2.2 Fractions, Decimals, Percents & Ratios

2.2 Fractions, Decimals, Percents & Ratios

Fractions, decimals, percents, and ratios are all different mathematical representations of proportional relationships and rational numbers. GRE Quantitative Reasoning tests your ability to seamlessly convert between these formats and apply them to real-world scenarios such as financial growth, mixture problems, and scaling rates.

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Fraction Arithmetic & Comparison

A fraction represents a division of two numbers, $\frac{a}{b}$, where $a$ is the numerator and $b$ ($b \neq 0$) is the denominator.

Core Operations with Fractions

• Addition & Subtraction: Find a Least Common Denominator (LCD). $$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$
• Multiplication: Multiply numerators together and denominators together. $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$
• Division: Multiply by the reciprocal of the divisor. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

Cross-Multiplication Method for Comparing Fractions

When comparing two positive fractions $\frac{a}{b}$ and $\frac{c}{d}$, cross-multiply the numerators by the opposite denominators:

$$\frac{a}{b} \text{ vs } \frac{c}{d} \implies a \times d \text{ vs } b \times c$$

• If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$.
• If $a \times d < b \times c$, then $\frac{a}{b} < \frac{c}{d}$.

Example Comparison: $\frac{7}{11}$ vs $\frac{5}{8}$

• Cross-products: $7 \times 8 = 56$ versus $11 \times 5 = 55$.
• Since $56 > 55$, it follows that $\frac{7}{11} > \frac{5}{8}$.

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Decimals & Decimal-Fraction Conversions

Decimals represent fractions with denominators that are powers of ten ($10, 100, 1000, \dots$).

Converting Repeating Decimals to Fractions

To convert a repeating decimal like $0.\overline{36} = 0.363636\dots$ into a fraction:

1. Let $x = 0.363636\dots$
2. Multiply by $10^k$ where $k$ is the period length ($k=2$ digits repeating): $100x = 36.363636\dots$
3. Subtract the original equation: $100x - x = 36.363636\dots - 0.363636\dots \implies 99x = 36$.
4. Solve for $x$: $x = \frac{36}{99} = \frac{4}{11}$.

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Percentages, Percent Change & Successive Percents

A percent simply means "per hundred" ($1\% = \frac{1}{100} = 0.01$).

The Universal Percent Change Formula

$$\text{Percent Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100\%$$

• If the result is positive, it is a percent increase.
• If the result is negative, it is a percent decrease.

GRE Trap - The Reference Base: The denominator in the percent change formula is ALWAYS the original ("Old") value, never the new value!

Successive Percent Changes

Consecutive percentage changes are multiplicative, NOT additive.

• An increase of $x\%$ corresponds to multiplying by a factor of $\left(1 + \frac{x}{100}\right)$.
• A decrease of $y\%$ corresponds to multiplying by a factor of $\left(1 - \frac{y}{100}\right)$.

Worked Example: Price Shift Analysis

An item priced at $\text{\textdollar}100$ receives a $20\%$ markup followed by a $20\%$ markdown:

$$\text{Final Price} = 100 \times (1 + 0.20) \times (1 - 0.20) = 100 \times 1.20 \times 0.80 = 100 \times 0.96 = \$96$$

Result: A net decrease of $4\%$, NOT $0\%$!

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Interest Calculations: Simple vs. Compound Interest

Compounding Frequency Definitions ($n$)

• Annual: $n = 1$
• Semi-annual: $n = 2$
• Quarterly: $n = 4$
• Monthly: $n = 12$

Worked Example: Compound Interest Calculation

Invest $\text{\textdollar}5,000$ at $8\%$ annual interest compounded semi-annually ($n=2$) for $2$ years ($t=2$):

$$A = 5000 \left(1 + \frac{0.08}{2}\right)^{2 \times 2} = 5000 (1.04)^4$$ $$(1.04)^2 = 1.0816 \implies (1.04)^4 = (1.0816)^2 \approx 1.16985856$$ $$A = 5000 \times 1.16985856 = \$5,849.29$$

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Ratios, Proportions & Direct/Inverse Variation

A ratio expresses the relative size of two or more quantities, written as $A:B$, $\frac{A}{B}$, or "$A$ to $B$".

Part-to-Part vs. Part-to-Whole

If the ratio of apples to oranges in a basket is $3:5$:

• Part-to-Part Ratio: Apples : Oranges $= 3:5$.
• Total Parts: $3 + 5 = 8$.
• Part-to-Whole Ratios: Apples $= \frac{3}{8}$ of total fruit; Oranges $= \frac{5}{8}$ of total fruit.

Combining Multiple Ratios

If Ratio $A:B = 2:3$ and Ratio $B:C = 4:5$, combine them by finding a common multiple for $B$ (LCM of $3$ and $4$ is $12$):

• Scale $A:B = 2:3$ by $4 \rightarrow 8:12$
• Scale $B:C = 4:5$ by $3 \rightarrow 12:15$
• Combined Ratio $A:B:C = 8:12:15$.

Direct vs. Inverse Variation

• Direct Proportion: $y = kx \implies \frac{y_1}{x_1} = \frac{y_2}{x_2}$ (as $x$ increases, $y$ increases proportionally).
• Inverse Proportion: $y = \frac{k}{x} \implies x_1 \cdot y_1 = x_2 \cdot y_2$ (as $x$ increases, $y$ decreases proportionally, such as worker-time problems).