2.1 Fractions, Decimals, & Mixed Numbers

2.1 Fractions, Decimals, & Mixed Numbers

Quick Summary: Understanding fractions, decimals, and mixed numbers is foundational to passing the ATI TEAS 7 Mathematics section. You must master converting between improper fractions and mixed numbers, finding Least Common Denominators (LCD), multiplying by reciprocals during division, and converting fractions to decimals using the on-screen 4-function calculator.

Fractions, decimals, and mixed numbers form the bedrock of clinical dosage calculations, fluid tracking, and laboratory measurements in nursing. On the ATI TEAS 7 exam, questions involving fractions and decimals test both basic operational fluency and applied problem-solving in healthcare scenarios.

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1. Anatomy of Fractions & Types of Rational Quantities

A fraction expresses a part of a whole quantity or a ratio between two numbers. It consists of two primary components:

• Numerator (top number): Represents how many equal parts are selected or being measured (the dividend).
• Denominator (bottom number): Represents the total number of equal parts that make up one whole unit (the divisor).

$\text{Fraction} = \frac{\text{Numerator (Dividend)}}{\text{Denominator (Divisor)}}$

Classification of Fraction Types

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2. Converting Between Improper Fractions and Mixed Numbers

Because ATI TEAS 7 questions often present answer choices as simplified mixed numbers while mathematical operations require improper fractions, seamless conversion between the two formats is essential.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number a b/c to an improper fraction:

1. Multiply the whole number integer (a) by the denominator (c).
2. Add the numerator (b) to that product.
3. Write the resulting sum over the original denominator (c).

$\text{Improper Fraction} = \frac{(a \times c) + b}{c}$

Worked Example 1: Convert 3 5/8 into an improper fraction.

1. Multiply whole number by denominator: 3 × 8 = 24.
2. Add the numerator: 24 + 5 = 29.
3. Place over original denominator: 29/8.

Result: 3 5/8 = 29/8.

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction n/d to a mixed number:

1. Divide the numerator (n) by the denominator (d) using integer division to find the whole number quotient.
2. The remainder becomes the new numerator.
3. The denominator remains unchanged.

Worked Example 2: Convert 23/5 into a mixed number.

1. Divide 23 ÷ 5: 5 goes into 23 four times (4 × 5 = 20). Quotient = 4.
2. Calculate the remainder: 23 - 20 = 3.
3. Assemble mixed number: Whole number 4, remainder 3, denominator 5.

Result: 23/5 = 4 3/5.

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3. Operations with Fractions

Addition and Subtraction (Finding the LCD)

Unlike multiplication and division, adding or subtracting fractions requires a common denominator. The Least Common Denominator (LCD) is the smallest positive integer that is divisible by all denominators involved.

Step-by-Step Algorithm for Addition/Subtraction:

1. Convert all mixed numbers into improper fractions.
2. Determine the LCD of all denominators.
3. Convert each fraction to an equivalent fraction with the LCD as its denominator.
4. Add or subtract the numerators while keeping the LCD constant.
5. Simplify the final fraction to lowest terms and convert to a mixed number if required.

Worked Example 3 (Clinical Fluid Addition): A pediatric patient receives 1/3 cup of water at breakfast, 1/2 cup of juice at lunch, and 1/4 cup of oral rehydration fluid at dinner. Calculate the total volume consumed.

1. Identify denominators: 3, 2, 4.
2. Find LCD: Multiples of 4 (4, 8, 12, 16...). 12 is divisible by 3, 2, and 4. So LCD = 12.
3. Convert fractions:
   • 1/3 = (1 × 4) / (3 × 4) = 4/12
   • 1/2 = (1 × 6) / (2 × 6) = 6/12
   • 1/4 = (1 × 3) / (4 × 3) = 3/12
4. Add numerators: (4 + 6 + 3) / 12 = 13/12.
5. Convert to mixed number: 13 ÷ 12 = 1 R 1 → 1 1/12 cups.

Multiplication of Fractions

Multiplication is straightforward: do NOT find a common denominator. Multiply numerators together and denominators together.

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

Tip: Simplify by cross-canceling common factors between any numerator and any denominator before multiplying to avoid dealing with large numbers.

Worked Example 4: Calculate 2 1/4 × 2/3.

