Mixed Numbers

According to the order of operations, we must 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, therefore:

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

Now we will get the following 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:

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

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 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:

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

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:

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

When we are presented with a fraction over a fraction (in this case one-third over one-sixth) We can convert it into a more manageable form.

$1/3 : 1/6$

It's important to remember that a fraction is actually another sign of division, hence the given exercise is in fact equivalent to one-third divided by one-sixth.
When dealing with division of fractions, the easiest method for solving them is by performing "multiplication by the reciprocal" as shown below:

$1/3\times6/1$

Multiply the numerator by the numerator and the denominator by the denominator to obtain the following result:

$\frac{6}{3}$

Which when reduced equals

$\frac{2}{1}$

Now let's return to the original exercise. In order to solve it we need to take the mixed fraction and convert it to an improper fraction.
We can achieve this by simply moving the whole numbers back to the numerator.

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

$3\times2=6$

$6+1=7$

Therefore the resulting fraction is:

$\frac{7}{2}$

We want to proceed to perform the subtraction exercise.
When both fractions have the same denominator we subtract them.
Therefore in order to achieve this we'll expand the fraction $\frac{2}{1}$ to a denominator of 2, and obtain the following:

$\frac{4}{2}$

We can now proceed to perform subtraction -

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

Convert this back to a mixed fraction in order to obtain the following result: