# Basic Arithmetic : Manipulation of Fractions

## Example Questions

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### Example Question #1 : Manipulation Of Fractions

What is the reciprocal of ?

Explanation:

To get the reciprocal of a fraction, you simply switch the numerator and the denominator.

In our case our numerator is  and our denominator is .

So  becomes .

### Example Question #1 : Manipulation Of Fractions

Compute:

Explanation:

We will need to rewrite this in order to eliminate the negative exponent in the problem.

Because the denominator has a negative exponent, that is the same as having a positive exponent in the numerator. Therefore we can rewrite the problem as follows and then multiply.

### Example Question #1 : Manipulation Of Fractions

Evaluate:

Explanation:

To divide a term by a fraction, take the reciprocal of the fraction.

Then mutiply both terms.

### Example Question #1 : Reciprocals

What is the sum of the reciprocal of  and ?

Explanation:

To find the reciprocal of a fraction, flip the numerator and the denominator.

Thus, the reciprocal of  is .

Then we need to find the sum of 4 and 7, which is 11.

### Example Question #1 : Manipulation Of Fractions

What is the reciprocal of  multiplied by the reciprocal of ?

Explanation:

To find the reciprocal of a fraction, we simply need to switch the numerator and the denominator: for example, the reciprocal of a fraction  is .

With integers, it helps to remember that all integers are really fractions with a denominator of :

, and

The reciprocals of these numbers are  and  respectively.

Therefore, to solve the problem, we first need to find the reciprocals of  and . If we keep in mind that , we can determine that the reciprocals are  and , respectively. The product of these two numbers is:

### Example Question #1 : Mixed Numbers

What is  in fraction form?

Explanation:

To rewrite the mixed fractions as improper fractions we multiply the whole number by the denominator and add the numerator to get our new numerator. Our denominator stays the same. With this in mind we get the following:

and

.

.

### Example Question #1 : Manipulation Of Fractions

Convert this mixed number  to an improper fraction.

Explanation:

The mixed number can be broken down as follows: 3 is the whole number, 1 is the numerator, and 4 is the denominator.

In order to convert the mixed number to an improper fraction, you must first multiply the denominator with the whole number and then add the numerator to it.

This answer is the numerator of the improper fraction:

.

The denominator of the fraction remains the same and the final answer is .

### Example Question #1 : Manipulation Of Fractions

What is  equivalent to?

Explanation:

This question is asking you to change an improper fraction into a mixed number. To do that we will need to divide 75 by 6.

Now, we place the remainder as the numerator of a fraction with a denominator that is 6 because we just divided by six.

So then, 12 remainder 3 becomes .

Simplify this mixed fraction.

### Example Question #2 : Manipulation Of Fractions

Find the sum.

Explanation:

To add mixed numbers, you need to first change both of them into improper fractions.

Then, add the two improper fractions like you would any other fraction.

We need both fractions to have the same denominator before we can add them. Since 27 is a multiple of 3, we only need to change 1 fraction.

Now, we can add these fractions.

Finally, change the improper fraction back into a mixed number.

### Example Question #1 : Manipulation Of Fractions

What is the result of the above equation?

Explanation:

So we first multiply the numerators:

We then multiply the denominators

Our resulting fraction is:

We reduce by dividing by 2 from the numerator and denominator.

For the numerator:

For the denominator:

The resulting fraction is:

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