# Basic Arithmetic : Addition with Fractions

## Example Questions

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### Example Question #1 : Addition With Fractions

Solve the following expression:

Explanation:

In order to start solving this problem, we need to get each term to have a common denominator.

The common denominator between 2 and 6 is 6. We can rewrite the expression and solve using the following method:

### Example Question #2 : Addition With Fractions

Determine the answer as a mixed number:

Explanation:

1. Find the least common denominator:

The lowest number that both 4 and 6 can go into is 12 meaning that it is the least common denominator.

2. Use the new least common denominator to create two equivalent fractions with a denominator of 12.

3. Add the two fractions together.

4. Convert the fraction to a mixed number:

### Example Question #3 : Addition With Fractions

Evaluate:

Explanation:

In fractions, to add or subtract the numerator, the denominators must be equal.

Find the least common denominator, multiply the top and bottom with what was multiplied on the denominator, and simplify.

=

### Example Question #2 : Addition With Fractions

If Billy has  of a gallon of water, and Bobby has  of a gallon of water, how much water do they have between the two of them?

They have  of a gallon of water.

They have  of a gallon of water.

They have of a gallon of water.

They have  of a gallon of water.

They have  of a gallon of water.

They have  of a gallon of water.

Explanation:

To complete this question, you need to find the least common denominator in order to add the fractions. Cross multiply the fractions so that they are in the same terms, and then add.  To do this, simply multiply each fraction by the denominator of the other, over itself (so that the fraction equals 1).  Follow the steps below:

### Example Question #32 : Operations With Fractions

What is the sum of  and ?

Explanation:

When adding two fractions, the denominators must be the same. When the denominators are the same, we merely add the numerators together to see what the sum is. The trickiest part is transforming the two fractions into fractions with the same denominator.

When we transform a fraction, we are multiplying it by . For example,  is the same fraction as  because . Because , we are not changing the value of the fraction, we are merely expressing the same value in a different form. This is essential when adding fractions with different denominators.

To rewrite these fractions with the same denominator, we need to find the Least Common Denominator (LCD). In this case, the least common denominator is , as  contains both  and  as factors. Our next step is to rewrite our two fractions as fractions with denominators of .

To transform our first fraction, , into a fraction with  as its denominator, we need to multiply the fraction by .

To transform our second fraction, , into a fraction with  as its denominator, we need to multiply the fraction by . Therefore, our answer becomes:

### Example Question #3 : Addition With Fractions

Solve:

Explanation:

The least common multiple of 2 and 9 is 18.

### Example Question #33 : Fractions

Simplify

Explanation:

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

### Example Question #6 : Addition With Fractions

Simplify the following fraction:

Explanation:

Determine the factors of both the numerator and the denominator:

We notice that 3 is a factor of both 12 and 39 so we can simplify by dividing both 12 and 39 by 3.

The result is therefore,

### Example Question #35 : Operations With Fractions

Find the simplified result:

Explanation:

Start by making both fractions into the same denominator. One option is

Then adjust the numerators by multiplying each fraction's numerator by the other fraction's denominator:

Then we simplify by dividing both numerator and denominator by 2:

which gives us the final result.

### Example Question #13 : Adding And Subtracting Fractions

Explanation:

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 7 over 7 and the second fraction by 5 over 5.