# ACT Math : Complex Fractions

## Example Questions

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### Example Question #1 : Complex Fractions

Which of the following is equal to ?

Explanation:

First we must take the numerator of our whole problem. There is a fraction in the numerator with  as the denominator. Therefore, we multiply the numerator of our whole problem by , giving us .

Now we look at the denominator of the whole problem, and we see that there is another fraction present with  as a denominator. So now, we multiply the denominator by , giving us .

Our fraction should now read . Now, we can factor our denominator, making the fraction .

Finally, we cancel out  from the top and the bottom, giving us .

### Example Question #2 : Complex Fractions

Simplify:

Explanation:

Rewrite  into the following form:

Change the division sign to a multiplication sign by flipping the 2nd term and simplify.

### Example Question #1 : Complex Fractions

Evaluate:

Explanation:

The expression  can be rewritten as:

Change the division sign to a multiplication sign and take the reciprocal of the second term.  Evaluate.

### Example Question #4 : Complex Fractions

Simplify:

Explanation:

The expression  can be simplified as follows:

We can simplify each fraction by multiplying the numerator by the reciprocal of the denominator.

From here we add our two new fractions together and simplify.

### Example Question #5 : Complex Fractions

Simplify the following:

Explanation:

Begin by simplifying your numerator. Thus, find the common denominator:

Next, combine the fractions in the numerator:

Next, remember that to divide fractions, you multiply the numerator by the reciprocal of the denominator:

Since nothing needs to be simplified, this is just:

### Example Question #1 : How To Subtract Complex Fractions

Simplify,

Explanation:

Convert the numerators and denominators into single fractions, then simplify.

Start by finding the lowest common denominator in both the numerator and denominator of the complex fraction.

Simplify. Divide complex fractions by multiplying the numerator by the reciprocal of the denominator.

Solve.

### Example Question #2 : How To Subtract Complex Fractions

Subtract:

Explanation:

The least common multiple can be found by multiplying the denominators: 2, 3, and 5. The common denominator of these numbers is 30. Multiply the numerator with what was multiplied to the denominator of each term, and then solve.

### Example Question #3 : How To Subtract Complex Fractions

What is ?

Explanation:

First, simplify both sides.  becomes  and  becomes . The LCF between  and  is 36. Thus,  This simplifies to .

### Example Question #4 : How To Subtract Complex Fractions

Simplify:

Explanation:

Begin by simplifying the denominator of the first fraction:

Now, remember that division of fractions is done by multiplying the numerator by the reciprocal of the denominator. Thus:

Simplify a bit:

Simplify