Dividing Mixed Numbers - Sixth Grade Mathematics
Dividing mixed numbers requires converting mixed numbers into improper fractions before applying division algorithms, ensuring precision across multi-step computations.
The Mandatory First Step: Conversion
When dividing mixed numbers, there is no shortcut that allows you to divide whole numbers and fractions separately. You CANNOT divide the wholes and then divide the fractions. Attempting to do so violates the distributive law and produces completely erroneous results.
The Fundamental Rule of Mixed Number Division:
EVERY mixed number must be converted into an IMPROPER FRACTION
before performing Keep-Change-Flip!
Common Error Demonstration:
Problem: 4 and 1/2 / 2 and 1/4
INCORRECT METHOD: Divide wholes: 4 / 2 = 2. Divide fractions: (1/2) / (1/4) = 2.
Guess: 2 and 2? (COMPLETELY WRONG)
CORRECT METHOD: Convert 4 and 1/2 to 9/2. Convert 2 and 1/4 to 9/4.
(9/2) / (9/4) = (9/2) * (4/9) = (1 * 2) / (1 * 1) = 2.
The exact correct answer is 2!
The Four-Step Operational Protocol
Dividing expressions involving mixed numbers follows a four-step framework.
+-------------------------------------------------------------------------+
| DIVIDING MIXED NUMBERS PROTOCOL |
| |
| Step 1: Convert ALL mixed numbers and whole numbers to improper |
| fractions: (Whole * Denom + Num) / Denom. |
| Step 2: Apply KEEP-CHANGE-FLIP (multiply by the reciprocal of the |
| divisor fraction). |
| Step 3: Cross-simplify common factors between numerators and |
| denominators. |
| Step 4: Multiply across and convert the improper result back into |
| a mixed number in simplest form. |
+-------------------------------------------------------------------------+
Comprehensive Step-by-Step Example
Evaluate: 3 and 3/5 divided by 1 and 1/2. Step 1: Convert to improper fractions: 3 and 3/5 = (3 * 5 + 3)/5 = 18/5. 1 and 1/2 = (1 * 2 + 1)/2 = 3/2. Problem is rewritten as: (18/5) / (3/2). Step 2: Apply KCF: Keep 18/5, change / to *, flip 3/2 to 2/3: (18/5) * (2/3). Step 3: Cross-simplify: Divide 18 and 3 by 3: 18 becomes 6, 3 becomes 1. Expression becomes: (6/5) * (2/1). Step 4: Multiply across: (6 * 2) / (5 * 1) = 12/5. Step 5: Convert back to a mixed number: 12 / 5 = 2 with a remainder of 2, giving 2 and 2/5.
Mixed Numbers Divided by Whole Numbers
When dividing a mixed number by a whole number, express the whole number as a fraction over 1 before inverting.
Evaluate: 4 and 2/3 / 7
Step 1: Convert 4 and 2/3 to improper fraction: (4 * 3 + 2)/3 = 14/3.
Convert 7 to whole fraction: 7/1.
Step 2: Apply KCF:
(14/3) * (1/7).
Step 3: Cross-simplify 14 and 7 by dividing by 7: 14 becomes 2, 7 becomes 1.
Step 4: Multiply:
(2 * 1) / (3 * 1) = 2/3.
Exercises
Exercise 1. Calculate each quotient and express as a mixed number or fraction in simplest form: (a) 2 and 1/4 / 1 and 1/8; (b) 5 and 1/3 / 2 and 2/5; (c) 3 and 1/2 / 6.
Exercise 2. A hiking trail is 11 and 1/4 miles long. A park ranger places distance markers every 1 and 7/8 miles along the trail. How many markers are placed along the trail?
Exercise 3. A container holds 9 and 3/4 liters of water. If small bottles holding 3/4 liter each are filled from this container, how many bottles can be filled completely?
Solutions
Solution to Exercise 1. For (a) 2 and 1/4 / 1 and 1/8: convert to improper fractions: 2 and 1/4 = 9/4; 1 and 1/8 = 9/8. Apply KCF: (9/4) * (8/9). Cross-simplify 9 and 9 to 1 and 1; cross-simplify 8 and 4 by dividing by 4 to get 2 and 1. Multiply: (1 * 2) / (1 * 1) = 2. For (b) 5 and 1/3 / 2 and 2/5: convert to improper fractions: 5 and 1/3 = 16/3; 2 and 2/5 = 12/5. Apply KCF: (16/3) * (5/12). Cross-simplify 16 and 12 by dividing by 4: 16 becomes 4, 12 becomes 3. Multiply: (4 * 5) / (3 * 3) = 20/9 = 2 and 2/9. For (c) 3 and 1/2 / 6: convert: 3 and 1/2 = 7/2, and 6 = 6/1. Apply KCF: (7/2) * (1/6) = 7/12.
Solution to Exercise 2. To find the number of markers, divide total trail distance by marker spacing: (11 and 1/4) / (1 and 7/8). Convert to improper fractions: 11 and 1/4 = 45/4; 1 and 7/8 = 15/8. Apply KCF: (45/4) * (8/15). Cross-simplify 45 and 15 by dividing by 15: 45 becomes 3, 15 becomes 1. Cross-simplify 8 and 4 by dividing by 4: 8 becomes 2, 4 becomes 1. Multiply: (3 * 2) / (1 * 1) = 6. Exactly 6 markers are placed along the trail.
Solution to Exercise 3. Divide total water by bottle capacity: (9 and 3/4) / (3/4). Convert 9 and 3/4 to an improper fraction: (9 * 4 + 3)/4 = 39/4. Apply KCF: (39/4) * (4/3). Cross-simplify 4 and 4 to 1 and 1. Cross-simplify 39 and 3 by dividing by 3: 39 becomes 13, 3 becomes 1. Multiply: (13 * 1) / (1 * 1) = 13. Exactly 13 bottles can be filled completely.