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Adding and Subtracting Mixed Numbers


This page looks at how we can approach adding and subtracting mixed numbers effectively.

Mixed numbers, also called mixed fractions, have a slightly different form from more standard fractions.
With standard fractions there is just a numerator and denominator present, and no whole number part.


Adding and Subtracting Mixed Numbers Sums

When we have a mixed number in a sum, we have to deal with a whole number part, as well as a fraction part.

So instead of a fractions sum lookijng like:       \bf{\frac{4}{5}} + \bf{\frac{1}{3}}

A sum featuring mixed numbers would look something like:       6\bf{\frac{4}{9}}  +  2\bf{\frac{3}{9}}


For such a situation we can separate the whole numbers and fractions parts up, then adding or subtracting accordingly. Which here would be:

6  +  2  +  \bf{\frac{4}{9}}  +  \bf{\frac{3}{9}}   =   8  +  \bf{\frac{7}{9}}     giving the result    8\bf{\frac{7}{9}}.




Examples




1.1

6\bf{\frac{5}{9}}  −  2\bf{\frac{4}{9}}

Solution

A subtraction sum such as this works a bit differently than a sum where we add mixed numbers.
But the same approach as seen above does give us a result.

Things do look slightly different at the first step in this example.

6\bf{\frac{5}{9}}  −  2\bf{\frac{5}{9}}    =    6  −  2  +  \bf{\frac{5}{9}}  −  \bf{\frac{4}{9}}


( Because the 1st fraction  \bf{\frac{5}{9}}  was positive in nature when it was part of the mixed number  6\bf{\frac{5}{9}}.

It must also be positive when the sum is separated up, and the fraction  \bf{\frac{5}{9}}  is moved further right.

So rather than adding a number to a number, PLUS a fraction to a fraction, like before.
We instead take away a number from a number, PLUS taking away a fraction from a fraction. )

To give us:

6  −  2  +  \bf{\frac{5}{9}}  −  \bf{\frac{4}{9}}   =   4  +  \bf{\frac{1}{9}}   =   4\bf{\frac{1}{9}}



1.2

4\bf{\frac{1}{5}}  −  1\bf{\frac{3}{5}}

Solution

4  −  1  +  \bf{\frac{1}{5}}  −  \bf{\frac{3}{5}}    =    3  +  (-\bf{\frac{2}{5}})    =    3  −  \bf{\frac{2}{5}}    =    2\bf{\frac{3}{5}}

There is also an alternative approach, which is to convert both mixed numbers to fractions, before carrying out the full subtraction.
Then converting the answer back to a mixed number.

4\bf{\frac{1}{5}}  −  1\bf{\frac{3}{5}}     =     \bf{\frac{4 \space \times \space 5 \space + \space 1}{5}}  −  \bf{\frac{1 \space \times \space 5 \space + \space 3}{5}}   =   \bf{\frac{21}{5}}  −  \bf{\frac{8}{5}}

=   \bf{\frac{13}{5}}   =   2\bf{\frac{3}{5}}



1.3

5\bf{\frac{1}{4}}  +  \bf{\frac{1}{6}}

Solution

First put into the following the form.

 5 + \bf{\frac{1}{4}} + \bf{\frac{1}{6}}

Then we look to write the fractions where there is a common denominator between them, and we can complete the sum.

5 + \bf{\frac{1}{4}} + \bf{\frac{1}{6}}   =   5 + \bf{\frac{6}{24}} + \bf{\frac{4}{24}}   =   5\bf{\frac{10}{24}}   =   5\bf{\frac{5}{12}}



1.4

–3\bf{\frac{5}{7}}  +  8\bf{\frac{6}{7}}

Solution

The first fraction in the sum  \bf{\frac{5}{7}}  is part of a negative mixed number.

Thus it will be negative when the sum is separated up, and the fraction  \bf{\frac{5}{7}}  is moved further right.

–3\bf{\frac{5}{7}}  +  8\bf{\frac{6}{7}}    =    –3  +  8  +  (–\bf{\frac{5}{7}})  +  \bf{\frac{6}{7}}    =    5  +  \bf{\frac{1}{7}}    =    5\bf{\frac{1}{7}}







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