3.3 Fractions, Decimals & Percentage Equivalences

Fractions, Decimals & Percentage Equivalences

Rational numbers can be expressed in three distinct notations: fractions, decimals, and percentages. Fluent mathematical reasoning requires recognizing that these are simply different expressions of the same underlying value. For example, half a dozen oranges can be described as $\frac{1}{2}$ of the set, $0.5$ of the set, or $50\%$ of the set.

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Concepts of Fractions: Types, Conversions, and Simplification

A fraction $\frac{a}{b}$ represents $a$ equal parts out of a total of $b$ equal parts forming a whole, where $a$ is the numerator and $b$ is the non-zero denominator.

1. Classifying Fractions

• Proper Fraction: The numerator is strictly less than the denominator (e.g., $\frac{3}{5}, \frac{7}{10}, \frac{1}{8}$). Its value is less than 1.
• Improper Fraction: The numerator is greater than or equal to the denominator (e.g., $\frac{7}{4}, \frac{11}{3}, \frac{8}{8}$). Its value is 1 or greater.
• Mixed Number: Combines a non-zero whole number and a proper fraction (e.g., $1 \frac{3}{4}, 3 \frac{2}{5}$).

2. Conversions Between Mixed Numbers and Improper Fractions

• Mixed Number to Improper Fraction: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator: $a \frac{b}{c} = \frac{(a \times c) + b}{c}$ Example: $3 \frac{5}{8} = \frac{(3 \times 8) + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8}$.
• Improper Fraction to Mixed Number: Divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains unchanged: Example: For $\frac{43}{6}$, $43 \div 6 = 7 \text{ R } 1$, so $\frac{43}{6} = 7 \frac{1}{6}$.

3. Equivalent Fractions and Simplification

Two fractions are equivalent if they represent the exact same value. Multiplying or dividing both numerator and denominator by the same non-zero integer generates equivalent fractions: $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

To simplify a fraction to lowest terms (simplest form), divide both the numerator and denominator by their Highest Common Factor (HCF): $\frac{48}{60} = \frac{48 \div 12}{60 \div 12} = \frac{4}{5}$

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Comparing and Ordering Fractions with Unlike Denominators

To compare fractions with different denominators, convert them to equivalent fractions sharing the Lowest Common Denominator (LCD), which is the Lowest Common Multiple (LCM) of the denominators.

Worked Example: Ordering Unlike Fractions

Problem: Arrange the fractions $\frac{3}{4}, \frac{5}{6}, \frac{7}{12},$ and $\frac{2}{3}$ in ascending order (from smallest to largest).

Solution Walkthrough:

1. Find the LCM of the denominators $4, 6, 12, 3$:
   • Multiples of 12: $12, 24, 36...$
   • Since 12 is divisible by 4 ($12 \div 4 = 3$), by 6 ($12 \div 6 = 2$), and by 3 ($12 \div 3 = 4$), the $\text{LCD} = 12$.
2. Convert each fraction to an equivalent fraction with denominator 12:
   • $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
   • $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
   • $\frac{7}{12} = \frac{7}{12}$
   • $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
3. Compare the numerators: $7 < 8 < 9 < 10$.
4. Ascending order: $\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}$.

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Fraction Operations: Addition, Subtraction, Multiplication & Division

1. Addition and Subtraction with Regrouping

Fractions must share a common denominator before adding or subtracting. When subtracting mixed numbers, regrouping 1 whole from the whole number portion is often necessary.

Worked Example: Mixed Number Subtraction with Regrouping

Problem: Calculate $5 \frac{1}{4} - 2 \frac{5}{6}$.

Solution Steps:

1. Find the LCD of 4 and 6: $\text{LCM}(4, 6) = 12$.
2. Convert fractional parts: $5 \frac{1}{4} = 5 \frac{3}{12}$ and $2 \frac{5}{6} = 2 \frac{10}{12}$.
3. Since $\frac{3}{12} < \frac{10}{12}$, borrow 1 whole from 5: $5 \frac{3}{12} = 4 + 1 + \frac{3}{12} = 4 + \frac{12}{12} + \frac{3}{12} = 4 \frac{15}{12}$
4. Subtract whole numbers and fractions separately: $4 \frac{15}{12} - 2 \frac{10}{12} = (4 - 2) + \left(\frac{15 - 10}{12}\right) = 2 \frac{5}{12}$

2. Multiplication of Fractions

• Fraction of a Quantity: Replace the word "of" with multiplication: $\frac{3}{8} \text{ of } 96 = \frac{3}{8} \times 96 = 3 \times (96 \div 8) = 3 \times 12 = 36$
• Multiplying Two Fractions: Multiply numerators together and denominators together, cross-simplifying common factors beforehand: $\frac{4}{9} \times \frac{15}{16} = \frac{4 \div 4}{9 \div 3} \times \frac{15 \div 3}{16 \div 4} = \frac{1}{3} \times \frac{5}{4} = \frac{5}{12}$
• Multiplying Mixed Numbers: Convert all mixed numbers into improper fractions before multiplying: $2 \frac{2}{5} \times 1 \frac{7}{8} = \frac{12}{5} \times \frac{15}{8} = \frac{3 \times 3}{1 \times 2} = \frac{9}{2} = 4 \frac{1}{2}$

3. Division of Fractions: The Reciprocal Rule

The reciprocal (or multiplicative inverse) of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. The product of any non-zero fraction and its reciprocal is always 1 ($\frac{a}{b} \times \frac{b}{a} = 1$).

