Why do we "flip and multiply" when dividing by fractions? Explaining the reasoning with diagrams
There are two ways to understand why we "flip and multiply" when dividing by fractions. First, division asks "how many divisors are in there" — for example, six divided by two asks how many twos there are in six. Second, dividing by a fraction is the same as multiplying by its reciprocal, because multiplication and division are inverse operations: treating "divide by a number" as "multiply by the reciprocal of that number" gives the same result. The article explains this step by step using three levels of examples — whole numbers, unit fractions and general fractions — and then demonstrates the calculation steps and checking methods.
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- Division can be understood as "how many divisors are in there", and fraction division is the same.
- When flipping and multiplying, only the divisor is flipped; the dividend stays unchanged.
- Dividing by a unit fraction is the same as multiplying by its denominator, which is easiest to understand with diagrams.
- General fractions can first be understood as "divide by the numerator first, then multiply by the denominator".
- Using multiplication to reverse the calculation and estimating are more effective ways to check answers than just comparing with the answer key.
- 3Three levels of examples explaining fraction division
Children's question: why do we need to "flip"?
Many Primary 5 and Primary 6 students feel that "flip and multiply" is a rule with no reason behind it when they learn fraction division: it is clearly division, so why does it turn into multiplication? If they only memorise the steps, they will easily make mistakes when a question varies slightly; only when they understand the reasoning can students judge for themselves when they can flip, and which number gets flipped.
First, one key point needs to be made clear: only the divisor is flipped, and the dividend stays the same. For example, for two-thirds divided by three-quarters, it is the three-quarters that gets flipped, becoming two-thirds multiplied by four-thirds. This point should be made clear first, and then we can explain why this can be done.
Level one: division originally asks "how many are in there"
It is easiest to understand by starting with whole numbers. Eight divided by two asks how many twos there are in eight; the answer is four. Twelve divided by three asks how many threes there are in twelve; the answer is four. This idea of "quotitive division" is the foundation for understanding fraction division.
Change the same question into fractions: two divided by one-half asks how many halves there are in two. Each whole can be split into two halves, so there are four halves in two altogether; the answer is four. Written as a calculation, this is two multiplied by the reciprocal of one-half, which is two, giving four. Children only need to draw two bars, split each into two halves, and count four half-portions to understand where the answer comes from.
Level two: dividing by a unit fraction is just "multiplying by how many"
A unit fraction is a fraction with numerator one, such as one-half, one-third and one-quarter. Dividing by this kind of fraction has a very intuitive explanation: how many halves there are in something is the same as multiplying it by two; how many thirds there are is the same as multiplying by three.
For example, two-thirds divided by one-fifth asks how many fifths there are in two-thirds. Split each of the two-thirds into five small parts, giving ten fifteenths in total, that is, ten fifteenths; and each fifth equals three fifteenths, so there are three and one-third fifths in it. This result is the same as two-thirds multiplied by five. By subdividing the bar in a diagram, children can see that "dividing by one-fifth" and "multiplying by five" are actually doing the same thing.
Level three: why general fractions can also be flipped
With a non-unit fraction such as three-quarters, the reasoning is the same, just with one more step. Dividing by three-quarters can first be thought of as "divide by three first, then multiply by four": because three-quarters equals three multiplied by one-quarter, and dividing by a product can be done by dividing by each factor in turn. Dividing by one-quarter is the same as multiplying by four, so the whole process is multiplying by four-thirds, which is flipping three-quarters.
Another way to put it is that multiplication and division are inverse operations. Any number multiplied by its reciprocal equals one, for example three-quarters multiplied by four-thirds equals one. Division and multiplication are operations that undo each other, so "dividing by three-quarters" and "multiplying by four-thirds" give the same result. This explanation may be a little abstract for primary students; you can first build intuition with diagrams and concrete examples, then move on to this explanation.
