Working with Fractions · Lesson 5 of 8
Division of Fractions and Reciprocals
“Build the division rule from missing-factor questions and the meaning of a reciprocal.”
• Connect division with a multiplication equation containing a missing factor. • Find reciprocals of nonzero fractions and whole numbers. • Derive and apply division by multiplying by the divisor’s reciprocal. • Check a quotient by multiplying it by the original divisor.
Division of Fractions
You know that 12 ÷ 4 = 3. Another way to ask the same question is: what must multiply 4 to make 12? Since 4 × 3 = 12, the missing factor is 3. This connection lets us understand division of fractions using the multiplication we have already learned.
In 12 ÷ 4 = 3, the dividend is the starting amount 12, the divisor is the number 4 by which we divide, and the quotient is the result 3. The divisor must not be zero.
Finding a Product of One
First ask what must multiply 2/3 to give 1. We need to undo the two-thirds scaling. Multiplying by 3/2 does that: the numerator product and denominator product are both 6. A pair of numbers with product 1 gives a useful tool for division.
Problem
Find 1 ÷ 2/3 by asking a multiplication question.
- 1.Write the missing-factor relationship: 2/3 × ? = 1.
- 2.Try 3/2. The product is (2 × 3)/(3 × 2) = 6/6 = 1.
- 3.Therefore the missing factor is 3/2, so 1 ÷ 2/3 = 3/2 = 1 1/2.
The reciprocal of a nonzero number is the number that multiplies it to give 1. For a nonzero fraction a/b, its reciprocal is b/a. A whole number n has reciprocal 1/n when n is nonzero.
Finding a reciprocal is not the same as changing a fraction into a mixed number or taking its negative. The reciprocal of 5/4 is 4/5, and their product is 1. For 2 1/3, first write 7/3; its reciprocal is then 3/7. Keeping this distinction clear prevents a common error when division begins.
From One to the Required Dividend
Knowing what produces 1 is enough to find what produces any other dividend. To make 3 from 2/3, first multiply by 3/2 to make 1, then multiply by 3. The combined multiplier is 3/2 × 3 = 9/2. That combined multiplier is the quotient.
Problem
Calculate 3 ÷ 2/3.
- 1.Rewrite the question as 2/3 × ? = 3.
- 2.The reciprocal 3/2 produces 1 from 2/3. Three copies of that reciprocal produce 3.
- 3.Thus the missing factor is 3 × 3/2 = 9/2. The quotient is 9/2 = 4 1/2.
- 4.Check: 2/3 × 9/2 = 3, which is the original dividend.
Problem
Find 2/3 ÷ 3/5.
- 1.The equivalent missing-factor question is 3/5 × ? = 2/3.
- 2.The reciprocal of the divisor 3/5 is 5/3. Multiplying 3/5 by 5/3 produces 1.
- 3.To produce 2/3 instead of 1, multiply the reciprocal by 2/3: 2/3 × 5/3 = 10/9.
- 4.Check the answer: 3/5 × 10/9 = 30/45 = 2/3. Therefore the quotient is 10/9 = 1 1/9.
The Rule for Dividing Fractions
Each calculation followed the same two steps: find the divisor’s reciprocal, then multiply the dividend by it. The dividend remains unchanged. Division becomes multiplication because we replace the divisor with a multiplier that undoes its effect.
Problem
Calculate 1/5 ÷ 1/2.
- 1.The divisor is 1/2, whose reciprocal is 2/1.
- 2.Multiply the dividend by this reciprocal: 1/5 × 2/1 = 2/5.
- 3.Check: 1/2 × 2/5 = 1/5. The quotient is therefore 2/5.
Problem
Calculate 7/4 ÷ 2.
- 1.Write the divisor as 2/1. Its reciprocal is 1/2.
- 2.Multiply: 7/4 × 1/2 = 7/8.
- 3.Check: 2 × 7/8 = 14/8 = 7/4. Sharing the original amount into two equal parts gives 7/8 per part.
Problem
Calculate 3 2/3 ÷ 1 3/8.
- 1.Convert both mixed numbers: 3 2/3 = 11/3 and 1 3/8 = 11/8.
- 2.Take only the divisor’s reciprocal: 11/3 ÷ 11/8 = 11/3 × 8/11.
- 3.Cancel the common factor 11 to get 8/3 = 2 2/3.
- 4.Check: 11/8 × 8/3 = 11/3, the original dividend.
Do not take the reciprocal of both fractions. For 2/3 ÷ 3/5, keep 2/3 and multiply it by 5/3. Also, do not swap the order of a division: 1/2 ÷ 1/4 = 2, while 1/4 ÷ 1/2 = 1/2. Division does not have multiplication’s order property.
A quotient answers the question “what multiplies the divisor to give the dividend?” Multiplying the proposed quotient by the original divisor is therefore a direct check, not a separate rule to memorise.
Quiz
Which equation asks the same question as 3/4 ÷ 2/5?
What is the reciprocal of 7/3?
Which product gives 2/3 ÷ 3/5?
What is 3/4 ÷ 1/8?
Why does zero have no reciprocal?
Practice Problems
- Find the reciprocals of 3/8, 5, 1, and 2 1/4. Explain why you must convert the mixed number first.
- Evaluate 3 ÷ 7/9; 14/4 ÷ 2; 2/3 ÷ 2/3; and 14/6 ÷ 7/3.
- Evaluate 4/3 ÷ 3/4; 7/4 ÷ 1/7; 8/2 ÷ 4/15; 1/5 ÷ 1/9; and 1/6 ÷ 11/12.
- Calculate 2 1/2 ÷ 1 1/4. Verify the answer by multiplication.
- A student changes 3/5 ÷ 2/7 into 5/3 × 7/2. Identify and correct the error.
- Give an example showing that changing the order of division changes the answer.
Reciprocals: 8/3, 1/5, 1, 4/9. First group: 27/7, 7/4, 1, 1. Second group: 16/9, 49/4, 15, 9/5, 2/11. Mixed division: 5/2 × 4/5 = 2; check 2 × 5/4 = 5/2. Correct the student’s work to 3/5 × 7/2 = 21/10. One order example is 1/2 ÷ 1/4 = 2 versus 1/4 ÷ 1/2 = 1/2.
Key Takeaways
• Division can be rewritten as a missing-factor multiplication question. • A nonzero number and its reciprocal have product 1. • To divide, keep the dividend and multiply by the reciprocal of the divisor. • Convert mixed numbers and whole numbers to fractions before using the rule. • Verify a quotient by multiplying it by the original divisor; division by zero is undefined.