What is 12 ÷ 4? You know this already. But can this problem be restated as a multiplication problem? What should be multiplied by 4 to get 12?
Answer
12 ÷ 4 = 3, and the same fact can be written the other way round.
12 ÷ 4 = 3
means 4 × ? = 12
4 × 3 = 12 ✔
Why this matters: every division question is a “missing factor” question. Turning ÷ into × is what lets us divide fractions without inventing a new method.
Why the answer is bigger than 3: you are asking how many two-thirds fit into 3. Since each piece is smaller than 1, more than 3 of them fit — four and a half, in fact.
Q4.
What is 1⁄5 ÷ 1⁄2?
Answer
Rewrite as a multiplication and use the reciprocal of the divisor.
In each of the division problems above, observe how we found the answer. Can we frame a rule that tells us how to divide two fractions?
Answer
Yes — two steps every time.
Find the reciprocal of the divisor (turn it upside down).
Multiply the dividend by that reciprocal.
a⁄b ÷ c⁄d = a⁄b × d⁄c = (a × d)/(b × c)
Division
Reciprocal of divisor
Quotient
1 ÷ 2/3
3/2
3/2
3 ÷ 2/3
3/2
9/2
1/5 ÷ 1/2
2/1
2/5
2/3 ÷ 3/5
5/3
10/9
Did you know? This rule was first stated in general form by Brahmagupta in the Brāhmasphuṭasiddhānta (628 CE), and Bhāskara II explained it through the idea of the reciprocal in the Līlāvatī (1150 CE).