Dividing Fractions by Fractions - Sixth Grade Mathematics
Dividing a fraction by another fraction determines how many times a divisor fraction fits inside a dividend fraction, using the powerful Keep-Change-Flip algorithm.
The Keep-Change-Flip (KCF) Protocol
The standard algorithm for dividing fractions transforms a division problem into an equivalent multiplication problem using the reciprocal of the divisor.
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| THE KEEP-CHANGE-FLIP (KCF) PROTOCOL |
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| Step 1: KEEP the first fraction (the dividend) exactly as it is. |
| Step 2: CHANGE the division symbol (/) to multiplication (*). |
| Step 3: FLIP the second fraction (the divisor) to its reciprocal. |
| Step 4: Multiply across numerators and denominators. |
| Step 5: Cross-simplify before multiplying whenever possible. |
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Visualizing KCF:
( 3 / 4 ) / ( 2 / 5 )
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KEEP CHANGE FLIP
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v v v
( 3 / 4 ) * ( 5 / 2 ) = 15 / 8 = 1 and 7/8
Why Keep-Change-Flip Works Mathematically
Many students memorize Keep-Change-Flip without understanding why flipping the divisor is valid. The mathematical justification relies on the properties of complex fractions and multiplying by 1 in disguise.
Any division problem can be written as a vertical complex fraction:
(a / b) / (c / d)
Multiply both top and bottom by the reciprocal of the denominator (d / c):
[ (a / b) * (d / c) ] / [ (c / d) * (d / c) ]
Notice what happens in the bottom denominator:
(c / d) * (d / c) = 1 (A fraction times its reciprocal equals 1!)
The entire expression simplifies to:
[ (a / b) * (d / c) ] / 1 = (a / b) * (d / c)
Because dividing by 1 leaves any quantity unchanged, dividing by a fraction is fundamentally identical to multiplying by its reciprocal.
Complete Division Walkthroughs with Cross-Simplification
Cross-simplifying before multiplying makes fraction division efficient and minimizes large multiplication steps.
Evaluate: (7/10) / (14/15)
Step 1 (KCF): Keep 7/10, Change / to *, Flip 14/15 to 15/14:
(7/10) * (15/14)
Step 2 (Cross-Simplify):
- Numerator 7 and Denominator 14: Divide both by 7 --> 7 becomes 1, 14 becomes 2.
- Denominator 10 and Numerator 15: Divide both by 5 --> 10 becomes 2, 15 becomes 3.
Step 3 (Multiply):
(1 * 3) / (2 * 2) = 3/4.
Final Result: 3/4.
Visualizing Fraction by Fraction Division
Consider (3/4) / (1/8). How many eighths are in three-fourths? Convert 3/4 to eighths: 3/4 = 6/8. The problem is now: How many 1/8 pieces are in 6/8? There are obviously 6 pieces! Applying KCF: (3/4) * (8/1) = (3 * 8) / (4 * 1) = 24/4 = 6. Both the visual model and the algorithm yield the identical result.
Exercises
Exercise 1. Calculate each quotient and express in simplest form: (a) (5/6) / (2/3); (b) (8/9) / (4/15); (c) (3/8) / (9/16).
Exercise 2. A piece of fabric measures 7/8 of a yard long. How many ribbon strips of length 1/16 of a yard can be cut from this fabric?
Exercise 3. Explain why (2/3) / (5/6) results in a fraction less than 1, while (5/6) / (2/3) results in a fraction greater than 1.
Solutions
Solution to Exercise 1. For (a) (5/6) / (2/3): apply KCF to get (5/6) * (3/2). Cross-simplify 6 and 3 by dividing by 3: 3 becomes 1, and 6 becomes 2. Multiply: (5 * 1) / (2 * 2) = 5/4 = 1 and 1/4. For (b) (8/9) / (4/15): apply KCF to get (8/9) * (15/4). Cross-simplify 8 and 4 by dividing by 4: 8 becomes 2, 4 becomes 1. Cross-simplify 9 and 15 by dividing by 3: 9 becomes 3, 15 becomes 5. Multiply: (2 * 5) / (3 * 1) = 10/3 = 3 and 1/3. For (c) (3/8) / (9/16): apply KCF to get (3/8) * (16/9). Cross-simplify 3 and 9 by dividing by 3 (gives 1 and 3). Cross-simplify 8 and 16 by dividing by 8 (gives 1 and 2). Multiply: (1 * 2) / (1 * 3) = 2/3.
Solution to Exercise 2. To find the number of strips, divide the total fabric by the strip length: (7/8) / (1/16). Apply KCF: (7/8) * (16/1). Cross-simplify 8 and 16 by dividing by 8: 8 becomes 1, and 16 becomes 2. Multiply: (7 * 2) / 1 = 14. Exactly 14 ribbon strips can be cut.
Solution to Exercise 3. In (2/3) / (5/6), compare the sizes of dividend and divisor: 2/3 = 4/6, which is smaller than 5/6. When dividing a smaller quantity by a larger quantity, the quotient is always strictly less than 1 (specifically, 4/6 divided by 5/6 equals 4/5). Conversely, in (5/6) / (2/3), the dividend 5/6 is greater than the divisor 4/6. Dividing a larger quantity by a smaller quantity means the divisor fits inside more than once, producing a quotient strictly greater than 1 (specifically, 5/4 = 1 and 1/4).