GRE Math : Fractions

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #1 : Fractions

Edward rolls three dice; two are six-sided, and one is twenty-sided, with 1 through 6 represented on the six-sided dice, and 1 through 20 represented on the twenty-sided die.

What is the probability that the sum of his roll will equal 5?

Possible Answers:

\displaystyle \frac{1}{720}

\displaystyle \frac{1}{36}

\displaystyle \frac{1}{120}

\displaystyle \frac{1}{144}

\displaystyle \frac{1}{240}

Correct answer:

\displaystyle \frac{1}{120}

Explanation:

The first step will be to calculate how many total different potential rolls are possible. This is given by the product of the number of possible rolls for each of the dice: 6, 6, 20:

\displaystyle Number of Possible Rolls = 6 \cdot 6\cdot 20=720\displaystyle Number of Possible Rolls = 6 * 6* 20=720

Next it is necessary to account for the total number of rolls that will sum up to 5. Writing them out can help, such as in the format of:

\displaystyle (1^{st} die, 2^{nd} die,3^{rd} die)

\displaystyle (3,1,1) ,(2,2,1),(2,1,2),(1,2,2),(1,3,1),(1,1,3)

Knowing that there are 6 different rolls that sum up to 5, the probability of rolling a 5 can be found by taking the number of events that satisfy this sum and dividing it by the number of total possible events:

\displaystyle Probability_{Rolling 5} =\frac{6}{720} = \frac{1}{120}

Example Question #411 : Arithmetic

\displaystyle z\neq3

Simplify the following:

\displaystyle \frac{z^5-9z^4+27z^3-27z^2}{z^3-6z^2+9z}

Possible Answers:

\displaystyle z^3-3

\displaystyle z-3

\displaystyle z^2-3

\displaystyle z^2-3z

\displaystyle z^3-3z^2

Correct answer:

\displaystyle z^2-3z

Explanation:

Looking at this equation, note that since all terms in the numerator and denominator contain a \displaystyle z, it is possible to rewrite it as follows:

\displaystyle \frac{z^5-9z^4+27z^3-27z^2}{z^3-6z^2+9z}=\frac{z^2(z^3-9z^2+27z-27z)}{z(z^2-6z+9)}

or

\displaystyle \frac{z(z^3-9z^2+27z-27z)}{(z^2-6z+9)}

Now the parenthetical terms must be addressed. The problem statement and answer choices give a clue that they are some sort of multiples of \displaystyle z-3.

In fact, Pascal's triangle reveals that the top and bottom are the cube and square of this term respectively:

\displaystyle \frac{z(z-3)^3}{(z-3)^2}

Cancelling terms, we are left with:

\displaystyle z(z-3)=z^2-3z

Example Question #1 : Fractions

\displaystyle 0.0132748501327485... is a repeating decimal. What digit is in the \displaystyle 57^{th} place?

Possible Answers:

\displaystyle 3

\displaystyle 1

\displaystyle 7

\displaystyle 2

\displaystyle 0

Correct answer:

\displaystyle 0

Explanation:

Examination of the value \displaystyle 0.0132748501327485... reveals that after the sequence \displaystyle 01327485, the decimal repeats, and that the sequence has a length of eight values.

A longer answer would be to write the sequence out and count down the digits until the \displaystyle 57th value was found. However, this is a time-consuming process and one that is prone to error.

Rather, notice how since the numbers repeat, it's possible to skip most of the counting:

\displaystyle 1^{st} digit: \displaystyle 0

\displaystyle 9^{th} digit: \displaystyle 0

And so on. Since \displaystyle 56 is the closest multiple to \displaystyle 57, we can subtract the two to find the digit that matches the \displaystyle 57^{th} digit.

\displaystyle 57^{th}-56^{th}=1^{st}

So the \displaystyle 57^{th} digit is \displaystyle 0.

Example Question #1 : Fractions

Which of the following is equal to  \displaystyle \frac{1}{4} of the reciprocal of \displaystyle .025 percent?

Possible Answers:

\displaystyle 1000

\displaystyle 10000

\displaystyle 0.625

\displaystyle 10

\displaystyle 100

Correct answer:

\displaystyle 1000

Explanation:

The first step will be to find the reciprocal of \displaystyle .025 percent. Note that as a percent, this should be converted to a decimal form: \displaystyle .00025.

The reciprocal of a number is given by \displaystyle 1 divided by that number, so the reciprocal of \displaystyle .00025 is given as:

\displaystyle \frac{1}{.00025}= 4000

Therefore:

\displaystyle (\frac{1}{4})4000=1000

Example Question #1 : Fractions

\displaystyle 20 percent of \displaystyle 200 is \displaystyle p.

