Complex Fractions - GRE Quantitative Reasoning

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.

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:

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:

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:

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:

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:

The simplest way to do this problem is visually. From looking at the graph, the greatest difference between Beetleton and Caterpilly is 2 billion US, and this difference is also observed where those points for Beetleton and Caterpilly are closest to the x-axis: 2008.

Of course, the percent differences can also be calculated using the formula:

and for each year, the percent differences are as follows:

Of course this method is much more time consuming.

Percent growth is given by the formula:

So the percent grown for Caterpilly from 2010 to 2011 is:

Conversely, if percent growth is known, a new value can be found as follows:

The GDP for 2014 is ( in billions US dollars), so the projected GDP in billions of US dollars for 2015 is:

For this problem, note that for the students elected from the running for either president, vice president, and hall monitor, position matters, and so this is dealing with a permution, with the following number of potential outcomes:

However, for the second election, in which students are competing for positions, since all the secretarial offices are equal, position does not matter, and so we are dealing with a combination:

The total potential outcomes is given by the product of these two values:

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 .

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 :

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.

The simplest way to do this problem is visually. From looking at the graph, the greatest difference between Beetleton and Caterpilly is 2 billion US, and this difference is also observed where those points for Beetleton and Caterpilly are closest to the x-axis: 2008.

Of course, the percent differences can also be calculated using the formula:

and for each year, the percent differences are as follows:

Of course this method is much more time consuming.

Percent growth is given by the formula:

So the percent grown for Caterpilly from 2010 to 2011 is:

Conversely, if percent growth is known, a new value can be found as follows:

The GDP for 2014 is ( in billions US dollars), so the projected GDP in billions of US dollars for 2015 is:

For this problem, note that for the students elected from the running for either president, vice president, and hall monitor, position matters, and so this is dealing with a permution, with the following number of potential outcomes:

However, for the second election, in which students are competing for positions, since all the secretarial offices are equal, position does not matter, and so we are dealing with a combination:

The total potential outcomes is given by the product of these two values:

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 .

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 :

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.

This problem merely requires careful working out of each part. Begin by simplifying the first fraction:

The numerator will be:

The denominator will be:

Thus, we have the following fraction:

Remember that you must multiply the numerator by the reciprocal of the denominator:

Now, work on the second fraction:

This fraction is much easier. After simplifying the numerator, you get:

This is the same as:

Thus, we come to our original expression! It is:

The common denominator of these fractions is . Thus, you have: