5.2 Fractions, Decimals, Ratios & Percentages

Fractions, Decimals, Ratios & Percentages

Fractions, decimals, ratios, and percentages represent four interconnected representations of rational proportional reasoning. On the FTCE General Knowledge Mathematics subtest, proportional reasoning forms the single largest category of applied arithmetic questions. Candidates must demonstrate seamless agility in interconverting between representations, executing fraction and mixed-number operations, solving direct and inverse variation models, and calculating multi-tier percentage changes.

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1. Fraction Concepts, Equivalence & Operations

A fraction $\frac{a}{b}$ consists of a numerator $a$ (representing the number of equal parts being considered) and a denominator $b$ (representing the total number of equal parts into which the whole is divided, where $b \neq 0$).

Fraction Types & Conversions

• Proper Fraction: Numerator is strictly less than denominator ($|a| < |b|$, e.g., $\frac{3}{7}, \frac{5}{8}$). Value is between $-1$ and $1$.
• Improper Fraction: Numerator is greater than or equal to denominator ($|a| \ge |b|$, e.g., $\frac{11}{4}, \frac{9}{9}$). Value is $\ge 1$ or $\le -1$.
• Mixed Number: A combination of a non-zero integer and a proper fraction ($c \frac{a}{b}$, e.g., $2 \frac{3}{4}$).
  • Converting Mixed Number to Improper Fraction: $$c \frac{a}{b} = \frac{(c \times b) + a}{b} \qquad \implies \quad 4 \frac{3}{5} = \frac{(4 \times 5) + 3}{5} = \frac{23}{5}$$
  • Converting Improper Fraction to Mixed Number: Divide numerator by denominator: $29 \div 6 = 4$ with a remainder of $5 \implies \frac{29}{6} = 4 \frac{5}{6}$.

Equivalent Fractions & Simplest Form

Multiplying or dividing both numerator and denominator by the same non-zero integer preserves fractional value:

$$\frac{a}{b} = \frac{a \cdot k}{b \cdot k} \qquad (k \neq 0)$$

A fraction is in simplest form (lowest terms) when $\text{GCF}(a, b) = 1$. To reduce: divide numerator and denominator by their greatest common factor:

$$\frac{48}{72} = \frac{48 \div 24}{72 \div 24} = \frac{2}{3}$$

Comparing and Ordering Fractions

1. Cross-Multiplication Comparison Method: $$\frac{a}{b} < \frac{c}{d} \iff a \cdot d < b \cdot c \qquad (\text{for positive } b, d)$$ Example: Compare $\frac{5}{8}$ and $\frac{7}{11}$: $$5 \times 11 = 55 \qquad \text{vs.} \qquad 8 \times 7 = 56$$ Since $55 < 56$, it follows that $\frac{5}{8} < \frac{7}{11}$.
2. Common Denominator Method: Find the Least Common Multiple of the denominators (LCD) and scale numerators.
3. Benchmark Comparison: Compare fractions against standard benchmarks ($0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$).

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2. Computational Operations with Fractions & Mixed Numbers

Addition & Subtraction of Fractions

Fractions must have identical denominators before adding or subtracting numerators:

$$\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$$

If denominators differ, determine the Least Common Denominator $\text{LCD} = \text{LCM}(d_1, d_2)$ and rewrite equivalent fractions:

$$\frac{5}{6} + \frac{3}{8} \implies \text{LCM}(6, 8) = 24 \implies \frac{5 \times 4}{6 \times 4} + \frac{3 \times 3}{8 \times 3} = \frac{20}{24} + \frac{9}{24} = \frac{29}{24} = 1 \frac{5}{24}$$

Mixed Number Subtraction with Regrouping (Borrowing)

Subtract $5 \frac{1}{4} - 2 \frac{5}{6}$:

1. Find common denominator: $\text{LCM}(4, 6) = 12$: $$5 \frac{3}{12} - 2 \frac{10}{12}$$
2. Since $\frac{3}{12} < \frac{10}{12}$, regroup $1$ from whole number $5$: $$5 \frac{3}{12} = 4 + 1 + \frac{3}{12} = 4 + \frac{12}{12} + \frac{3}{12} = 4 \frac{15}{12}$$
3. Execute subtraction: $$4 \frac{15}{12} - 2 \frac{10}{12} = (4 - 2) + \left(\frac{15 - 10}{12}\right) = 2 \frac{5}{12}$$

Multiplication of Fractions

Multiply numerators directly and denominators directly. Always cross-cancel common factors prior to multiplying to maintain minimal computation:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

Example with cross-cancellation:

$$\frac{14}{25} \times \frac{15}{28} = \frac{14^1}{25^5} \times \frac{15^3}{28^2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10}$$

Rule for Mixed Numbers: Convert all mixed numbers to improper fractions before multiplying:

$$2 \frac{1}{3} \times 1 \frac{4}{7} = \frac{7}{3} \times \frac{11}{7} = \frac{7^1 \times 11}{3 \times 7^1} = \frac{11}{3} = 3 \frac{2}{3}$$

Division of Fractions

Dividing by a fraction is algebraically identical to multiplying by its reciprocal (multiplicative inverse):

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \qquad \left(b, c, d \neq 0\right)$$

Example:

$$3 \frac{3}{4} \div 1 \frac{7}{8} = \frac{15}{4} \div \frac{15}{8} = \frac{15}{4} \times \frac{8}{15} = \frac{15^1}{4^1} \times \frac{8^2}{15^1} = \frac{2}{1} = 2$$

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3. Decimal Arithmetic & Interconversions

Decimal Operations

• Addition & Subtraction: Align decimal points vertically. Pad trailing zeros as place holders: $$\begin{aligned} 42.800 \\ + 7.649 \\ \hline 50.449 \end{aligned}$$
• Multiplication: Multiply numbers ignoring decimal points; then sum the total decimal places in both factors to place the decimal point in the final product: $$3.14 \; (2\text{ decimal places}) \times 0.05 \; (2\text{ decimal places}) = 0.1570 \; (4\text{ decimal places}) = 0.157$$
• Division: If the divisor is a decimal, multiply both divisor and dividend by $10^k$ to convert divisor into an integer: $$14.28 \div 0.4 \implies \frac{14.28 \times 10}{0.4 \times 10} = \frac{142.8}{4} = 35.7$$

Algebraic Conversion of Repeating Decimals to Fractions

Every periodic repeating decimal can be transformed into an exact rational fraction using algebra:

Pure Repeating Decimal Example

Convert $N = 0.\overline{45} = 0.454545\dots$ to a fraction:

1. Let $x = 0.\overline{45}$.
2. Because the repeating block has $2$ digits, multiply by $10^2 = 100$: $$100x = 45.\overline{45}$$
3. Subtract the original equation: $$\begin{aligned} 100x &= 45.4545\dots \\ - x &= \;\;0.4545\dots \\ \hline 99x &= 45 \end{aligned}$$
4. Solve for $x$ and simplify: $$x = \frac{45}{99} = \frac{45 \div 9}{99 \div 9} = \frac{5}{11}$$

Delayed Repeating Decimal Example

Convert $N = 0.1\overline{6} = 0.16666\dots$ to a fraction:

1. Let $x = 0.1\overline{6}$.
2. Multiply by $10$ to isolate the non-repeating part: $10x = 1.\overline{6}$.
3. Multiply by $100$ to include one repeating period: $100x = 16.\overline{6}$.
4. Subtract equations: $$100x - 10x = 16.\overline{6} - 1.\overline{6} \implies 90x = 15 \implies x = \frac{15}{90} = \frac{1}{6}$$

Master Fraction-Decimal-Percent Conversion Reference

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4. Ratios, Rates, Proportions & Variations

Ratios & Rates

• Ratio: A comparison of two quantities by division, written as $a:b$, $a\text{ to }b$, or $\frac{a}{b}$.
  • Part-to-Part Ratio: Compares one component to another (e.g., boys to girls $= 12:15 = 4:5$).
  • Part-to-Whole Ratio: Compares one component to the total aggregate (e.g., boys to total students $= 12:27 = 4:9$).
• Unit Rate: A rate normalized so the denominator equals $1$ unit (e.g., speed in miles per hour, unit cost in dollars per ounce): $$\text{Unit Price} = \frac{\text{Total Cost}}{\text{Total Quantity}}$$

Solving Proportions

A proportion is an equation stating that two ratios are equal:

$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c \qquad (\text{Cross-Multiplication Property})$$

Direct vs. Inverse Variation Models

Inverse Variation Applied Example

If $6$ educators can grade a standardized assessment batch in $8$ hours, how many hours would it take $10$ educators working at the same pace?