To divide by a fraction, multiply by its reciprocal ("Invert and Multiply"): $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Worked Example: Dividing Mixed Numbers

Problem: Compute $3 \frac{3}{4} \div \frac{5}{8}$.

1. Convert mixed number to improper fraction: $3 \frac{3}{4} = \frac{15}{4}$.
2. Invert the divisor $\frac{5}{8}$ to get its reciprocal $\frac{8}{5}$.
3. Multiply and simplify: $\frac{15}{4} \times \frac{8}{5} = \frac{15 \div 5}{4 \div 4} \times \frac{8 \div 4}{5 \div 5} = \frac{3}{1} \times \frac{2}{1} = 6$

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Decimal Place Value and Arithmetic

Digits to the right of the decimal point represent fractional parts of powers of 10:

• Tenths ($0.1 = \frac{1}{10} = 10^{-1}$): First place right of decimal point.
• Hundredths ($0.01 = \frac{1}{100} = 10^{-2}$): Second place right of decimal point.
• Thousandths ($0.001 = \frac{1}{1,000} = 10^{-3}$): Third place right of decimal point.

Decimal Operations

1. Addition and Subtraction: Align numbers strictly by their decimal points vertically. Add placeholder zeros so all numbers have the same number of decimal places.
   • $14.6 - 8.75$: Rewrite as $14.60 - 8.75 = 5.85$.
2. Multiplication: Ignore the decimal points and multiply as whole numbers. Then, count the total number of decimal places across both factors and place the decimal point that many places from the right in the product.
   • $4.25 \times 0.6$: Compute $425 \times 6 = 2,550$. Factor 4.25 has 2 decimal places, and 0.6 has 1 decimal place ($2 + 1 = 3$ total). Place point 3 digits from right: $2.550 = 2.55$.
3. Division: If the divisor has decimal digits, multiply both divisor and dividend by $10, 100,$ or $1,000$ to turn the divisor into a whole number before using standard long division.
   • $14.4 \div 0.12$: Multiply both by 100 $\rightarrow 1,440 \div 12 = 120$.

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The Equivalence Triad: Fractions, Decimals & Percentages

A percentage expresses a ratio or quantity as a fraction of 100 ("per centum"). Thus, $p\% = \frac{p}{100} = p \div 100$; for example, $45\% = \frac{45}{100} = 0.45$.

Conversion Mechanics

• Fraction to Decimal: Divide the numerator by the denominator ($3 \div 4 = 0.75$).
• Decimal to Percentage: Multiply by $100\%$ (shift decimal point 2 places right): $0.75 \times 100\% = 75\%$.
• Percentage to Fraction: Place the percentage over 100 and reduce to lowest terms: $45\% = \frac{45}{100} = \frac{9}{20}$.
• Fraction to Percentage: Multiply the fraction by $100\%$: $\frac{3}{5} \times 100\% = \frac{300}{5}\% = 60\%$.

Benchmark Equivalence Reference Table

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Practical Multi-Step Problem Solving

Worked Example 1: Finding Percentage of a Quantity & Net Price

Problem: A school uniform retailer in Spanish Town marks a blazer at J$6,800. For an annual Back-to-School sale, the shop offers a $25\%$ discount. If a customer holds a promotional coupon for an extra J$500 off the discounted price, what is the customer's final price?

Solution Walkthrough:

1. Convert discount to a fraction or decimal: $25\% = \frac{1}{4} = 0.25$.
2. Calculate discount value: $\frac{1}{4} \times \text{J\textdollar}6,800 = \text{J\textdollar}1,700$.
3. Compute discounted sale price: $\text{J\textdollar}6,800 - \text{J\textdollar}1,700 = \text{J\textdollar}5,100$.
4. Apply promotional voucher: $\text{J\textdollar}5,100 - \text{J\textdollar}500 = \text{J\textdollar}4,600$.
5. Final price: J$4,600.

Worked Example 2: Comparing Proportions and Test Scores

Problem: In a mock assessment, Shenelle scored $\frac{17}{20}$ on a 20-item Mathematics paper and $82\%$ on a 50-item Science paper. On which paper did she achieve a higher percentage score, and how many questions did she answer correctly across both papers combined?

Solution Walkthrough:

1. Convert the Mathematics score to a percentage: $\frac{17}{20} = \frac{17 \times 5}{20 \times 5} = \frac{85}{100} = 85\%$
2. Compare percentages: $85\% \text{ (Mathematics)} > 82\% \text{ (Science)}$.
   • Shenelle achieved the higher percentage score on the Mathematics paper.
3. Compute the number of correct questions on each paper:
   • Mathematics: 17 correct.
   • Science: $82\% \text{ of } 50 = \frac{82}{100} \times 50 = \frac{82}{2} = 41$ correct.
4. Total questions answered correctly: $17 + 41 = 58$ out of $20 + 50 = 70$.
5. Combined accuracy percentage: $\frac{58}{70} = \frac{29}{35} \approx 82.86\%$.
6. Result: Mathematics had the higher percentage ($85\%$ vs. $82\%$), and she answered 58 questions correctly in total.