Adding frac­tions (if they have dif­fer­ent denom­i­na­tors) is not some­thing you can eas­ily work out how to do. You have to know a method. The method is not dif­fi­cult and becomes sec­ond nature with practice.

Frac­tions with dif­fer­ent denom­i­na­tors are incom­pat­i­ble, you can­not add them before you have made them com­pat­i­ble by con­vert­ing them so that have the same (com­mon) denom­i­na­tor. A fool­proof way to find a com­mon denom­i­na­tor is to mul­ti­ply them. You could say that, to make frac­tions com­pat­i­ble, go forth and mul­ti­ply the denominators!

For exam­ple:

5/4 + 4/6 — a com­mon denom­i­na­tor is 4 x 6 = 24

You will have pos­si­bly noticed that 4 & 6 have a lower com­mon denom­i­na­tor, namely 12. If you notice a lower com­mon denom­i­na­tor, go ahead and use it; it will save you hav­ing to sim­plify your answer at the end. BUT don’t get hung up about find­ing the low­est com­mon denom­i­na­tor. Just be assured that by mul­ti­ply­ing the denom­i­na­tors you have a secure way to find a com­mon denom­i­na­tor. You can always sim­plify your answer at the end.

Adding Frac­tions — Some Definitions

  • Mixed num­ber: a whole num­ber and a frac­tion.For exam­ple 1¼.
  • Improper Frac­tion:  a frac­tion where the top num­ber (the numer­a­tor) is greater than the bot­tom num­ber (the denom­i­na­tor). For exam­ple 5/4.
  • Numer­a­tor: The top num­ber of a frac­tion. For exam­ple, the numer­a­tor of 5/4 is 5.
  • Denom­i­na­tor: The bot­tom num­ber of a frac­tion. For exam­ple, the denom­i­na­tor of 5/4 is 4.
  • Here is a video that walks you through the 5 sim­ple steps to add any two frac­tions. I sug­gest you read the rest of this arti­cle first and then use the video to make sure you under­stand and, more impor­tantly, remem­ber the 5 steps to add fractions:

Video Tuto­r­ial: Adding Fractions

Adding Frac­tions in 5 Sim­ple Steps

We know now the under­ly­ing prin­ci­ples of adding frac­tions and have defined some key terms. Here’s an addi­tion of frac­tions ques­tion to show the 5 sim­ple steps that can be used to add any two fractions.

Ques­tion:   1¼ + 4/6

Step 1: Con­vert Any Mixed Num­bers to Improper Fractions

In this ques­tion 1¼ is a mixed num­ber. To con­vert a mixed num­ber to improper frac­tions, mul­ti­ply the whole units (in this case, 1) by the denom­i­na­tor of the frac­tion (in this case, 4). Add the answer to the numer­a­tor of the frac­tion (in this case, 1) and place this answer over the denominator:

((1 x4 ) + 1 ) / 4  = 5/4

So we have revised the frac­tions to add to:-

5/4 + 4/6

Step 2: Find a Com­mon Denominator

5/4 + 4/6

To find a com­mon denom­i­na­tor sim­ply mul­ti­ply the 2 denominators:

4 x 6 =24

Step 3: Con­vert the Numerators

To do this mul­ti­ply the numer­a­tors by the same amount as you mul­ti­plied the denominators.

Tak­ing the first frac­tion 5/4

to get the com­mon denom­i­na­tor we mul­ti­plied by 6:  4 x 6 =24

So we need to mul­ti­ply the numer­a­tor by the same amount:

5 x 6 = 30

So our first frac­tion becomes (5 x 6)/(4 x 6) = 30/24

Sim­i­larly for the sec­ond frac­tion 4/6

to get the com­mon denom­i­na­tor we mul­ti­plied by 4: 6 x 4 =24

So we need to mul­ti­ply the numer­a­tor by the same amount:

4 x 4 = 16

So our sec­ond frac­tion becomes (4 x 4)/(6 x 4) = 16/24

And we now have com­pat­i­ble frac­tions to add:-

30/24 + 16/24

Step 4: Add the Numerators

We can now sim­ply add the numerators:

30/24 + 16/24 = 46/24

Step 5: If Pos­si­ble Sim­plify the Frac­tion and If Answer is an Improper Frac­tion Con­vert to a Mixed Number

So from step 4 we have 30/24 + 16/24 = 46/24

First sim­plify  46/24 = 23/12

Now we have an improper frac­tion (the numer­a­tor is greater than the denom­i­na­tor) so we need to con­vert it to a mixed num­ber. To do this divide the numer­a­tor by the denom­i­na­tor (23 ÷ 12) and show the answer as a whole num­ber with the remain­der as a fraction:

23/12 =  1 11/12