3.2 Addition & Subtraction of Fractions and Mixed Numbers (LCD & Borrowing)

3.2 Addition & Subtraction of Fractions and Mixed Numbers (LCD & Borrowing)

Addition and subtraction of fractions and mixed numbers are daily computational routines in healthcare settings. Nurses regularly calculate cumulative fluid intake and output (I&O), compute total nutritional intake across fractional meal portions, determine remaining volumes in multidose medication vials, and calculate total changes in patient body weight. On the HESI A2 exam, fraction addition and subtraction questions frequently incorporate mixed numbers and unlike denominators that test your understanding of the Least Common Denominator (LCD) and borrowing techniques.

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Operations with Like Denominators

When two or more fractions share identical denominators, they represent parts of equal size. Under these conditions, addition and subtraction are straightforward: simply combine the numerators while keeping the common denominator unchanged.

$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \quad \text{and} \quad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$

Step-by-Step Worked Examples

Worked Example 1: Add $\frac{3}{10} + \frac{1}{10}$.

1. The denominators are identical ($10$).
2. Add the numerators: $3 + 1 = 4$.
3. Keep the denominator: $\frac{4}{10}$.
4. Simplify to lowest terms: Divide numerator and denominator by their GCF ($2$): $\frac{4 \div 2}{10 \div 2} = \mathbf{\frac{2}{5}}$

Worked Example 2: Subtract $\frac{7}{12} - \frac{1}{12}$.

1. Denominators are identical ($12$).
2. Subtract the numerators: $7 - 1 = 6$.
3. Place over denominator: $\frac{6}{12}$.
4. Simplify: $\frac{6 \div 6}{12 \div 6} = \mathbf{\frac{1}{2}}$.

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Operations with Unlike Denominators: Finding the LCD

Fractions with different denominators represent parts of different sizes (for instance, eighths versus twelfths). Before adding or subtracting, you must convert all fractions into equivalent fractions sharing a single Least Common Denominator (LCD). The LCD is the Least Common Multiple (LCM) of the individual denominators.

Method 1: Listing Multiples (Best for Small Numbers)

List consecutive multiples of each denominator until you locate the smallest number appearing in both lists.

• For denominators $6$ and $8$:
  • Multiples of $6$: $6, 12, 18, \mathbf{24}, 30, 36, 42, 48$
  • Multiples of $8$: $8, 16, \mathbf{24}, 32, 40, 48$
  • Smallest shared multiple: $24$ $\implies \text{LCD} = 24$.

Method 2: Prime Factorization (Best for Larger Numbers)

1. Break each denominator down into its prime factors.
2. For each unique prime factor, identify the highest exponent that appears in any single factorization.
3. Multiply these highest prime powers together.

Worked Example: Find the LCD of $12$, $18$, and $20$.

1. Prime factorizations:
   • $12 = 2^2 \times 3$
   • $18 = 2 \times 3^2$
   • $20 = 2^2 \times 5$
2. Collect unique prime factors: $2$, $3$, and $5$.
3. Select maximum powers:
   • For $2$: highest power is $2^2 = 4$
   • For $3$: highest power is $3^2 = 9$
   • For $5$: highest power is $5^1 = 5$
4. Multiply: $\text{LCD} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = \mathbf{180}$.

High-Frequency Clinical Denominators Reference

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Addition of Fractions with Unlike Denominators

Once the LCD is identified, convert each fraction by multiplying both numerator and denominator by the factor required to produce the LCD, then add the numerators.

Worked Example: Calculate $\frac{5}{6} + \frac{3}{8}$.

1. Find the LCD: As determined above, the LCD of $6$ and $8$ is $24$.
2. Convert $\frac{5}{6}$: Since $24 \div 6 = 4$, multiply top and bottom by $4$: $\frac{5 \times 4}{6 \times 4} = \frac{20}{24}$
3. Convert $\frac{3}{8}$: Since $24 \div 8 = 3$, multiply top and bottom by $3$: $\frac{3 \times 3}{8 \times 3} = \frac{9}{24}$
4. Add the numerators: $\frac{20}{24} + \frac{9}{24} = \frac{29}{24}$.
5. Convert improper fraction to mixed number: $29 \div 24 = 1$ with remainder $5$, giving $\mathbf{1\frac{5}{24}}$.

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Subtraction of Fractions with Unlike Denominators

Subtraction follows the exact same LCD conversion process as addition, culminating in subtracting the second numerator from the first.

Worked Example: Calculate $\frac{7}{10} - \frac{4}{15}$.

1. Find the LCD of $10$ and $15$: Multiples of $10$ are $10, 20, 30$; multiples of $15$ are $15, 30$. The LCD is $30$.
2. Convert $\frac{7}{10}$: Multiply by $3$: $\frac{7 \times 3}{10 \times 3} = \frac{21}{30}$.
3. Convert $\frac{4}{15}$: Multiply by $2$: $\frac{4 \times 2}{15 \times 2} = \frac{8}{30}$.
4. Subtract numerators: $\frac{21}{30} - \frac{8}{30} = \frac{21 - 8}{30} = \mathbf{\frac{13}{30}}$.
5. Simplify: $13$ is prime and does not divide $30$; the fraction is already in lowest terms.

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Addition of Mixed Numbers

There are two reliable methods for adding mixed numbers: (1) adding the whole numbers and fractional parts separately, or (2) converting all mixed numbers into improper fractions. For addition, separate grouping is generally faster and less prone to arithmetic error.

The Separate Grouping Algorithm

1. Find the LCD of the fractional parts and convert each fraction.
2. Add the whole numbers together.
3. Add the converted fractions together.
4. If the fractional sum is an improper fraction, convert it into a mixed number and carry the whole number into the integer total.
5. Simplify the final fraction if necessary.

