To add fractions with different denominators, first the fractions should be rewritten as equivalent fractions with the same denominators. The simplest common denominator to get is the product of the denominators (i.e. , in this question). However, the best denominator to use is the least common denominator, which is the least common multiple of the denominators (which is 12 in this case).

Now the denominators are equal and we can add the numerators:

which can be written as:

To add fractions with different denominators, first the fractions should be rewritten as equivalent fractions with the same denominators. The simplest common denominator to get is the product of the denominators (i.e. , in this case). However, the best denominator to use is the least common denominator, which is 12 in this question. So we can write:

As the denominators are equal, we can add the numerators:

or:

At first, we should add the first two mixed numbers. This is best done by writing writing them vertically and adding integers and fraction parts (Note: denominators are the same).

So we have:

We can rewrite them vertically:

Denominators are the same, so we can add and subtract the numerators:

The fraction can be reduced to lowest terms by dividing the numerator and the denominator by their greatest common factor which is 5 in this case. So we get:

Since the greatest common factor of 1 and 3 is 1,  is in lowest terms.

In this story problem we are able to see that we are having to do something with fractions. Story problems can be tricky so let's try to transform this story into an equation. First lets take out the fractions we see in the problem.

One-half we know can be shown as .

One-fifth we also know is shown as .

Those are the only two fractions we are having to work with. Next let's examine what the story wants us to do with the fractinos. The key word in this problem is combine, that means we are going to add our fractions together. 

This is what the story problem looks like when we turn it into an equation. 

Next, since we are adding fractions we need to have a common denominator. Our denominators are 2 and 5. To have a common denominator we must have a multiple of 2 and 5 that both are able to go into. Think of a number that both 2 and 5 can both go into... Was 10 the number you came up with?

Let's use 10 as our common denominator. So how many times does 2 have to be multiplied to give us 10? (5). We need to multiply the numerator and the denominator of one half by five.

The fraction is still the same, but we had to do this for a common denominator. 

We need to still change our other fraction, how many times does 5 go into 10? (2) 

We now have common denominators so all we have to do is add the numerators across and keep the denominator as 10. So , and the denominator remains 10, giving us the answer... 

The least common denominator of 5 and 7 is 35. Use this as the denominator for both fractions and change the numerators accordingly:

Rewrite the fractions with the common denominator , and add numerators:

The least common denominator of the three fractions is , so write each fraction in terms of this denominator:

By order of operations, subtract, then add:

,

which is the correct result.

Rewrite each fraction with the least common denominator, which is :

We can add vertically, adding the fractions and the whole numbers:

Rewrite using the least common denominator, which is ; then add numerators: