Mixed Fractions - Examples, Exercises and Solutions

Note that the right-hand side of the addition exercise between the fractions gives a result of a whole number, so we'll start with that:

$6\frac{2}{3}+\frac{1}{3}=7$

Giving us:

$7\frac{5}{6}+7=14\frac{5}{6}$

According to the order of operations, we will solve the exercise from left to right.

Let's note that in the first addition exercise, we have an addition between two halves that will give us a whole number, so:

$\frac{1}{2}+3\frac{1}{2}=4$

Now we will get the exercise:

$4+4\frac{2}{4}=$

Let's note that we can simplify the mixed fraction:

$\frac{2}{4}=\frac{1}{2}$

Now the exercise we get is:

$4+4\frac{1}{2}=8\frac{1}{2}$

According to the rules of the order of operations in arithmetic, we solve the exercise from left to right.

Let's note that:

$\frac{1}{3}+\frac{2}{3}=\frac{3}{3}=1$

We should obtain the following exercise:

$1+2\frac{3}{4}=3\frac{3}{4}$

When we have a fraction over a fraction, in this case one-third over one-sixth, we can convert it to a form that might be more familiar to us:

$1/3 : 1/6$

It's important to remember that a fraction is actually another sign of division, so the exercise we have is one-third divided by one-sixth.
When dealing with division of fractions, the easiest method for solving is performing "multiplication by the reciprocal", meaning:

$1/3\times6/1$

Multiply numerator by numerator and denominator by denominator and get:

$\frac{6}{3}$

Which when reduced equals

$\frac{2}{1}$

Now let's return to the original exercise, to solve it we need to take the mixed fraction and convert it to an improper fraction,
meaning move the whole numbers back to the numerator.

To do this we'll multiply the whole number by the denominator and add to the numerator

$3\times2=6$

$6+1=7$

And therefore the fraction is:

$\frac{7}{2}$

Now we want to do the subtraction exercise, but we see that we have another step on the way.
We subtract fractions when both fractions have the same denominator,
so we'll expand the fraction $\frac{2}{1}$ to a denominator of 2, and we'll get:

$\frac{4}{2}$

And now we can perform subtraction -

$\frac{7}{2}-\frac{4}{2}=\frac{3}{2}$

We'll convert this back to a mixed fraction and we'll see that the result is

Let's solve the exercise from left to right.

We will combine the left expression in the following way:

$\frac{6+8}{7}x=\frac{14}{7}x=2x$

Now we get:

$2x+3\frac{2}{3}x=5\frac{2}{3}x$