7.1 Equivalent Fractions, Mixed Numbers & Comparison

Fractions connect number, measurement and proportional thinking. Learn to explain equal parts, move between equivalent forms and use fractions in quantities. The examples below are original teaching exercises, not evidence of topic frequency or school scoring weights.

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1. Anatomy of a Fraction & Core Concepts

A fraction is an expression representing a part of an equal whole, a division operation, or a ratio between two quantities. It is written in the form:

$$\frac{a}{b}$$

• Numerator ($a$): The top integer, indicating how many equal parts are selected or under consideration.
• Denominator ($b$): The bottom integer ($b \neq 0$), indicating the total number of equal parts into which the whole is divided.
• Vinculum (Fraction Bar): The horizontal line separating numerator and denominator, representing mathematical division: $\frac{a}{b} = a \div b$.

The Three Classes of Fractions

1. Proper Fractions: The numerator is strictly less than the denominator ($a < b$). The value of the fraction is strictly less than $1$ (e.g., $\frac{3}{5}, \frac{7}{12}, \frac{1}{9}$).
2. Improper (Top-Heavy) Fractions: The numerator is greater than or equal to the denominator ($a \ge b$). The value is greater than or equal to $1$ (e.g., $\frac{7}{4}, \frac{11}{3}, \frac{5}{5}$).
3. Mixed Numbers: An expression combining a non-zero whole integer and a proper fraction (e.g., $1\frac{3}{4}, 3\frac{2}{3}, 5\frac{1}{6}$).

┌───────────────────────────────────────────────────────────────┐
│                     Fraction Types Matrix                     │
├──────────────────┬──────────────────┬─────────────────────────┤
│ Proper Fraction  │ Improper Fraction│      Mixed Number       │
│     (a < b)      │     (a ≥ b)      │   (Whole + Fraction)    │
├──────────────────┼──────────────────┼─────────────────────────┤
│  3/4  (Value < 1)│  7/4  (Value ≥ 1)│  1 3/4  (Value = 7/4)   │
│  5/8  (Value < 1)│ 11/3  (Value ≥ 1)│  3 2/3  (Value = 11/3)  │
│  2/5  (Value < 1)│ 19/6  (Value ≥ 1)│  3 1/6  (Value = 19/6)  │
└──────────────────┴──────────────────┴─────────────────────────┘

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2. Converting Between Improper Fractions and Mixed Numbers

Fluency in converting between improper fractions and mixed numbers is a prerequisite for executing arithmetic operations under timed conditions.

Improper Fraction to Mixed Number

To convert an improper fraction $\frac{a}{b}$ into a mixed number:

1. Divide the numerator ($a$) by the denominator ($b$) using integer division.
2. The quotient becomes the whole number part ($W$).
3. The remainder becomes the new numerator ($r$).
4. The denominator ($b$) remains unchanged.

$$\frac{a}{b} = W\frac{r}{b} \quad \text{where } a = (b \times W) + r$$

Worked Example: Convert $\frac{38}{7}$ to a mixed number.

• Calculate $38 \div 7$:
• $7 \times 5 = 35$, remainder $38 - 35 = 3$.
• Quotient = $5$, Remainder = $3$.
• Therefore, $\frac{38}{7} = 5\frac{3}{7}$.

Mixed Number to Improper Fraction

To convert a mixed number $W\frac{n}{d}$ into an improper fraction:

1. Multiply the whole number ($W$) by the denominator ($d$).
2. Add the numerator ($n$) to this product.
3. Place the resulting total over the original denominator ($d$).

$$W\frac{n}{d} = \frac{(W \times d) + n}{d}$$

Worked Example: Convert $6\frac{4}{9}$ to an improper fraction.

• Multiply whole part by denominator: $6 \times 9 = 54$.
• Add numerator: $54 + 4 = 58$.
• Keep the denominator: $\frac{58}{9}$.

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3. Equivalent Fractions & Simplifying to Lowest Terms

Two fractions are equivalent if they represent the exact same proportion or numerical value, even though they possess different numerators and denominators.

