GRE Math : Complex Fractions

Example Questions

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Example Question #1 : Complex Fractions

Solve:

Explanation:

To simplify a complex fraction, simply invert the denomenator and multiply by the numerator:

Multiplying the numerator by the reciprocal of the denominator for each term we get:

Since we have a common denominator we can now add these two terms.

Example Question #1 : Complex Fractions

Simplify:

Explanation:

Although you could look for the common denominator of the fraction as it has been written, it is probably easiest to rewrite the fraction in slightly simpler terms. Thus, recall that you can rewrite your fraction as:

Using the rule for dividing fractions, you can rewrite your expression as:

Then, you can multiply each set of fractions, getting:

This makes things very easy, for then your value is:

Example Question #3 : Complex Fractions

Simplify:

Explanation:

For this problem, begin by rewriting the complex fraction, using the rule for dividing fractions:

This is much easier to work on. Cancel out the s and the  and the , this gives you:

, which is merely .  Thus, your problem is:

The common denominator is , so you can rewrite this as:

Example Question #4 : Complex Fractions

Explanation:

Begin by simplifying all terms inside the parentheses. Begin with the innermost set. Find a common denominator for the two terms. In this case, the common denominator will be twenty:

Simplify  to  and convert  to not a mixed fraction:

Multiply the two fractions in the parentheses by multiplying straight across (A quick shortcut would be to factor out the 10 on top and bottom).

Now convert  to a non-mixed fraction. It will become .

In order to subtract the two fractions, find a common denominator. In this case, it will be 70.

Now subtract, and find the answer!

Example Question #1 : How To Divide Complex Fractions

A cistern containing  gallons of water has sprung two leaks. One leaks at a rate of  of a gallon every half hour. The second one leaks at a rate of  a gallon every fifth of an hour. In how many hours will the cistern be empty (presuming that the leaks will empty it eventually)?

Explanation:

It is best to figure out what each of the leaks are per hour. We can figure this out by adding together the two fractional rates of leaking. For the first leak, we can do this as follows:

This is the same as:

For the second leak, we use the same sort of procedure:

Thus, our two leaks combined are:

The common denominator for these is ; thus, we can solve:

Now, our equation can be set up:

, where  is the time it will take for the cistern to be emptied.

Multiply by  on both sides:

Solve for :

Divide by :

Example Question #1 : How To Divide Complex Fractions

Which of the following answer choices is a value for  in the following equation?

Explanation:

Begin by simplifying the left side of the equation. You can do this by multiplying the numerator of the fraction by the reciprocal of its denominator:

Now, we know that our equation is:

Multiply both sides by  and you get:

Thus, by taking the square root of both sides, you get:

Among your answers,  is the only one that matches these.

Example Question #2 : How To Divide Complex Fractions

Explanation:

Begin by converting both top and bottom into non-mixed fractions:

So now we have:

In order to divide, take the fraction on the bottom, flip it, and multiply it by the fraction up top:

Multiply straight across:

Now reduce the fraction. Both top and bottom are divisible by 9 (an easy way to tell this is to see that in the original fractions we are multiplying both 9 and 18 are divisible by 9), so reduce each side by a factor of 9:

The answer is .

Example Question #1 : How To Multiply Complex Fractions

Simplify:

Explanation:

Remember that fraction multiplication is the easiest of the arithmetical operations we can use on fractions. We can merely multiply the numerators and denominators by each other. As you will see, this is the easiest way to do this problem, for the numerators and denominators can be cancelled. Thus, we know:

Now, the parts of this fraction can be cancelled, giving us a much simpler expression:

, which is the same as

To simplify this, you just need to multiply the numerator by the reciprocal of the denominator; thus, we have:

Example Question #1 : Complex Fractions

Simplify the following equation:

Explanation:

The most important element of this question is attention to detail. It may help to rewrite the equation by cancelling out like terms in the fraction, starting with the removal of an equivalent number of zeroes from the numerator and denomerator, followed by shifting the decimals an equivalent number of spaces in the numerator and denomerator:

Following this, like factors can be cancelled from the numerator and denominator, facilitating calculation of the answer:

Example Question #1 : Complex Fractions

It is known that  of the athletes at a convention are volleyball players, and that  of the volley ball players are female. If there are 54 female volleyball players at the convention, how many of the athletes at the meet are not volleyball players?

Explanation:

The first step to this problem will be to find the total number of volleyball players, since the total number of athletes is related to this value. Since we know how many female volleyball players there are, we can find the total number of volleyball players by relating the proportion:

This in turn allows us to find the total number of athletes:

And finally, from this, we can find the total number of athletes that aren't volleyball players:

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