Why Do You Flip the Fraction When Dividing?

In the United States, the Common Core standards place division of fractions by fractions in Grade 6. Standard 6.NS.A.1 asks students to “interpret and compute quotients of fractions”, and it gives an example that matters for this post: explain that (2/3) ÷ (3/4) = 8/9 “because 3/4 of 8/9 is 2/3” (Common Core State Standards, Grade 6, The Number System). Japan teaches the same topic in the sixth year of primary school.

The method most of us remember is a rule. Keep the first fraction, change division to multiplication, flip the second fraction. Some classrooms call it “keep, change, flip”. There is also an old rhyme: “Ours is not to reason why, just invert and multiply.”

The rule works. The trouble starts when a child asks why. This post gives three ways to see the reason. One of them clicking is enough.

The example throughout is one I chose: 2/3 ÷ 1/4. The answer is 8/3, or 2 and 2/3.

What a reciprocal is

First, one word. Two numbers are reciprocals of each other when they multiply to 1. The reciprocal of 3/4 is 4/3, because 3/4 × 4/3 = 1. A whole number like 5 can be written 5/1, so its reciprocal is 1/5.

For a fraction, you get the reciprocal by swapping the top and bottom. “Flip” is just an informal name for this. There is one exception. Zero has no reciprocal, since nothing times 0 gives 1. That is the same reason you cannot divide by 0.

View 1: count how many fit

One meaning of division is “how many of these fit in that”. 6 ÷ 2 asks how many 2s fit in 6.

Read the same way, 2/3 ÷ 1/4 asks how many quarters fit in two thirds.

The two are hard to compare as they are, so give them a common denominator. 2/3 is 8/12, and 1/4 is 3/12.

2/3 = 8/12  ■■■■■■■■□□□□
1/4 = 3/12  ■■■□□□□□□□□□

Treat each twelfth as one piece. Now the question is how many groups of 3 pieces fit in 8 pieces, which is 8 ÷ 3. The answer is 8/3. Two full groups fit, with 2 of the 3 pieces of a third group left over, so 2 and 2/3 groups.

That is the same answer the flip rule gives, and nothing was flipped. With a common denominator, fraction division turns back into whole number division.

Finding a common denominator every time is slow, though. You can think of “multiply by the reciprocal” as the shortcut for this.

View 2: turn the divisor into 1

Division has a useful property. Multiply both numbers by the same thing and the answer stays the same. 12 ÷ 4 = 3, and 120 ÷ 40 is also 3.

Use it to turn the divisor into 1. To make 1/4 into 1, multiply by 4, which is the reciprocal of 1/4. Multiply the first number by 4 as well.

2/3 ÷ 1/4
= (2/3 × 4) ÷ (1/4 × 4)
= (2/3 × 4) ÷ 1
= 2/3 × 4

Dividing by 1 changes nothing, so all that is left is 2/3 × 4. Multiplying by the reciprocal of the divisor falls out on its own.

If the divisor is 3/5, the number that turns it into 1 is 5/3, so you multiply both by 5/3 and are left with the first number × 5/3. Whatever the fraction, its reciprocal turns it into 1. I think this is the shortest honest explanation of the flip.

View 3: find the amount for one

Many textbooks use a measuring problem.

“With 1/4 of a litre of paint you can cover 2/3 of a square metre. How much can you cover with 1 litre?”

One litre is 4 times a quarter litre, so you can cover 4 times as much: 2/3 × 4 = 8/3 square metres. Written as a sum, it is 2/3 ÷ 1/4, so here too ÷ 1/4 became × 4.

This view helps most when the top of the divisor is not 1. Try “2/5 of a litre covers 3/4 of a square metre. How much does 1 litre cover?”

1. First find what 1/5 of a litre covers. That is half of 2/5, so halve the area: 3/4 ÷ 2.
2. One litre is 5 times 1/5 of a litre, so multiply by 5: 3/4 ÷ 2 × 5.

“Divide by 2, then multiply by 5” is the same as “multiply by 5/2”. You divide by the top of 2/5 and multiply by its bottom. That is why the fraction looks flipped. If you draw it as an area model, the two steps are visible in the picture.

Checking with multiplication

The Common Core example above uses one more idea. Division undoes multiplication, so 2/3 ÷ 1/4 = □ means finding the □ for which □ × 1/4 = 2/3.

Put 8/3 in the box: 8/3 × 1/4 = 8/12 = 2/3. It works. Taking a quarter makes a number smaller, so to undo it you multiply by 4. Again, ÷ 1/4 and × 4 are the same thing.

You can compute without knowing why, but

A child can get fraction division problems right without knowing the reason. That is true.

I still think the reason is worth one pass. A child who knows the aim is to turn the divisor into 1 will not flip the wrong fraction. And a child who understands “how many fit” and “amount for one” finds it easier to write the division in a word problem.

When explaining it to a child, I would start with View 1. It gets the answer with the same idea as whole number division, so it does not feel like a new rule. Then show View 2 as the shortcut that saves finding a common denominator every time. In that order, the flip looks less like a trick and more like a convenience.

Where Math Gates fits

Math Gates is a circuit puzzle game I make. You place gates that change a number and mirrors that turn its path 90 degrees on the empty cells of a board, then press Run. A bit carrying a number travels in a straight line. The port at the end accepts only values that meet its condition.

The fifth of the seven worlds, Divide Circuit, is about fractions and reciprocals. From there on, a ÷ gate lets uneven results through as fractions, and a 1/v gate that takes the reciprocal appears. Numbers are kept as exact fractions, so there is no drift like 0.333… When you place a gate, the board draws where the bit will go and with what value. You can compare what a ÷ gate and a × gate do to the same number and check the ideas in this post with your own hands. Clearing a puzzle shows a discovery card that gives a name and a formula to what you just did.

The design rule is not to explain the math but to let players try it. The game does not tell you the reason in words first, so if a child wants the explanation, it works best alongside something like this post.

There are 140 puzzles across seven worlds, aimed at ages 8 and up, on iPhone and iPad. It is free to play with ads between puzzles, and a one-time purchase removes the ads.

For why the order of steps changes the answer, see Why the Order of Operations Matters. For what puzzles can and cannot teach, see What Is Computational Thinking, and What Can Puzzles Teach?.

You flip the divisor because you want it to become 1. A common denominator and the amount for one lead to the same place. Several roads arriving at one answer is, in my view, one of the nicer things about arithmetic.