponents

  • $M$ = Multiplication

  • $D$ = Division

  • $A$ = Addition

  • $S$ = Subtraction

• Standard Operation Example:

  • Quantity A: $1 + 4(5-3)^2 - 3 \times 6$

  • Quantity B: $-1$

  • Step 1: Parentheses and Exponents: $1 + 4(2)^2 - 3 \times 6 \rightarrow 1 + 4(4) - 3 \times 6$

  • Step 2: Multiplication (Left to Right): $1 + 16 - 18$

  • Step 3: Addition and Subtraction (Left to Right): $17 - 18 = -1$

  • Answer: C (Quantity A = Quantity B).

• The "PEMDAS Misconception": The acronym is slightly misleading. Multiplication does not always come before division, nor addition before subtraction.

• Equal Operations Rule:

  • Multiplication and Division are equal operations (division is multiplication by the reciprocal).

  • Addition and Subtraction are equal operations (subtraction is adding a negative number).

  • Rule: Perform equal operations whichever comes first as you read from left to right.

• Priority Example:

  • Quantity A: $4 \div 2 \times 4$

  • Quantity B: $12$

  • Incorrect interpretation: $4 \div (2 \times 4) = 4 \div 8 = \frac{1}{2}$

  • Correct interpretation: $(4 \div 2) \times 4 = 2 \times 4 = 8$

  • Answer: A (Quantity A is greater).

The GRE Calculator

• Format: Revised GRE features an onscreen calculator for computerized tests; paper-based test-takers are provided calculators at the center. Personal calculators are prohibited.

• Core Functions: Addition, subtraction, multiplication, division, and square root.

• Memory Buttons:

  • M+ (Memory Sum): Inserts a number into memory.

  • MR (Memory Recall): Displays the number stored in memory.

  • MC (Memory Clear): Erases the calculator's memory.

• Navigation: The Transfer Display button moves values from the calculator to the answer box for Numeric Entry questions.

• Significant Limitations:

  • Time Cost: Using the calculator takes an average of $20\text{-}30$ seconds per problem. Constant use could consume $4\text{-}6$ minutes of the $21$ or $26$ minute math sections.

  • Typing Errors: The interface is awkward and facilitates mistyping numbers or symbols.

  • Display Limit: It cannot display more than eight digits. If a result is greater than $99,999,999$ or smaller than $0.0000001$, the screen displays "Error".

  • Functional Gap: It cannot handle fractions, exponents, variables, or store formulas.

• Calculator Trap Example 1 (Units Digit):

  • Question: What is the units digit of $123,456,789 \times 123$?

  • Calculator Result: Error (overflow).

  • Manual Analysis: Multiply the units digits only ($9 \times 3 = 27$). The units digit must be $7$.

• Calculator Trap Example 2 (Remainders):

  • Question: If $a = 6 \times 7$ and $b = 3 \times 5$, what is the remainder of $a \div b$?

  • Calculator Calculation: $42 \div 15 = 2.8$.

  • Correct Analysis: $42 \div 15 = 2$ with a remainder of $12$ (since $(15 \times 2) + 12 = 42$). A remainder is a whole number.

• Best Practices: Estimate answers beforehand to spot key-entry errors. Keep the calculator open on the screen at all times to minimize clicking.

Time: The Hidden Topic

• Efficiency Difficulty: Difficulty often stems from the time limit ($21$ or $26$ minutes for $15$ problems), not just the math itself.

• Winning the Speed Competition:

  • Improve speed of routine calculations.

  • Know the best frameworks/setups for problems (e.g., a $30\text{-second}$ solution vs. a $3\text{-minute}$ solution).

• Breaking Down Numbers (The Smart Way): Instead of long division or "guess and check," represent division as a fraction and cancel common factors.

  • Example: $81 \div 27 \rightarrow \frac{81}{27} = \frac{9(9)}{9(3)} = \frac{9}{3} = 3$.

• Strategic Goal: Making numbers smaller makes them easier to work with; working with large numbers is often a mistake.

• Substitution Example:

  • $x = \frac{350}{14} = \frac{7(50)}{7(2)} = 25$

  • $y = \frac{900}{18} = \frac{9(100)}{9(2)} = 50$

  • $x + y = 75$.

• Multiplication Breakdown Example:

  • $a = 35 \times 5 \times 28 = (7 \times 5) \times 5 \times (4 \times 7) = 49 \times 100 = 4,900$

  • $b = 12 \times 25 \times 13 = (3 \times 4) \times 25 \times 13 = 39 \times 100 = 3,900$

  • $\frac{a-b}{100} = \frac{4,900 - 3,900}{100} = \frac{1,000}{100} = 10$.

Multiplication Tables and Relationships

• Requirement: Memorize multiplication tables through $12 \times 12$.

• Bidirectional Memory: Know that $7 \times 12 = 84$ and that $84 = 7 \times 12$.

• The Multiplication Symmetrical Table: Every multiplication has a "twin" ($9 \times 7$ is the same as $7 \times 9$). If one is hard to remember, use the other.

• Perfect Square Diagonal: Numbers like $25$, $64$, and $121$ cut the table in half.

• Numbers Separated by 2 Trick: To multiply two numbers separated by $2$ (e.g., $n-1$ and $n+1$), square the number between them and subtract one.

  • $5 \times 7 = 6^2 - 1 = 35$

  • $11 \times 13 = 12^2 - 1 = 143$

  • $14 \times 16 = 15^2 - 1 = 224$

  • $19 \times 21 = 20^2 - 1 = 399$

Squares, Cubes, and Special Powers

• Squares Memory List:

  • $10^2 = 100$

  • $11^2 = 121$

  • $12^2 = 144$

  • $13^2 = 169$

  • $14^2 = 196$

  • $15^2 = 225$

  • $20^2 = 400$

  • $25^2 = 625$

  • $30^2 = 900$

• Cubes Memory List:

  • $1^3 = 1$

  • $2^3 = 8$

  • $3^3 = 27$

  • $4^3 = 64$

  • $5^3 = 125$

  • $6^3 = 216$

  • $10^3 = 1,000$

• Special Powers List:

  • 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

• Key Roots for Geometry (Triangles):

  • Right Isosceles ($45\text{-}45\text{-}90$): Sides $x : x : x\sqrt{2}$

  • $30\text{-}60\text{-}90$ Triangle: Sides $x : x\sqrt{3} : 2x$

• Roots Approximation List ($1$ to $10$):

  • $\sqrt{1} = 1.0$

  • $\sqrt{2} \approx 1.4$

  • $\sqrt{3} \approx 1.7$

  • $\sqrt{4} = 2.0$

  • $\sqrt{5} \approx 2.2$

  • $\sqrt{6} \approx 2.4$

  • $\sqrt{7} \approx 2.6$

  • $\sqrt{8} \approx 2.8$

  • $\sqrt{9} = 3.0$

  • $\sqrt{10} \approx 3.2$

• Memory Rule: For values larger than $\sqrt{4}$, add approximately $0.2$ per digit. For values smaller than $\sqrt{4}$, subtract approximately $0.3$. (Valid from $2$ to $10$).

• Sample Problem Comparison:

  • Quantity A: $\sqrt{2} + \sqrt{6} + \sqrt{8}$

  • Quantity B: $4$

  • Analysis: Roots cannot be added under the radical ($\sqrt{2}+\sqrt{6}+\sqrt{8} \neq \sqrt{16}$). Using approximations: $1.4 + 2.4 + 2.8 = 6.6$. Result: A is greater.

Fraction to Decimal Equivalents (The Conversion List)

• Basic Conversions:

  • $\frac{1}{2} = 0.5$

  • $\frac{1}{3} = 0.\overline{3}$

  • $\frac{1}{4} = 0.25$

  • $\frac{1}{5} = 0.2$

  • $\frac{1}{6} \approx 0.16$

  • $\frac{1}{8} = 0.125$

  • $\frac{1}{9} = 0.\overline{1}$

  • $\frac{1}{10} = 0.1$

  • $\frac{1}{11} = 0.\overline{09}$

  • $\frac{1}{12} \approx 0.08$

  • $\frac{1}{99} = 0.\overline{01}$

  • $\frac{1}{100} = 0.01$

  • $\frac{1}{\pi} \approx 0.3$

• Derived Values Example: If $\frac{1}{11} = 0.0909...$, then $\frac{3}{11} = 3 \times 0.0909 = 0.2727...$.

• Digit Position Example: In the decimal equivalent of $\frac{3}{11}$, the $16\text{th}$ digit after the decimal is $7$ because all even-placed digits are $7$ ($0.[2]7[2]7...$ where indices $2, 4, 6$… are $7$).

Arithmetic Shortcuts

• The Multiplication Trick: Split a complicated number into two simpler parts.

  • $9 \times 21 = 9(20 + 1) = 180 + 9 = 189$

  • $12 \times 12.5 = 12(12 + 0.5) = 144 + 6 = 150$

• Chunking (Division Shortcut): Break numbers into manageable chunks that are multiples of the divisor.

  • $168 \div 14 \rightarrow (140 + 28) \div 14 = 10 + 2 = 12$

  • $108 \div 7 \rightarrow (70 + 38) \div 7 = 10 + 5 \text{ rem. } 3$

• Number Tricks (4 and 8):

  • Multiply by $4$: Double the number twice ($13 \times 4: 13 \rightarrow 26 \rightarrow 52$).

  • Divide by $4$: Halve the number twice ($72 \div 4: 72 \rightarrow 36 \rightarrow 18$).

  • Multiply/Divide by $8$: Double or halve the number three times.

• Multiplying by 9, 11, and 99:

  • $x \times 9 = (x \times 10) - x$

  • $x \times 11 = (x \times 10) + x$

  • $x \times 99 = (x \times 100) - x$

• Multiplying and Dividing by 5:

  • Multiply by $5$: Multiplied by $10$ then divide by $2$ ($14 \times 5 = 140 \div 2 = 70$).

  • Divide by $5$: Divide by $10$ then multiply by $2$ ($80 \div 5 = 8 \times 2 = 16$).

• Addition Shortcut: Add the tens digits and the units digits separately.

  • $47 + 38 \rightarrow (40+30) + (7+8) = 70 + 15 = 85$

• Subtraction by Addition (Number Line Distance): Find distances from a midpoint (like $100$ or $50$).

  • $123 - 87 \rightarrow (87 \rightarrow 100) = 13$ and $(100 \rightarrow 123) = 23$. Sum: $13 + 23 = 36$.

  • Estimation Method: Add a guess to the lower number. $86 - 29 \rightarrow 29 + 50 = 79$. Since $79$ is $7$ below $86$, the answer is $50 + 7 = 57$.

Quick Percents and Divisibility Rules

• 10% Shortcut: Slide the decimal one space to the left.

  • $5\%$ of a number is half of $10\%$.

  • $15\%$ of a number is $10\% + 5\%$.

  • $20\%, 30\%, 40\%$ are multiples of the $10\%$ value.

• 1% Shortcut: Slide the decimal two spaces to the left. Used for specific percents like $6\%$ ($5\% + 1\%$) or $31\%$ ($30\% + 1\%$).

• Divisibility Rules:

  • $2$: 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 $3$.

  • $8$: Last three digits divisible by $8$ (can be cut in half thrice).

  • $9$: Sum of digits is divisible by $9$.

  • $10$: Ends in $0$.

  • $7$: Use "Chunking." (Example: $224 \rightarrow 210 + 14$, both divisible by $7$).

Practice Problems

1. Breaking down numbers:

   • (a) Does $16 \times 25 = 400$?

   • (b) Does $18 \times 9 = 162$?

   • (c) Does $14 \times 35 = 500$?

   • (d) Does $35 \times 15 \times 12 = 6,300$?

2. Conversion List (no calculator):

   • (a) $\frac{2}{7}$

   • (b) $\frac{2}{5}$

   • (c) $0.\overline{5}$

   • (d) $0.375$

   • (e) $\frac{3}{11}$

   • (f) $\frac{5}{99}$

3. Numerical equivalents:

   • (a) $\sqrt{2}$

   • (b) $\sqrt{7}$

   • (c) $15^2$

   • (d) $5^3$

   • (e) $16$

   • (f) $64$

4. Multiplication trick:

   • (a) $8(17)$

   • (b) $11(14)$

   • (c) $31(13)$

   • (d) $7(123)$

5. Number tricks:

   • (a) $16 \times 9$

   • (b) $21 \times 11$

   • (c) $7 \times 99$

   • (d) $14 \times 16$

6. Number tricks (Division):

   • (a) $90 \div 5$

   • (b) $92 \div 4$

   • (c) $144 \div 8$

   • (d) $210 \div 6$

7. Addition Shortcut:

   • (a) $47 + 75$

   • (b) $93 + 89$

   • (c) $23 + 54 + 38$

   • (d) $42 + 33 + 57$

8. Subtraction by Addition:

   • (a) $63 - 27$

   • (b) $81 - 39$

   • (c) $123 - 64$

   • (d) $231 - 87$

9. Chunking:

   • (a) $165 \div 3$

   • (b) $108 \div 9$

   • (c) $322 \div 14$

   • (d) $255 \div 11$

10. 10% Shortcut:

    • (a) $5\%, 10\%, 15\%$ of $40$

    • (b) $20\%, 40\%, 60\%$ of $120$

    • (c) $1\%, 10\%, 12\%$ of $240$

    • (d) $1\%, 10\%, 98\%$ of $110$

11. Divisibility Rules:

    • (a) Is $189$ divisible by $3$ and $9$?

    • (b) Is $480$ divisible by $3, 6,$ and $9$?

    • (c) Is $108$ divisible by $3, 4, 6,$ and $9$?

    • (d) Is $405$ divisible by $3, 5,$ and $9$?

Solutions to Practice Questions

• Drill 1: a. Yes; b. Yes; c. No (it's 490); d. Yes.

• Drill 2: a. $\approx 0.28$; b. $0.4$; c. $\frac{5}{9}$; d. $\frac{3}{8}$; e. $0.\overline{27}$; f. $0.0505...$.

• Drill 3: a. $\approx 1.4$; b. $\approx 2.6$; c. $225$; d. $125$; e. $4^2$ and $2^4$; f. $8^2, 4^3, 2^6$.

• Drill 4: a. $136$; b. $154$; c. $403$; d. $861$.

• Drill 5: a. $144$; b. $231$; c. $693$; d. $224$.

• Drill 6: a. $18$; b. $23$; c. $18$; d. $35$.

• Drill 7: a. $122$; b. $182$; c. $115$; d. $132$.

• Drill 8: a. $36$; b. $42$; c. $59$; d. $144$.

• Drill 9: a. $55$; b. $12$; c. $23$; d. $23 \text{ rem. } 2$.

• Drill 10: a. $2, 4, 6$; b. $24, 48, 72$; c. $2.4, 24, 28.8$; d. $1.1, 11, 107.8$.

• Drill 11: a. Yes; b. Only by $3$ and $6$; c. Yes; d. Yes.

• Quantitative Questions: 12. C; 13. B ($840$); 14. A; 15. A, B, D, E, F; 16. C; 17. B; 18. D.