1. Identify relationship: Workforce size ($w$) and grading time ($t$) vary inversely $\implies w_1 t_1 = w_2 t_2$.
2. Substitute known values: $$6 \times 8 = 10 \times t_2 \implies 48 = 10 t_2 \implies t_2 = \frac{48}{10} = 4.8\text{ hours}$$
3. Convert decimal time: $0.8\text{ hr} \times 60\text{ min/hr} = 48\text{ minutes} \implies 4\text{ hours } 48\text{ minutes}$.

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5. Percent Applications, Markups, Discounts & Financial Calculations

Percent Fundamentals

A percent represents a fraction with a denominator of $100$ ($\text{per cent} = \text{per hundred}$):

$$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\% \qquad \iff \qquad \text{Part} = \text{Percent (in decimal form)} \times \text{Whole}$$

Percent Change (Increase / Decrease)

$$\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$$

⚠️ CRITICAL FTCE TRAP: Always divide the absolute difference by the ORIGINAL (initial) value, NEVER the new or discounted value!

Example: A school district budget increases from $\text{\textdollar}4,000,000$ to $\text{\textdollar}4,600,000$. What is the percent increase?

$$\text{Percent Increase} = \frac{4,600,000 - 4,000,000}{4,000,000} \times 100\% = \frac{600,000}{4,000,000} \times 100\% = 0.15 \times 100\% = 15\%$$

Successive (Compound) Percentage Changes

When an item undergoes multiple consecutive percentage adjustments, you cannot simply add or subtract the percentages. Each adjustment applies to the updated intermediate base.

Multiplier Method

• A markup of $p\%$ corresponds to multiplying by $(1 + r)$ where $r = p/100$.
• A discount of $d\%$ corresponds to multiplying by $(1 - d/100)$.

Example: A graphing calculator wholesale cost is $\text{\textdollar}80$. The bookstore marks it up by $25\%$. Later, during a back-to-school promotion, the bookstore offers a $20\%$ discount off the retail price. What is the final sale price?

1. Markup multiplier: $1 + 0.25 = 1.25$. $$\text{Retail Price} = 80 \times 1.25 = \$100$$
2. Discount multiplier: $1 - 0.20 = 0.80$. $$\text{Final Sale Price} = 100 \times 0.80 = \$80$$
3. Insight: A $25\%$ markup followed by a $20\%$ discount returned the price to the original $\text{\textdollar}80$ because $1.25 \times 0.80 = 1.00$.

Example 2 (Consecutive Discounts): An item priced at $\text{\textdollar}200$ receives a $30\%$ store discount and an additional $10\%$ loyalty coupon.

$$\text{Final Price} = 200 \times (1 - 0.30) \times (1 - 0.10) = 200 \times 0.70 \times 0.90 = 200 \times 0.63 = \$126$$

The effective total discount is $1 - 0.63 = 0.37 = 37\%$ (NOT $30\% + 10\% = 40\%$!).

Simple Interest & Compound Interest

• Simple Interest Formula: $$I = P \cdot r \cdot t$$ Where:
  • $I = \text{Interest earned or paid}$
  • $P = \text{Principal amount invested or borrowed}$
  • $r = \text{Annual interest rate as a decimal (e.g., } 6\% = 0.06)$
  • $t = \text{Time in YEARS (e.g., } 18\text{ months } = 1.5\text{ years)}$
• Total Accumulated Balance ($A$): $$A = P + I = P + Prt = P(1 + rt)$$
• Compound Interest (Compounded $n$ times per year): $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
  • Annually: $n = 1$
  • Semiannually: $n = 2$
  • Quarterly: $n = 4$
  • Monthly: $n = 12$