Sherpa Prep GRE Chapter 4: Fractions Study Notes

Introduction to Fractions

• General Significance: Like arithmetic, fractions are a vital component of the revised GRE. While few questions focus exclusively on fractions, many require the ability to manipulate them quickly and accurately. Fractions are particularly advantageous for word problems because they often contain shortcuts that decimals lack.

• Exam Philosophy: Exam-makers are not assessing your ability to act as a human calculator; rather, they want to see if you can find smart, simple solutions to seemingly complex problems. Proficiency with fractions allows for these shortcuts.

• Basic Definition: A fraction is a number in the form $\frac{a}{b}$, where the horizontal bar signifies division.

• Terminology:

  • Numerator: The number on top (e.g., in $\frac{2}{3}$, the numerator is $2$).

  • Denominator: The number on the bottom (e.g., in $\frac{2}{3}$, the denominator is $3$).

• Zero in Fractions:

  • If the numerator is zero, the fraction equals zero: $\frac{0}{3} = 0$.

  • If the denominator is zero, the fraction is "undefined" because division by zero is impossible: $\frac{3}{0} = \text{undefined}$.

• Whole Numbers: Any whole number can be expressed as a fraction by placing it over $1$. Examples include $\frac{6}{1}$, $\frac{15}{1}$, and $\frac{-3}{1}$.

• Reduction Rule: Always reduce fractions as much as possible. For example, while $\frac{10}{20}$ and $\frac{1}{2}$ are equivalent, $\frac{1}{2}$ is much easier to use in calculations.

Adding and Subtracting Fractions

• Requirement: Fractions must have a common denominator to be combined through addition or subtraction.

• Same Denominator: Combine the numerators while keeping the denominator the same.

  • Example: $\frac{1}{3} + \frac{2}{3} = \frac{1+2}{3} = \frac{3}{3} = 1$

  • Example: $\frac{4}{5} - \frac{1}{5} = \frac{4-1}{5} = \frac{3}{5}$

• Different Denominators (Common Denominator Method):

  • Step 1: Find the Least Common Denominator (LCD), which is the smallest number cleanly divisible by all denominators. For $\frac{3}{4}$ and $\frac{1}{6}$, the LCD is $12$.

  • Step 2: Multiply the numerator and denominator of each fraction by the value needed to reach the LCD. For $\frac{3}{4}$, multiplying top and bottom by $3$ yields $\frac{9}{12}$.

  • Example: To add $\frac{2}{9} + \frac{1}{4}$, find the LCD ($36$). Multiply $\frac{2}{9}$ by $\frac{4}{4}$ to get $\frac{8}{36}$ and multiply $\frac{1}{4}$ by $\frac{9}{9}$ to get $\frac{9}{36}$. Sum: $\frac{8+9}{36} = \frac{17}{36}$.

• The Fast Fraction Shortcut:

  • Procedure: Cross-multiply the two fractions and add (or subtract) the resulting products. Place this result over the product of the original denominators.

  • Addition Example: $\frac{2}{9} + \frac{1}{4}$. Cross-multiply ($4 \times 2 = 8$ and $9 \times 1 = 9$). Sum: $8+9=17$. Denominator: $9 \times 4=36$. Result: $\frac{17}{36}$.

  • Subtraction Example: $\frac{1}{7} - \frac{2}{5}$. Cross-multiply ($5 \times 1 = 5$ and $7 \times 2 = 14$). Difference: $5-14=-9$. Denominator: $7 \times 5=35$. Result: $\frac{-9}{35}$.

• Combining Whole Numbers and Fractions: Place the whole number over $1$ to avoid confusion.

  • Example: $4 - \frac{3}{5} = \frac{4}{1} - \frac{3}{5}$. Cross-multiply ($5 \times 4 = 20$ and $1 \times 3 = 3$). Result: $\frac{20-3}{5} = \frac{17}{5}$.

Multiplying Fractions

• Basic Procedure: Multiply numerators together and denominators together.

  • Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2(4)}{3(5)} = \frac{8}{15}$.

• The Breakdown Strategy: Never multiply large numbers directly if you can avoid it. Break numerators and denominators into small elements to cancel common factors before multiplying.

  • Example: $\frac{14}{15} \times \frac{18}{42}$.

  • Factorization: $\frac{2 \times 7}{3 \times 5} \times \frac{3 \times 6}{6 \times 7}$.

  • Cancellation: The $7$s and $3$s and $6$s cancel out. Result: $\frac{2}{5}$.

• Identification of Factors: When looking to break down a number (e.g., $48$), any product works ($6 \times 8$, $4 \times 12$, or $2 \times 24$). It is most efficient to look for common factors shared with the other fractions in the problem.

