The right method when dividing fractions is using the concept of reciprocal. Reciprocals are flipped fractions.

Well, the reciprocal of the number \(\frac{2}{3}\) is \(\frac{3}{2}\). When dealing with fractions, practically all reciprocals are improper.

Dividing is mostly just as division, but here, we must come up with the reciprocal of our divisor (so the \(2nd\) value). And when we’ve got our reciprocal, we just multiply. Let’s look at an example: \(\frac{1}{3} \div \frac{1}{2}\) equals

• The reciprocal of the divisor is \(\frac{2}{1}\)

• Then, rewrite as a multiplication problem. So \(\frac{1}{3}*\frac{2}{1}\) equals?

• Then, multiply the numerators. So \(1*2 = 2\)

• And multiply the denominators. So \(3*1 = 3\)

• Now write out our new fraction. So \(\frac{2}{3}\)

• Simplification needed? No.

The answer is: \(\frac{1}{3}\,\div\) (divided by) \(\frac{1}{2}\) equals \(\frac{2}{3}\)

Now, if you would like to do this shorter, just flip the \(2nd\) term to multiply.

Now let’s look at “Dividing Simple Fractions”

Well, let’s look at an example that has simple numbers. This works in the same way as the problem above. In this example, we will need to convert the improper fraction because our answer is going to be bigger than \(1\) (one). \(\frac{4}{5}\) (divided by) \(\div\,\frac{3}{7}\) equals?

• The reciprocal of our divisor is \(\frac{7}{3}\)

• So rewrite this as a multiplication problem. So \(\frac{4}{5}*\frac{7}{3} =\) ?

• Now, multiply the numerators. So \(4*7 = 28\)

• Then, multiply the denominators. So \(5*3 = 15\)

• Write out our new fraction. So \(\frac{28}{15}\)

• Then, convert our improper fraction into a new mixed number.

So: \(\frac{28}{15}\) equals \(28\,\div\,15\) which equals \(1r13\), which equals \(1\,\frac{13}{15}\)

• Simplification needed? No.

The answer is: \(\frac{4}{5}\,\div\) (divided by) \(\frac{3}{7}\) equals \(1\,\frac{3}{5}\)

Now how could we end up with an answer greater than \(1\) (one)? Well, as with a lot of our earlier division problems, our dividend can go into our divisor more times than just \(1\) (one time).

Example: \(45\) (divided by) \(\div\, 9\) equals \(5\).

With fractions, it works the same. We may see big fractions divided by little fractions. There are a whole lot of little pieces or segments that will go into that bigger value. So let’s look at the problem below that has \(\frac{9}{10}\) (which is close to \(1\)) and \(\frac{1}{20}\) (which is close to zero). \(\frac{1}{20}\) goes into our \(\frac{9}{10}\) segment a lot of times, as it is so very tiny. \(\frac{9}{10}\) (divided by) \(\div\, \frac{1}{20}\) equals?

• The reciprocal of the divisor is \(\frac{20}{1}\)

• Now, rewrite this as a multiplication problem. So: \(\frac{9}{10}*\frac{20}{1}\,=\)?

• Then multiply the numerators. So: \(9*20 = 180\)

• Then multiply the denominators. So: \(10*1 = 10\)

• Now, write out our new fraction. So: \(\frac{180}{10}\)

• Then, convert our improper fraction into a new mixed number.

So: \(\frac{180}{10}\) equals \(180÷10\) which equals \(18\)

• Do we need to simplify? No.

The answer is: \(\frac{9}{10}\) (divided by) \(\div\,\frac{1}{20}\) equals \(18\)

This answer is meaning that we will need \(18\) pieces of our \(\frac{1}{20}\) number if we want to fill \(1\) space with the size of nine-tenths \((\frac{9}{10})\).

Now, let’s look at “Dividing Mixed Numbers”

Well, it’s time to do a few mixed numbers. Now we will be making a few improper fractions just like we did in the multiplication section.

We’ll begin by converting the whole thing into improper fractions. Then, we’ll do our flip, after which we’ll do our multiplication. If needed, finish it all off with some simplification. \(5\,\frac{1}{3}\) (divide d by) \( \div\frac{24}{9}\) equals?

• First, convert both the dividend and the divisor to improper fractions.

So: \( 5\,\frac{1}{3}\) equals \(5+\,\frac{1}{3}\) which equals \(\frac{15}{3}+\frac{1}{3}\) which again equals \(\frac{16}{3}\)

And: \(2\,\frac{4}{9}\) equals \(2+\,\frac{4}{9}\) which equals \(\frac{18}{9}+\frac{4}{9}\) which again equals \(\frac{22}{9}\)

• Now, rewrite this problem. So \(\frac{16}{3}\) (divided by) \(\div\frac{22}{9}=\)?

• The reciprocal of our divisor is: \(\frac{9}{22}\)

• Now, rewrite it as a new multiplication problem. So: \(\frac{16}{3}\) times \(\frac{9}{22}\) equals?

• First, multiply the two numerators. So \(16\) times \(9\) equals \(144\)

• Then, multiply the two denominators. So \(3\) times \(22\) equals \(66\)

• Then, write out our new fraction. So \(\frac{144}{66}\)

• Now, convert our improper fraction into a new mixed number.

So: \(\frac{144}{66}\) equals \(144\,\div\,66\) which equals \(2r12\) which, again, equals \(2\) and \(\frac{12}{66}\)

• Then, if needed, simplify our fraction. We have \(12\) and \(66\). Well, we notice that both numbers have a common factor, six \((6)\). So we will divide both the top and the bottom number by \(6\) to get to \(\frac{2}{11}\).

The answer is: \(5\,\frac{1}{3}\) (divided by) \(\div\, 2\,\frac{4}{9}\) equals \(2\) and \(\frac{2}{11}\)

So here’s one extra step within our process and our numbers may get very big. Generally, we’ll see rather easy values in the examples but we love challenging you a bit and if you’re able to manage this so far, you will be able to deal with \(3-digit\) divisions as well.

Keep in mind to take enough time for your answers and don’t forget to go through each and every step of the process. There will be times that you will not need to do anything to complete a step, however, you still need to check. And there won’t always be a need for simplification of your answers, but you must always check.

Next Lesson: Adding and Subtracting Fractions

*Last Updated on April 12, 2021.*