# Fractions Part 7 – Arithmetic on Mixed Fractions

So now let’s use the skills from the last two posts to work on mixed fractions. Consider:

${3}\frac{3}{8}\hspace{0.33em}{+}\hspace{0.33em}{6}\frac{7}{8}$

As the denominators are the same, one can do this problem by adding the whole parts first to get 9 then add the fractions together to get $\frac{10}{8}$ which after simplifying and converting to a mixed fraction is equal to $1\frac{1}{4}$. Then you add this to the 9 to get $10\frac{1}{4}$. But I want to show a more general method that is particularly useful when the denominators are different and/or the problem is a subtraction.

The first step is to convert the mixed fractions to improper ones as discussed in my last post:

${3}\frac{3}{8}\hspace{0.33em}{+}\hspace{0.33em}{6}\frac{7}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{{(}{8}\hspace{0.33em}\times\hspace{0.33em}{3}{)}\hspace{0.33em}{+}\hspace{0.33em}{3}}{8}\hspace{0.33em}{+}\hspace{0.33em}\frac{{(}{8}\hspace{0.33em}\times\hspace{0.33em}{6}{)}\hspace{0.33em}{+}\hspace{0.33em}{7}}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{27}{8}\hspace{0.33em}{+}\hspace{0.33em}\frac{55}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{82}{8}$

The addition in the last step was easy as the denominators are the same. Now all that is left is to simplify and convert to a mixed fraction. You can convert first then simplify which has the advantage of having smaller numbers to factor, so let’s do that:

$\frac{82}{8}\hspace{0.33em}{=}\hspace{0.33em}{82}\hspace{0.33em}\div\hspace{0.33em}{8}\hspace{0.33em}{=}\hspace{0.33em}{10}$ with remainder of 2 so

$\frac{82}{8}\hspace{0.33em}{=}\hspace{0.33em}{10}\frac{2}{8}\hspace{0.33em}$

So now all that is left is to simplify the fractional part:

$\frac{2}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{1}}{\rlap{/}{2}\hspace{0.33em}\times\hspace{0.33em}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{4}$

So ${3}\frac{3}{8}\hspace{0.33em}{+}\hspace{0.33em}{6}\frac{7}{8}\hspace{0.33em}{=}\hspace{0.33em}{10}\frac{1}{4}$ which is the same answer we got before. Isn’t maths consistent? (and fun!).

Now let’s do one with different denominators:

${6}\frac{3}{8}\hspace{0.33em}{-}\hspace{0.33em}{3}\frac{7}{12}$

We start by converting the problem into one with improper fractions:

${6}\frac{3}{8}\hspace{0.33em}{-}\hspace{0.33em}{3}\frac{7}{12}\hspace{0.33em}{=}\hspace{0.33em}\frac{{(}{8}\hspace{0.33em}\times\hspace{0.33em}{6}{)}\hspace{0.33em}{+}\hspace{0.33em}{3}}{8}\hspace{0.33em}{-}\hspace{0.33em}\frac{{(}{12}\hspace{0.33em}\times\hspace{0.33em}{3}{)}\hspace{0.33em}{+}\hspace{0.33em}{7}}{12}\hspace{0.33em}{=}\hspace{0.33em}\frac{51}{8}\hspace{0.33em}{-}\hspace{0.33em}\frac{43}{12}$

Now we need to find a common denominator between 8 and 12. To find the least common denominator (LCD), we first factor both numbers:

8 = 2 × 2 × 2,  12 = 2 × 2 × 3

As 2 and 3 are the only factors present, we now combine them, using them only the maximum number of times each appear in the above factorisation:

LCD = 2 × 2 × 2 × 3 = 24

So the common denominator we will use is 24. We want to convert each of the fractions in the problem into equivalent ones that have 24 as the denominator. For the first fraction, we will multiply top and bottom by 3 to get the 24 in the denominator. We will use 2 for the second fraction to get 24  in the denominator there as well:

$\begin{array}{l} {\frac{51}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{{51}\hspace{0.33em}\times\hspace{0.33em}{3}}{{8}\hspace{0.33em}\times\hspace{0.33em}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{153}{24}}\\ {\frac{43}{12}\hspace{0.33em}{=}\hspace{0.33em}\frac{{43}\hspace{0.33em}\times\hspace{0.33em}{2}}{{12}\hspace{0.33em}\times\hspace{0.33em}{2}}\hspace{0.33em}{=}\hspace{0.33em}\frac{86}{24}} \end{array}$

So now the problem becomes:

$\frac{51}{8}\hspace{0.33em}{-}\hspace{0.33em}\frac{43}{12}\hspace{0.33em}{=}\hspace{0.33em}\frac{153}{24}\hspace{0.33em}{-}\hspace{0.33em}\frac{86}{24}\hspace{0.33em}{=}\hspace{0.33em}\frac{67}{24}$

Now convert back to a mixed fraction:

$\frac{67}{24}\hspace{0.33em}{=}\hspace{0.33em}{67}\hspace{0.33em}\div\hspace{0.33em}{24}\hspace{0.33em}{=}\hspace{0.33em}{2}$ with a remainder of 19.

So the final answer is $2\frac{19}{24}$. The fractional part cannot be simplified any further.

So the steps to do multiplication would be just to convert the mixed fractions to improper ones, multiply these, and then convert back to a mixed fraction.

The general steps to do arithmetic on mixed fractions are:

1.  Convert all fractions to improper fractions.
2. If the problem is a multiplication one or if the denominators are the same, skip to step 5.
3. Find the LCD for the denominators.
4. Convert the improper fractions into equivalent ones with the LCD as the denominator.
5. Do the indicated arithmetic (multiplication, addition, or subtraction) on the improper fractions.
6. Convert the answer back to a mixed fraction if the numerator is greater than the denominator.
7. Simplify the fractional part if needed.