Step-by-step demonstration: two-thirds divided by three-quarters
Step one, look carefully at the question: the dividend is two-thirds and the divisor is three-quarters. Step two, flip only the divisor: three-quarters becomes four-thirds. Step three, change the division sign to a multiplication sign, writing two-thirds multiplied by four-thirds. Step four, multiply numerators together and denominators together: two times four equals eight, three times three equals nine, so the answer is eight-ninths. Step five, check whether the answer is reasonable: two-thirds is about 0.67 and three-quarters is 0.75. Dividing by a number close to one, the result should be slightly less than two-thirds; eight-ninths is about 0.89, which does not match the estimate, so you need to go back and check.
This is exactly where checking matters: dividing by three-quarters is the same as multiplying by four-thirds, and four-thirds is greater than one, so the result should be greater than two-thirds. Eight-ninths is about 0.89, which is greater than 0.67, so the direction of the estimate is correct. The estimate in step five should use the judgement that "multiplying by a number greater than one makes it bigger", rather than directly comparing with the original number.
Common mistakes and checking methods
The most common mistake is to flip the dividend as well, or to flip only part of it. Another common problem is cancelling at the wrong time: you should cancel common factors in the numerator and denominator before multiplying, to make the numbers smaller and reduce calculation errors. Some students also forget to change the division sign to a multiplication sign, or treat the denominator of a whole number as one. A whole number can be written as a fraction with denominator one, for example five equals five over one, so this will not be missed out.
There are three checking methods: first, estimate whether the answer should be bigger or smaller than the dividend; second, use multiplication to reverse it, multiplying the answer by the divisor to see whether it equals the dividend; third, check with a diagram, drawing the bar and counting whether the number of portions is reasonable. Of the three, reversing by multiplication is the most direct and is also the most suitable for upper primary students to check by themselves.
How parents can practise with their children
First let children do two or three questions with diagrams, and do not rush to memorise the formula. You can prepare strips of paper or squared paper, drawing whole numbers and fractions as countable portions, letting children count the answer themselves and then compare it with the calculation. When children find that "the result counted out" and "the result of flipping and multiplying" are the same, the rule is no longer a formula without reason.
Only then move on to pure calculation practice, starting with unit fractions, then general fractions, and finally mixing whole numbers and fractions. Each practice session should focus on only one difficulty, for example today only practising "flip only the divisor", and tomorrow practising cancelling. Ask children to explain their thinking where they went wrong; this is more effective at finding misunderstandings than simply giving the answer. If parents themselves are rusty on fraction operations, they can first look at the textbook or the key points of the curriculum, then do it together with their children.
When it is worth considering extra support
Fraction division is a key topic in Primary 5 and Primary 6, and is also the foundation for learning algebra and ratio later. If a child still repeatedly makes the same kind of mistake after doing diagram practice, for example always flipping the wrong number, or often making mistakes when cancelling, extra support can be considered. Common forms include tutoring centres, private tutoring, home tutoring and online tutoring, each with different arrangements and fees, usually charged monthly or per session, with prices varying quite a lot; it is advisable to understand the details clearly before deciding.
When choosing, you can pay attention to a few points: whether the tutor first identifies the child's specific misunderstanding, rather than just giving lots of practice; whether they are willing to explain with diagrams or concrete examples; and whether parents can regularly find out about progress. The Consumer Council has always reminded consumers to understand fees, refund and change arrangements clearly before enrolling on a course, and to keep receipts. Any extra support should be a supplement, not a replacement for the process of children drawing diagrams and checking by themselves.
Frequently asked questions
Why do we flip the divisor when dividing by a fraction?
Because dividing by a fraction is the same as multiplying by its reciprocal. Flipping and multiplying is not a new rule, but a way of rewriting division as multiplication, giving the same result as the original division. For example, two-thirds divided by three-quarters is the same as two-thirds multiplied by four-thirds.
Which number gets flipped?
Only the divisor is flipped; the dividend stays unchanged. For example, for four-fifths divided by two-thirds, it is the two-thirds that gets flipped, becoming four-fifths multiplied by three-halves. Flipping the dividend as well is a common mistake.
Why is the answer to dividing by three-quarters bigger than the original number?
Because three-quarters is less than one, dividing by a number less than one is the same as multiplying by a number greater than one, so the result gets bigger. You can check by estimating: if the answer is smaller than the dividend, it usually means the wrong number was flipped during the calculation.