Dividing Fractions by Fractions
Students use models, equations, and the reciprocal procedure to divide fractions and solve contextual problems.
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Connect Division and Fractions
Division can ask how many groups of one number fit into another number. This meaning works with fractions, too. Consider 3/4 ÷ 1/8. The question is, “How many groups of 1/8 are in 3/4?” Rewrite 3/4 in eighths: 3/4 = 6/8. Because six groups of 1/8 make 6/8, the quotient is 6. We can check with a related multiplication equation: 6 × 1/8 = 6/8 = 3/4. In general, a ÷ b = c means that c × b = a. Thinking about the size of the numbers also helps. Since 1/8 is smaller than 3/4, several groups should fit, so a quotient greater than 1 is reasonable.
Model Fraction Division
A visual model can show a quotient even when the answer is not a whole number. Find 2/3 ÷ 1/4 by asking how many lengths of 1/4 fit into 2/3. Use twelfths because 12 is a common denominator for 3 and 4. Then 2/3 = 8/12 and 1/4 = 3/12. Two complete groups of 3/12 use 6/12. The remaining 2/12 is two-thirds of another 3/12 group. Therefore, 8/12 contains 2 2/3 groups of 3/12, so 2/3 ÷ 1/4 = 2 2/3. The model also explains why a fractional quotient is possible: part of one divisor-sized group can remain. Common partitions make the dividend and divisor easy to compare.
Multiply by the Reciprocal
A quick way to divide fractions is to multiply by the reciprocal of the divisor. The reciprocal of a nonzero fraction is found by switching its numerator and denominator. For example, the reciprocal of 2/3 is 3/2. To calculate 5/6 ÷ 2/3, keep 5/6, change division to multiplication, and use 3/2: 5/6 × 3/2 = 15/12 = 5/4, or 1 1/4. This procedure works because multiplying 2/3 by 3/2 gives 1. Dividing by 2/3 is therefore equivalent to multiplying by the number that undoes multiplication by 2/3. Remember that only the divisor is replaced by its reciprocal. The dividend stays the same. Check the result with multiplication: 5/4 × 2/3 = 5/6.
Simplify Quotients
Simplify a fraction quotient by canceling common factors before or after multiplying. Consider 7/10 ÷ 14/15. Replace the divisor with its reciprocal: 7/10 × 15/14. Before multiplying, divide 7 and 14 by their common factor 7, leaving 1 and 2. Divide 15 and 10 by their common factor 5, leaving 3 and 2. Now multiply: 1/2 × 3/2 = 3/4. Canceling first keeps the numbers small, but it does not change the value. If you multiply first, you get 105/140, which also simplifies to 3/4. A quotient should be written in simplest form, with no common factor greater than 1 in the numerator and denominator. If the result is improper, it may also be written as a mixed number when the context calls for one.
Solve Real-World Problems
Fraction division can determine how many equal portions can be made. Suppose a craft club has 2 1/4 yards of ribbon and uses 3/8 yard for each project. To find the number of projects, divide the total ribbon by the ribbon per project. First write 2 1/4 as 9/4. Then calculate 9/4 ÷ 3/8 = 9/4 × 8/3. Simplify before multiplying: 9 ÷ 3 = 3 and 8 ÷ 4 = 2. The product is 3 × 2 = 6. The club has enough ribbon for 6 projects. Check the units and meaning: six groups of 3/8 yard use 18/8 yards, which equals 9/4 or 2 1/4 yards. A labeled model and equation connect the quotient to the situation.