1. Convert mixed number to improper fraction: 2 1/4 = (2 × 4 + 1) / 4 = 9/4.
2. Write product: 9/4 × 2/3.
3. Cross-cancel: Divide 9 and 3 by 3 (9 → 3, 3 → 1). Divide 2 and 4 by 2 (2 → 1, 4 → 2).
4. Multiply remaining terms: (3 × 1) / (2 × 1) = 3/2 = 1 1/2.

Division of Fractions ("Keep, Change, Flip")

To divide by a fraction, multiply by its reciprocal (invert the numerator and denominator of the divisor).

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

Worked Example 5 (Medication Vial Division): A nurse has 7/8 L of sterile antiseptic liquid. Each surgical site skin prep requires 1/16 L. How many complete preps can the nurse perform?

1. Setup division: 7/8 ÷ 1/16.
2. Apply Keep, Change, Flip:
   • Keep 7/8
   • Change ÷ to ×
   • Flip 1/16 to 16/1
3. Expression: 7/8 × 16/1.
4. Cross-cancel 16 ÷ 8 = 2 and 8 ÷ 8 = 1.
5. Multiply: 7 × 2 = 14.

Result: The nurse can perform 14 complete procedure preps.

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4. Converting Fractions to Decimals and Decimals to Fractions

Converting Fractions to Decimals

To convert any fraction to a decimal, divide the numerator by the denominator using long division or your calculator.

• Terminating Decimals: End after a finite number of decimal places (e.g., 3/8 = 3 ÷ 8 = 0.375).
• Repeating Decimals: Pattern repeats infinitely; indicated by a bar over repeating digits (e.g., 2/3 = 2 ÷ 3 = 0.6666... = 0.6̄). On TEAS questions, round repeating decimals to the place value specified in the prompt.

Common Conversion Benchmark Reference Table

Converting Decimals to Fractions

1. Identify the place value of the final digit (e.g., tenths, hundredths, thousandths).
2. Write the digits as the numerator over the place value power of 10 (10, 100, 1000).
3. Reduce to lowest terms by dividing by the greatest common factor (GCF).

Worked Example 6: Convert 0.065 to a simplified fraction.

1. The last digit (5) is in the thousandths place, so denominator is 1000.
2. Write fraction: 65/1000.
3. Simplify by dividing numerator and denominator by GCF 5:
   • 65 ÷ 5 = 13
   • 1000 ÷ 5 = 200

Result: 0.065 = 13/200.

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5. On-Screen 4-Function Calculator Strategies for TEAS 7

The computer-based ATI TEAS 7 provides an on-screen basic 4-function calculator (+, -, ×, ÷). It does not have fraction keys (a b/c), parentheses, or scientific functions.

Key Calculator Hacks for Fraction Problems:

1. Convert Fractions to Decimals Instantly: When asked to evaluate complex expressions involving mixed operations, convert all fractions to decimals first (3 ÷ 8 = 0.375). Perform calculations in decimal form, then match the decimal result to the fraction answer choices.
2. Handling Mixed Numbers: Convert 4 3/16 to decimal by doing 3 ÷ 16 = 0.1875, then adding 4 to get 4.1875.
3. Verifying Fraction Equality: If an answer choice is 17/40, check if it matches your calculated decimal 0.425 by running 17 ÷ 40 = 0.425 on the keypad.

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6. Common Student Traps & Healthcare Applications

Common Student Traps on the TEAS

• Trap 1: Adding/Subtracting Denominators directly. (1/3 + 1/2 ≠ 2/5). You MUST find an LCD first!
• Trap 2: Forgetting to invert when dividing. (3/4 ÷ 1/2 ≠ 3/8). Remember to flip the second fraction to 2/1.
• Trap 3: Intermediate Rounding Errors. Rounding decimals during middle steps of dosage calculations causes severe drift in the final answer. Keep full precision in your calculator until the final answer.

Healthcare Application: Oral Tablet Dosage

Clinical Scenario: A physician prescribes 0.375 mg of digoxin daily. The pharmacy supplies scored tablets labeled 0.25 mg per tablet. How many tablets should the nurse administer?

Calculation: $\text{Tablets} = \frac{\text{Desired Dose}}{\text{Have Dose}} = \frac{0.375\text{ mg}}{0.25\text{ mg}} = 1.5\text{ tablets} = 1\frac{1}{2}\text{ tablets}$