The Fundamental Law of Equivalence

Multiplying or dividing both the numerator and the denominator of a fraction by the exact same non-zero integer leaves the value of the fraction unchanged:

$$\frac{a}{b} = \frac{a \times k}{b \times k} \quad \text{and} \quad \frac{a}{b} = \frac{a \div k}{b \div k} \quad (k \neq 0)$$

Simplifying (Cancelling Down) to Lowest Terms

A fraction is in its simplest form (or lowest terms) when its numerator and denominator are coprime—meaning their Highest Common Factor (HCF) is $1$.

To simplify a fraction in a single step:

1. Identify the HCF of the numerator and denominator.
2. Divide both by that factor.

Worked Example: Simplify $\frac{48}{72}$ to lowest terms.

• Find the factors of 48: $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$.
• Find the factors of 72: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$.
• $\text{HCF}(48, 72) = 24$.
• Divide numerator and denominator by 24:

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

Alternative Sequential Method: If the HCF is not immediately obvious, divide by obvious shared prime factors sequentially ($\frac{48}{72}=\frac{24}{36}=\frac{4}{6}=\frac{2}{3}$, dividing both numerator and denominator at each step).

Missing Term Equivalence Problems

A missing-term equation can check whether both parts of a fraction have been scaled equally:

$$\frac{5}{8} = \frac{\square}{56}$$

1. Inspect the relationship between the known denominators: $56 \div 8 = 7$. The scaling factor $k = 7$.
2. Apply the identical multiplier to the numerator: $\square = 5 \times 7 = 35$.
3. Therefore, $\frac{5}{8} = \frac{35}{56}$.

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4. Comparing & Ordering Fractions

To compare or arrange a collection of fractions in ascending (smallest to largest) or descending (largest to smallest) order, candidates must use structured comparison strategies.

Method 1: Finding a Common Denominator via LCM

The most dependable method involves converting all fractions to a common denominator equal to the Lowest Common Multiple (LCM) of their original denominators.

Step-by-Step Problem: Arrange the following fractions in ascending order:

$$\frac{2}{3}, \quad \frac{5}{6}, \quad \frac{7}{12}, \quad \frac{3}{4}$$

1. Determine the LCM of the denominators ($3, 6, 12, 4$):
   • Multiples of 12: $12, 24, 36, \dots$
   • Since 12 is divisible by 3, 6, 4, and 12, the $\text{LCM} = 12$.
2. Convert each fraction into equivalent twelfths:
   • $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
   • $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
   • $\frac{7}{12} = \frac{7}{12}$
   • $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
3. Compare numerators: $$7 < 8 < 9 < 10 \implies \frac{7}{12} < \frac{8}{12} < \frac{9}{12} < \frac{10}{12}$$
4. Restore original fractions in ascending order: $$\frac{7}{12}, \quad \frac{2}{3}, \quad \frac{3}{4}, \quad \frac{5}{6}$$

Method 2: Benchmark Reasoning ($\frac{1}{2}$ and $1$)

When denominators are large, converting to a common denominator can consume excessive time. High-scoring candidates use benchmark fractions to eliminate options rapidly:

• Benchmark $\frac{1}{2}$: If the numerator is less than half the denominator, the fraction is $< \frac{1}{2}$ (e.g., $\frac{5}{12} < \frac{1}{2}$ because $5 < 6$). If greater than half, it is $> \frac{1}{2}$ (e.g., $\frac{5}{8} > \frac{1}{2}$ because $5 > 4$).
• Benchmark $1$ (Distance to Whole): When comparing fractions that are each one unit fraction away from $1$, the fraction with the larger denominator is closer to $1$:
  • Compare $\frac{7}{8}$ and $\frac{9}{10}$:
  • $1 - \frac{7}{8} = \frac{1}{8}$, whereas $1 - \frac{9}{10} = \frac{1}{10}$.
  • Because $\frac{1}{10} < \frac{1}{8}$, $\frac{9}{10}$ is closer to a full whole, so $\frac{9}{10} > \frac{7}{8}$.

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