3.1 Fractions: Types, Equivalencies & Operations

3.1 Fractions: Types, Equivalencies & Operations

Quick Summary: Rational numbers are numbers that can be written as a ratio of two integers $\frac{a}{b}$ with $b \neq 0$. Mastering fraction arithmetic requires a fluent understanding of equivalent fractions, least common denominators for addition and subtraction, and reciprocal multiplication for division. In Ontario's Grade 7–9 mathematics curriculum, teachers must address common student misconceptions—such as adding numerators and denominators directly—by reinforcing visual area models, fraction strips, and algebraic field properties.

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1. Classifying Rational Numbers & Fraction Forms

A rational number is any real number that can be expressed as the quotient of two integers $\frac{a}{b}$, where ( a ) is the numerator, ( b ) is the denominator, and $b \neq 0$. Rational numbers encompass integers, finite (terminating) decimals, repeating decimals, and fractions.

Fractions occur in three primary structural formats:

1. Proper Fractions: The absolute value of the numerator is strictly less than the absolute value of the denominator (( |a| < b )). A proper fraction represents a quantity strictly between ( 0 ) and ( 1 ) (or ( -1 ) and ( 0 )). Examples: $\frac{3}{4}, \frac{7}{10}, -\frac{5}{12}$.

2. Improper Fractions: The absolute value of the numerator is greater than or equal to the absolute value of the denominator ($|a| \ge b$). An improper fraction represents a quantity greater than or equal to ( 1 ). Examples: $\frac{9}{4}, \frac{15}{5}, \frac{22}{7}$.

3. Mixed Numbers: An expression combining a non-zero integer and a proper fraction, written in the form $w\frac{a}{b}$ (which implicitly denotes $w + \frac{a}{b}$ for positive values). Examples: $2\frac{1}{4}, 3\frac{5}{8}$.

Conversion Algorithms

• Converting Mixed Numbers to Improper Fractions: Multiply the whole number ( w ) by the denominator ( b ), add the numerator ( a ), and place the result over the original denominator:

  $$w\frac{a}{b} = \frac{w \cdot b + a}{b}$$

  Example: Convert $4\frac{3}{7}$ to an improper fraction: $\frac{4 \cdot 7 + 3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$.

• Converting Improper Fractions to Mixed Numbers: Perform integer division of numerator ( a ) by denominator ( b ). The quotient becomes the whole number ( w ), and the remainder ( r ) becomes the new numerator over ( b ):

  $$\frac{a}{b} = q + \frac{r}{b} = q\frac{r}{b} \quad \text{where } a = q \cdot b + r \text{ and } 0 \le r < b.$$

  Example: Convert $\frac{38}{5}$ to a mixed number: $38 \div 5 = 7$ with remainder ( 3 ), yielding $7\frac{3}{5}$.

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2. Equivalent Fractions & Simplifying to Lowest Terms

The Fundamental Property of Fractions states that multiplying or dividing both the numerator and denominator of a fraction by the same non-zero real number ( k ) produces an equivalent fraction:

$$\frac{a}{b} = \frac{a \cdot k}{b \cdot k} = \frac{a \div k}{b \div k} \quad (b, k \neq 0)$$

Simplifying to Lowest Terms (Irreducible Form)

A fraction $\frac{a}{b}$ is in lowest terms (or irreducible) when $\operatorname{gcd}(a, b) = 1$; that is, the greatest common divisor of numerator ( a ) and denominator ( b ) is 1.

To simplify a fraction:

1. Find the Greatest Common Divisor (GCD) of the numerator and denominator using prime factorization or the Euclidean Algorithm.
2. Divide both numerator and denominator by the GCD.

Example: Simplify $\frac{84}{126}$.

• Prime factorization of 84: $2^2 \cdot 3 \cdot 7$.
• Prime factorization of 126: $2 \cdot 3^2 \cdot 7$.
• $\operatorname{gcd}(84, 126) = 2^1 \cdot 3^1 \cdot 7^1 = 42$.
• Divide numerator and denominator by 42: $\frac{84 \div 42}{126 \div 42} = \frac{2}{3}$.

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3. Addition and Subtraction of Fractions

Like Denominators

When denominators are identical, add or subtract the numerators while keeping the denominator unchanged:

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

Unlike Denominators & Finding the Least Common Denominator (LCD)

When adding or subtracting fractions with different denominators:

1. Determine the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators: $\text{LCD} = \operatorname{lcm}(b, d)$.
2. Rewrite each fraction as an equivalent fraction with the LCD as its denominator.
3. Add or subtract the numerators.
4. Simplify the resulting fraction to lowest terms or a mixed number.

$$\frac{a}{b} \pm \frac{c}{d} = \frac{a \cdot \left(\frac{\text{LCD}}{b}\right) \pm c \cdot \left(\frac{\text{LCD}}{d}\right)}{\text{LCD}}$$

Example: Compute $\frac{5}{6} - \frac{3}{8}$.

• $\operatorname{lcm}(6, 8) = 24$.
• Rewrite: $\frac{5 \cdot 4}{6 \cdot 4} - \frac{3 \cdot 3}{8 \cdot 3} = \frac{20}{24} - \frac{9}{24} = \frac{11}{24}$.

Mixed Number Addition & Subtraction Strategies

When operating on mixed numbers, candidate teachers can choose between two methods:

• Method 1 (Improper Fraction Conversion): Convert all mixed numbers to improper fractions, compute using common denominators, and convert back. This method eliminates errors caused by regrouping/borrowing.
• Method 2 (Whole and Fractional Part Separation): Separate whole numbers and fractions. When subtracting, if the fractional minuend is smaller than the fractional subtrahend, borrow ( 1 ) (written as $\frac{d}{d}$) from the whole number component.

