How to Divide Fractions, Including Mixed Numbers and Improper Fractions

Most peo­ple imag­ine that divid­ing by a frac­tion will be very com­pli­cated but it’s really quite straight­for­ward. You just have to “flip” (a tech­ni­cal maths term) the divid­ing frac­tion by swap­ping the top num­ber (the numer­a­tor) with the bot­tom num­ber (the denom­i­na­tor). You then just have to mul­ti­ply, which we’ve already seen is simple.

You don’t need to know this, but the fol­low­ing proof explains why flip­ping the divid­ing frac­tion and then mul­ti­ply­ing works. It’s not dif­fi­cult to fol­low and if you under­stand it, you’ll never for­get how to divide by a frac­tion. It’s also, I think, more sat­is­fy­ing to know why some­thing works rather than just know how to do it. The proof is also quite ele­gant (math­e­mati­cians some­times say that proofs are “beau­ti­ful”, that’s a bit over the top). Any­way, here’s the proof:-

Proof: To Divide by a Frac­tion, Just Flip it and Multiply

Proof to Divide a Fraction, Flip and Multiply

Proof to Divide a Frac­tion, Flip and Multiply

Just as saw when we were mul­ti­ply­ing frac­tions, you will prob­a­bly have to deal with mixed num­bers and improper frac­tions. This video walks through the five sim­ple steps that will cover all pos­si­bilites:-

The video looks at the fol­low­ing division:-

Dividing by Fractions Question




These are the five steps that are shown in the video and that can be used to answer divide by a frac­tion question:-

Step 1:-  Con­vert any Mixed Num­bers to Improper Fractions

A mixed num­ber is a com­bi­na­tion of an inte­ger (or whole num­ber) and a frac­tion. Any mixed num­bers need to be con­verted to an improper frac­tion, a frac­tion where the top num­ber (numer­a­tor) is greater then the bot­tom num­ber (denominator):-

Convert Mixed Number to Improper Fraction

Step 2:- Take the divid­ing num­ber and switch or flip its numer­a­tor and denominator.

The proof shown above shows why this works. Don’t for­get to change the divide sign into the mul­ti­ply sign:-

Flip the dividing fraction

Step 3:- If it’s pos­si­ble sim­plify any of the fractions

Again, this is not always pos­si­ble, but if you can sim­plify it makes the remain­ing steps easier:-

Simplify the Fractions

Step 4:- Multiply

Now we’re ready to just mul­ti­ply the numer­a­tors and the denominators:-

Multiply Numerators and Denominators

Step 5:- If Required Change your answer from an Improper Frac­tion to a Mixed Number

Your ini­tial answer may or may not be an improper frac­tion (numer­a­tor greater than the denom­i­na­tor) but if it is you should con­vert it to a mixed num­ber. You may lose marks (or to be more pos­i­tive gain extra marks if you do) if you don’t com­plete this final step:-

Convert Improper Fraction to Mixed Number

Re-cap: How to Divide by a Frac­tion in Five Sim­ple Steps

Here’s all five steps summarized:-

How to Divide By a Fraction in 5 Simple Ste

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