How to Add & Subtract Improper Fractions

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The term "improper fraction" means that the numerator (the top number of the fraction) is bigger than the denominator (the bottom number of the fraction). Improper fractions are actually mixed numbers in disguise, so the last step of your math problem will usually be to convert that improper fraction into a mixed number. But if you're still performing operations like addition and subtraction, it's easiest to leave the numbers in improper fraction form for now.

Adding Improper Fractions

The process for adding improper fractions works exactly the same as the process for adding proper fractions. (In a proper fraction, the numerator is smaller than the denominator.)

    Start by making sure that both fractions you're dealing with have the same denominator. If they don't have the same denominator, you'll have to convert one or both fractions to a new denominator, so that they match.

    For example, if you're asked to add the fractions:

    \frac{5}{4} + \frac{13}{12}

    they don't have the same denominator. But if you have sharp eyes, you might notice that 4 × 3 = 12. You can't just multiply the denominator of 5/4 by 3 to turn it into a 12, because that would change the value of the fraction. But you can multiply the fraction by 3/3, which is just another way of writing 1. This changes it to a new denominator without altering its value:

    \frac{5}{4} × \frac{3}{3} = \frac{15}{12}

    Now you have two fractions with the same denominator: 15/12 and 13/12.

    Once you have two fractions with the same denominator, you can simply add the numerators, and then write the answer over the same denominator. To continue the example, to add the improper fractions 15/12 and 13/12, you'll first add the numerators:

    15 + 13 = 28

    Then write the answer over the same denominator:

    \frac{28}{12}

    Or to write it out another way:

    \frac{15}{12} + \frac{13}{12} = \frac{28}{12}

    If your answer from the previous step is already in lowest terms, you can consider the problem done. But if you can simplify the result any further, you should – and since you're dealing with at least one improper fraction, you may also be able to convert the answer to a mixed number. In this case, you can do both. Start by identifying common factors in the numerator and denominator, and then canceling them out:

    \frac{28}{12} = \frac{7(4)}{3(4)} = \frac{7}{3}

    (Four is a common factor in both numerator and denominator; canceling that out gives you a result of 7/3.)

    Next, convert the improper fraction to a mixed number by performing the division indicated by the fraction: 7 ÷ 3. But you shouldn't divide all the way through the decimal places; instead, stop when you have a whole-number result and a remainder. In this case,

    7 ÷ 3 = 2 \text{r}1

    or two with a remainder of 1.

    Write the whole number on its own – 2 – followed by a fraction with the remainder as the numerator and the denominator you last had – in this case, 3 – as the denominator still. To conclude the example, you have a mixed-number answer of

    2 \, \frac{1}{3}

Subtracting Improper Fractions

To subtract improper fractions, you use the same steps as adding. Consider another example:

\frac{6}{4} - \frac{5}{4}

    In this case both fractions already have the same denominator, so you can go straight on to the next step.

    Subtract the numerators from each other as originally directed, and then write the answer over the same numerator as both fractions you're dealing with. Keep in mind that while the order of your numbers didn't matter for addition, it does matter for subtraction – so don't swap the numbers around. In this case, you have:

    6 - 5 = 1

    Writing that over your denominator gives you an answer of:

    \frac{1}{4}

    In this case, your answer – 1/4 – is already in lowest terms, so you can't reduce or simplify it. And because it's no longer an improper fraction, you also can't convert it to a mixed number. So all you have to do to finish the problem is write your answer clearly:

    \frac{6}{4} - \frac{5}{4} = \frac{1}{4}

Adding Mixed Numbers With Improper Fractions

If you're asked to add mixed numbers together, or to add a mixed number to a fraction, the easiest method is almost always converting the mixed number into a fraction; this makes it easier to manipulate. For example, if you're asked to add

2 \, \frac{1}{6} + \frac{8}{6}

you'd first multiply the whole number portion of 2 1/6 by 6/6 to convert it into fraction form:

2 × \frac{6}{6} = \frac{12}{6}

Don't forget to add in the extra 1/6 from the mixed number:

\frac{12}{6} + \frac{1}{6} = \frac{13}{6}

Now your original problem becomes

\frac{13}{6} + \frac{8}{6}

Because both fractions have the same denominator, you can go ahead and add the numerators, and then write the answer over the existing denominator:

\frac{13}{6} + \frac{8}{6} = \frac{21}{6}

While some teachers may let you leave the answer in this form, it's always good practice to convert the answer back to a mixed number:

3 \, \frac{3}{6}

And then, using your eagle eyes, you've probably already spotted that you can cancel factors to simplify the fraction 3/6 to 1/2, which gives you a final answer of:

2 \, \frac{1}{6} + \frac{8}{6} = 3 \, \frac{1}{2}

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