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4. Multiplication and Division of Fractions

Multiplication of Fractions

To multiply two fractions, multiply their numerators together and their denominators together:

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

Pedagogical Tip (Cross-Simplification): Simplify common factors between any numerator and any denominator before multiplying to keep calculations efficient and avoid large intermediate numbers.

Division of Fractions and Reciprocals

The multiplicative inverse or reciprocal of a non-zero fraction $\frac{a}{b}$ is $\frac{b}{a}$, because $\frac{a}{b} \times \frac{b}{a} = 1$.

To divide by a fraction, multiply by its reciprocal ("Invert and Multiply"):

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c} \quad (b, c, d \neq 0)$$

Mathematical Proof/Justification: Multiply both the numerator fraction and denominator fraction of the complex fraction $\frac{\frac{a}{b}}{\frac{c}{d}}$ by the reciprocal of the denominator $\frac{d}{c}$:

$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b} \times \frac{d}{c}}{\frac{c}{d} \times \frac{d}{c}} = \frac{\frac{a \cdot d}{b \cdot c}}{1} = \frac{a}{b} \times \frac{d}{c}$$

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5. Fraction Operations Summary Matrix

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6. Step-by-Step Worked Problems

Worked Problem 1: Mixed Number Subtraction with Borrowing

Problem: Evaluate $7\frac{1}{6} - 3\frac{3}{4}$ and express the result as a mixed number in lowest terms.

Step-by-Step Solution:

1. Method 1: Convert to Improper Fractions

   • Convert $7\frac{1}{6}$: $\frac{7 \cdot 6 + 1}{6} = \frac{43}{6}$.
   • Convert $3\frac{3}{4}$: $\frac{3 \cdot 4 + 3}{4} = \frac{15}{4}$.

2. Find the Least Common Denominator (LCD)

   • Denominators are 6 and 4. $\operatorname{lcm}(6, 4) = 12$.

3. Convert to Equivalent Fractions with Denominator 12

   • $\frac{43}{6} = \frac{43 \cdot 2}{6 \cdot 2} = \frac{86}{12}$.
   • $\frac{15}{4} = \frac{15 \cdot 3}{4 \cdot 3} = \frac{45}{12}$.

4. Subtract Numerators

   $$\frac{86}{12} - \frac{45}{12} = \frac{86 - 45}{12} = \frac{41}{12}$$

5. Convert Back to a Mixed Number

   • $41 \div 12 = 3$ with a remainder of ( 5 ).
   • Final Answer: $3\frac{5}{12}$.

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Worked Problem 2: Recipe Scaling Multi-Step Problem

Problem: A catering recipe that serves 6 people requires $2\frac{1}{4}$ cups of milk. A chef needs to scale the recipe to serve 15 people. The chef has a container with 5 cups of milk. How much additional milk is needed, or how much milk will be left over?

Step-by-Step Solution:

1. Calculate the Scaling Factor

   $$\text{Scale Factor} = \frac{\text{Target Servings}}{\text{Original Servings}} = \frac{15}{6} = \frac{5}{2} = 2.5$$

2. Calculate Total Milk Required

   • Convert $2\frac{1}{4}$ to improper fraction: $\frac{9}{4}$.
   • Multiply by scale factor $\frac{5}{2}$: $$\text{Milk Required} = \frac{9}{4} \times \frac{5}{2} = \frac{45}{8} \text{ cups}$$
   • Express as a mixed number: $45 \div 8 = 5\frac{5}{8}$ cups.

3. Compare Available Milk to Required Milk

   • Milk available: ( 5 ) cups $= \frac{40}{8}$ cups.
   • Difference: $\frac{40}{8} - \frac{45}{8} = -\frac{5}{8}$ cups.
   • Final Answer: The chef does not have enough milk; an additional $\frac{5}{8}$ cup of milk is needed.

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Worked Problem 3: Shared Measurement Allocation Word Problem

Problem: An art teacher has a spool of decorative ribbon measuring $18\frac{3}{4}$ metres. She cuts pieces measuring $1\frac{3}{8}$ metres each for a student project. How many complete pieces can she cut, and what is the exact length of the remaining ribbon leftover on the spool?

Step-by-Step Solution:

1. Set Up Division for Number of Pieces

   $$\text{Number of pieces} = 18\frac{3}{4} \div 1\frac{3}{8}$$

2. Convert to Improper Fractions

   • $18\frac{3}{4} = \frac{18 \cdot 4 + 3}{4} = \frac{75}{4}$.
   • $1\frac{3}{8} = \frac{1 \cdot 8 + 3}{8} = \frac{11}{8}$.

3. Apply Invert and Multiply Rule

   $$\frac{75}{4} \div \frac{11}{8} = \frac{75}{4} \times \frac{8}{11} = \frac{75 \cdot 2}{1 \cdot 11} = \frac{150}{11}$$

4. Extract Complete Pieces and Fractional Part

   • $150 \div 11 = 13$ with remainder ( 7 ), so $\frac{150}{11} = 13\frac{7}{11}$.
   • This yields 13 complete pieces.

5. Calculate Remaining Length in Metres

   • Length used for 13 pieces: $13 \times 1\frac{3}{8} = 13 \times \frac{11}{8} = \frac{143}{8} = 17\frac{7}{8}$ metres.
   • Remaining ribbon on spool: $18\frac{3}{4} - 17\frac{7}{8} = \frac{75}{4} - \frac{143}{8} = \frac{150}{8} - \frac{143}{8} = \frac{7}{8}$ metre.
   • Final Answer: 13 complete pieces can be cut, with $\frac{7}{8}$ metre of ribbon remaining.