Complex Fractions - GRE Quantitative Reasoning

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!

is the answer

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:

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!

is the answer

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:

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!

is the answer

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:

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.

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.

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 :