• Example with Three Fractions: $\frac{48}{10} \times \frac{15}{18} = \frac{6 \times 8}{2 \times 5} \times \frac{3 \times 5}{6 \times 3}$. Canceling the $6$s, $5$s, and $3$s leaves $\frac{8}{2} = 4$.

• Rule of Thumb: Never make numbers bigger unless you have to. If you multiply before canceling, you erase shortcuts and increase the risk of computational errors.

Dividing Fractions

• Fundamental Rule: Division is just multiplication in another form. To divide two fractions, flip the second fraction (find its reciprocal) and multiply it by the first.

  • Example: If $x = \frac{4}{25} \div \frac{8}{15}$, then $x = \frac{4}{25} \times \frac{15}{8}$.

  • Breakdown: $\frac{4 \times (3 \times 5)}{(5 \times 5) \times (2 \times 4)} = \frac{3}{10}$.

• Order of Operations (PEMDAS): When facing multiple divisions, work from left to right.

  • Example: $\frac{21}{5} \div \frac{14}{9} \div \frac{18}{15}$.

  • First flip: $\frac{21}{5} \times \frac{9}{14} \div \frac{18}{15}$.

  • Second flip: $\frac{21}{5} \times \frac{9}{14} \times \frac{15}{18}$.

  • Reduction: $\frac{(3 \times 7) \times 9 \times (3 \times 5)}{5 \times (2 \times 7) \times (2 \times 9)} = \frac{3 \times 3}{2 \times 2} = \frac{9}{4}$.

• Fractions within Fractions: To simplify a fraction that has a fraction in the numerator or denominator, multiply the numerator by the "flip" of the denominator.

  • Example: $\frac{\frac{2}{5}}{\frac{3}{8}} = \frac{2}{5} \times \frac{8}{3} = \frac{16}{15}$.

  • Example for whole numbers: $\frac{6}{\frac{2}{3}} = \frac{6}{1} \times \frac{3}{2} = 9$.

Simple vs. Complex Fractions

• Simple Fraction: Neither the numerator nor the denominator contains addition or subtraction (e.g., $\frac{4 \times 8}{2 \div 2}$). Factors can be canceled immediately.

• Complex Fraction: Contains addition or subtraction in the numerator or denominator (e.g., $\frac{1+2}{4-2}$).

• Operational Rule: Addition or subtraction must be performed BEFORE breaking down or canceling terms. Canceling terms before adding/subtracting lead to incorrect results.

  • Correct: $\frac{4+8}{2+2} = \frac{12}{4} = 3$.

  • Incorrect: Attempting to cancel individual numbers before summing.

• The Complex Numerator Shortcut: A fraction with a complex numerator can be split into two separate fractions.

  • Example: $\frac{1+2}{6} = \frac{1}{6} + \frac{2}{6}$.

  • This is NOT true for complex denominators: $\frac{16}{4+8} \neq \frac{16}{4} + \frac{16}{8}$.

• Application Example: Evaluating $\frac{9,999^2 + 9,999}{9,999}$.

  • Split the numerator: $\frac{9,999^2}{9,999} + \frac{9,999}{9,999}$.

  • Simplify: $9,999 + 1 = 10,000$.

Mixed Numerals

• Definition: A mixed numeral is the sum of a whole number and a fraction (e.g., $1 \frac{2}{3}$ is actually $1 + \frac{2}{3}$).

• Conversion to Fraction:

  • Method 1: Express the whole number as a fraction over $1$ and add it to the other fraction using common denominators.

  • Method 2 (Multiplier Method): Multiply the denominator by the whole number, add the numerator, and place the result over the original denominator. Example: $8 \frac{3}{5} = \frac{5 \times 8 + 3}{5} = \frac{43}{5}$.

• Conversion from Fraction to Mixed Numeral: Divide the numerator by the denominator. Use the quotient as the whole number and the remainder as the new numerator over the original denominator.

  • Example: $\frac{43}{5}$. $43 \div 5 = 8$ remainder $3 \rightarrow 8 \frac{3}{5}$.

• Golden Rule for Operations: Always convert mixed numerals into improper fractions before adding, subtracting, multiplying, or dividing them.

Reciprocals

• Condition: Any two numbers whose product equals exactly $+1$ are reciprocals.

  • Standard example: $2$ and $\frac{1}{2}$ ($2 \times \frac{1}{2} = 1$).

• Nuances: Reciprocals are often thought of as "flips," but not all reciprocals look like simple flips.

  • Irregular Example: $\frac{\sqrt{3}}{3}$ and $\sqrt{3}$ are reciprocals because $\frac{\sqrt{3}}{3} \times \frac{\sqrt{3}}{1} = \frac{3}{3} = 1$.