\displaystyle p percent of \displaystyle q is \displaystyle 16.

Quantity A: \displaystyle p

Quantity B: \displaystyle q

Possible Answers:

The relationship between A and B cannot be determined.

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

Correct answer:

The two quantities are equal.

Explanation:

To make the comparison, the values of \displaystyle p and \displaystyle q must be determined.

We are told that \displaystyle 20 percent of \displaystyle 200 is \displaystyle p, so its value can be determined as follows:

\displaystyle p=(.20)(200)=40

With \displaystyle p known, it is possible to find \displaystyle q, since \displaystyle p percent of \displaystyle q is \displaystyle 16:

\displaystyle q(.01p)=16

\displaystyle q=\frac{16}{.01p}=\frac{16}{(.01)(40)}=\frac{16}{.4}=40

\displaystyle p=q=40

The two quantities are equal.

Example Question #1 : Fractions

\displaystyle 383490102847190 can be rewritten as \displaystyle 3.83490102847190 times what?

Possible Answers:

\displaystyle 10^{13}

\displaystyle 10^{15}

\displaystyle 10^{14}

\displaystyle 10^{11}

\displaystyle 10^{12}

Correct answer:

\displaystyle 10^{14}

Explanation:

To solve this problem, realize that a decimal may be placed at the very end of this integer:

\displaystyle 383490102847190=383490102847190.

Now, count how many spaces the decimal will need to move to the left to reach:

 \displaystyle 3.83490102847190

It must move a total of \displaystyle 14 spaces, so:

 \displaystyle 3.83490102847190(10^{14})=383490102847190

Example Question #2 : Fractions

Simplify: \displaystyle \frac{\frac{x}{2}+\frac{x}{3}+\frac{x}{4}-\frac{x}{5}}{6}

Possible Answers:

\displaystyle \frac{53}{360}x

\displaystyle \frac{77}{360}x

\displaystyle \frac{53}{60}x

\displaystyle \frac{77}{10}x

\displaystyle \frac{77}{60}x

Correct answer:

\displaystyle \frac{53}{360}x

Explanation:

To solve this problem, begin with simplifying the numerator. This can be done by first finding a common denominator. For

a common denominator would be \displaystyle 60:

or

Which combines into:

\displaystyle \frac{53}{60}x

But recall that this is just the numerator, and there is still a \displaystyle 6 in the denominator:

\displaystyle \frac{\frac{53}{60}x}{6}

So, the final answer is:

\displaystyle \frac{53}{360}x

Example Question #1 : Fractions

0.3 < 1/3

4 > √17

1/1/8

–|–6| = 6

Which of the above statements is true?

Possible Answers:

4 > √17

–|–6| = 6

1/< 1/8

0.3 < 1/3

Correct answer:

0.3 < 1/3

Explanation:

The best approach to this equation is to evaluate each of the equations and inequalities.  The absolute value of –6 is 6, but the opposite of that value indicated by the “–“ is –6, which does not equal 6.

1/2 is 0.5, while 1/8 is 0.125 so 0.5 > 0.125.

√17 has to be slightly more than the √16, which equals 4, so“>” should be “<”.

Finally, the fraction 1/3 has repeating 3s which makes it larger than 3/10 so it is true.

Example Question #1 : Fractions

Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.

         10 < n < 15

Quantity A          Quantity B

   7/13                    4/n

Possible Answers:

Quantity A is greater.

The answer cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Correct answer:

Quantity A is greater.

Explanation:

To determine which quantity is greater, we must first determine the range of potential values for Quantity B. Let's call this quantity m. This is most efficiently done by dividing 4 by the highest and lowest possible values for n.

4/10 = 0.4

4/15 = 0.267

So the possible values for m are 0.267 < m < 0.4

Now let's find the value for 7/13, to make comparison easier.

7/13 = 0.538

Given this, no matter what the value of n is, 7/13 will still be a higher proportion, so Quantity B is greater.

Example Question #1 : Fractions

Quantity A: \displaystyle \frac{1}{3}+0.43+\frac{1}{5}
 

Quantity B: \displaystyle \frac{1}{7}+0.5+\frac{1}{3}

 

Possible Answers:

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

The relationship cannot be determined from the given information.

Correct answer:

Quantity B is greater.

Explanation:

The GRE test now has a built-in calculator.  Simply convert the fractions to decimals and compare:

Quantity A = 0.333 +0.43 + 0.2 = 0.963

Quantity B = 0.1429 + 0.5 + 0.3333 = 0.976

Thus, Quantity B is larger.

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