Worked Example: Add $3\frac{3}{4} + 2\frac{5}{6}$.

1. Find LCD for denominators $4$ and $6$: LCD is $12$.
2. Convert fractions:
   • $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
   • $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
3. Add whole numbers: $3 + 2 = 5$.
4. Add fractions: $\frac{9}{12} + \frac{10}{12} = \frac{19}{12}$.
5. Convert improper fraction: $\frac{19}{12} = 1\frac{7}{12}$.
6. Combine: $5 + 1\frac{7}{12} = \mathbf{6\frac{7}{12}}$.

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Subtraction of Mixed Numbers & The Borrowing Technique

When subtracting mixed numbers, a major challenge arises when the fraction being subtracted (subtrahend) is larger than the starting fraction (minuend). In this situation, you must borrow (regroup) $1$ whole unit from the whole number and add it to the top fraction before subtracting.

The Step-by-Step Borrowing Algorithm

1. Convert both fractional parts to equivalent fractions with the LCD.
2. Compare numerators. If the top numerator is smaller than the bottom numerator, borrow $1$ from the top whole number ($W_1 \rightarrow W_1 - 1$).
3. Convert that borrowed $1$ into a fraction whose numerator and denominator equal the LCD ($\frac{\text{LCD}}{\text{LCD}}$).
4. Add this borrowed fraction to the top fraction: new numerator $= N_1 + \text{LCD}$.
5. Subtract the whole numbers, then subtract the numerators.
6. Simplify the remaining fraction.

$$W_1\frac{N_1}{D} = (W_1 - 1) + \left(\frac{D}{D} + \frac{N_1}{D}\right) = (W_1 - 1)\frac{N_1 + D}{D}$$

Detailed Worked Examples with Borrowing

Worked Example 1 (Like Denominators): Evaluate $5\frac{1}{4} - 2\frac{3}{4}$.

1. The fractions have identical denominators ($4$).
2. Compare numerators: $1 < 3$. You cannot subtract $3$ from $1$ without borrowing.
3. Borrow $1$ from the whole number $5$, leaving $4$.
4. Express the borrowed $1$ as $\frac{4}{4}$ and add it to $\frac{1}{4}$: $\frac{4}{4} + \frac{1}{4} = \frac{5}{4}$ The top mixed number is now rewritten as $4\frac{5}{4}$.
5. Subtract whole numbers and numerators: $4\frac{5}{4} - 2\frac{3}{4} = (4 - 2) + \frac{5 - 3}{4} = 2\frac{2}{4}$
6. Simplify the fractional part: $\frac{2}{4} = \frac{1}{2}$.
7. Final Answer: $\mathbf{2\frac{1}{2}}$.

Worked Example 2 (Unlike Denominators): Evaluate $7\frac{1}{6} - 3\frac{5}{8}$.

1. Find the LCD of $6$ and $8$: $\text{LCD} = 24$.
2. Convert both mixed numbers:
   • $7\frac{1}{6} = 7\frac{4}{24}$
   • $3\frac{5}{8} = 3\frac{15}{24}$
3. Compare numerators: $4 < 15$. Borrow $1$ from the whole number $7$, leaving $6$.
4. Convert the borrowed $1$ into $\frac{24}{24}$ and add to $\frac{4}{24}$: $6 + \left(\frac{24}{24} + \frac{4}{24}\right) = 6\frac{28}{24}$
5. Subtract whole numbers and numerators: $6\frac{28}{24} - 3\frac{15}{24} = (6 - 3) + \frac{28 - 15}{24} = 3\frac{13}{24}$
6. Final Answer: $\mathbf{3\frac{13}{24}}$ (since $13$ is prime, this is fully simplified).

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Clinical Application: Fluid Intake and Output (I&O)

In acute care nursing, tracking a patient's fluid balance is critical for patients with congestive heart failure, renal impairment, or post-surgical recovery. Fluid measurements logged in standard household containers must often be summed as fractions before total metric conversion.

Clinical Scenario: During an 8-hour shift, a patient on strict fluid tracking drinks:

• Breakfast: $\frac{3}{4}\text{ cup}$ of water
• Lunch: $1\frac{1}{3}\text{ cups}$ of chicken broth
• Afternoon snack: $\frac{5}{6}\text{ cup}$ of apple juice

Calculate the total volume of fluid consumed in cups as a mixed number in simplest form.

Step-by-Step Clinical Calculation:

1. Set up the addition expression: $\frac{3}{4} + 1\frac{1}{3} + \frac{5}{6}$.
2. Find the LCD for denominators $4$, $3$, and $6$:
   • Multiples of $4$: $4, 8, \mathbf{12}, 16$
   • Multiples of $3$: $3, 6, 9, \mathbf{12}$
   • Multiples of $6$: $6, \mathbf{12}$
   • $\text{LCD} = 12$.
3. Convert all fractions to twelfths:
   • $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
   • $1\frac{1}{3} = 1 + \frac{1 \times 4}{3 \times 4} = 1\frac{4}{12}$
   • $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
4. Combine whole numbers: $1$.
5. Sum the numerators: $\frac{9 + 4 + 10}{12} = \frac{23}{12}$.
6. Convert improper fraction: $\frac{23}{12} = 1\frac{11}{12}$.
7. Add to the whole number: $1 + 1\frac{11}{12} = \mathbf{2\frac{11}{12}\text{ cups}}$.
8. If converting to milliliters ($1\text{ cup} = 240\text{ mL}$): $2\frac{11}{12} \times 240 = \frac{35}{12} \times 240 = 35 \times 20 = 700\text{ mL}$.