  • A fraction may have more than one reciprocal form: $\frac{1}{\sqrt{3}}$ is also a reciprocal of $\sqrt{3}$.

• Sign Check: $-2 \frac{1}{3}$ and $\frac{3}{7}$ are NOT reciprocals because their product is $-1$, not $+1$.

The "Comparison Trick"

• Context: Comparing fractions is frequent in GRE Quantitative Comparison questions.

• Mechanism: Cross-multiply the numerators and denominators. Multiply from the BOTTOM UP and write the product over the corresponding numerator.

• Rule: The fraction under the larger product is the larger fraction.

  • Example: Compare $\frac{3}{5}$ and $\frac{4}{7}$.

  • $7 \times 3 = 21$.

  • $5 \times 4 = 20$.

  • Since 21 > 20, \frac{3}{5} > \frac{4}{7}.

• Internal Logic: This technique effectively finds the numerators of the fractions once they are converted to the same LCD ($35$ in the example above) without the labor of full conversion.

Properties of Zero

• Properties:

  1. If numerator is $0$, the fraction is $0$.

  2. If denominator is $0$, the fraction is undefined.

• GRE Restrictions: Questions often include terms like $x \neq -a$, $ab \neq 0$, or xy > 0. These are usually technicalities provided to ensure denominators do not equal zero (closing loopholes) and typically do not add difficulty to the logic of the problem.

Fractions Between 0 and 1 (Proper Fractions)

• The Opposite Property: Fractions between $0$ and $1$ behave in the opposite manner to numbers greater than $1$ when combined with other positive values.

• Multiplication/Division:

  • Multiplying by a value >1 makes the value bigger ($10 \times 2 = 20$).

  • Multiplying by a proper fraction makes the value smaller ($10 \times \frac{1}{2} = 5$).

  • Dividing by a value >1 makes the value smaller ($10 \div 2 = 5$).

  • Dividing by a proper fraction makes the value bigger ($10 \div \frac{1}{2} = 20$).

• Squares and Square Roots:

  • Squaring a proper fraction makes it smaller: $(\frac{1}{2})^2 = \frac{1}{4}$.

  • Taking the square root of a proper fraction makes it larger: $\sqrt{\frac{1}{2}} \approx 0.71$.

• GRE Strategy: Recognizing these behaviors allows you to solve comparison problems quickly without calculating specific difficult values (e.g., $\frac{1}{5}^6$ vs $\frac{1}{5}^7$).

FUQ's: Fractions with Unspecified Quantities

• Definition: A problem involving fractional quantities but no actual specified values for the whole.

• Strategy: Pick a number to represent the total. To keep calculations as simple whole numbers, pick a number equal to the PRODUCT of EVERY denominator mentioned in the problem.

• Example (The Pie Problem):

  • Adam eats $\frac{1}{3}$, Billy eats $\frac{1}{4}$ of what's left, Cletus eats $\frac{1}{2}$ of the remainder.

  • Product of denominators: $3 \times 4 \times 2 = 24$ slices.

  • Adam: $1/3 \text{ of } 24 = 8$ slices (leaves $16$).

  • Billy: $1/4 \text{ of } 16 = 4$ slices (leaves $12$).

  • Cletus: $1/2 \text{ of } 12 = 6$ slices (leaves $6$).

  • Fraction left: $\frac{6}{24} = \frac{1}{4}$.

• Limitation: You CANNOT pick numbers if the problem contains any real numbers (e.g., "the shop has $80$ horror movies"). In those cases, you must use algebra (let Total $= x$).

"Multi-Level" Fractions

• Definition: Complex fractions that contain fractions within their numerators or denominators, creating multiple "levels."

• Simplification Strategy: Rewrite one level at a time, moving from the bottom-most level upwards.

• Example Task: Simplify $\frac{a}{d + \frac{b}{c}}$.

  • Bottom level: $d + \frac{b}{c} = \frac{cd + b}{c}$.

  • Full fraction: $\frac{a}{\frac{cd + b}{c}}$.

  • Final step: $a \times \frac{c}{cd + b} = \frac{ac}{cd + b}$.

Properties of Numerators and Denominators

1. Numerator Change: Increasing the numerator makes a fraction bigger.

2. Denominator Change: Increasing the denominator makes a fraction smaller.

3. Same-Amount Change: Adding the same amount to both the numerator and denominator brings the fraction's value closer to $1$. Conversely, decreasing both by the same amount moves the value further from $1$.

   • If current value < 1, adding same amount increases the value (e.g., $\frac{1}{2} \rightarrow \frac{2}{3} \rightarrow \frac{999}{1,000} \approx 1$).

   • If current value > 1, adding same amount decreases the value (e.g., $\frac{3}{2} \rightarrow \frac{4}{3}$).