GRE Quantitative Reasoning - GRE Quantitative Reasoning

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Question

$2\frac{9}{10}-\left(1\frac{3}{7}\cdot \left(\frac{1}{4}+\frac{1}{20}\right)\right)=?$

Answer

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:

$2\frac{9}{10}-\left(1\frac{3}{7}\cdot \left(\frac{1}{4}+\frac{1}{20}\right)\right)$

$2\frac{9}{10}-\left(1\frac{3}{7}\cdot \left(\frac{5}{20}+\frac{1}{20}\right)\right)$

$2\frac{9}{10}-\left(1\frac{3}{7}\cdot \left(\frac{6}{20}\right)\right)$

Simplify $\frac{6}{20}$ to $\frac{3}{10}$ and convert $1\frac{3}{7}$ to not a mixed fraction:

$1\frac{3}{7}=\frac{10}{7}$

$2\frac{9}{10}-\left(\frac{10}{7}\cdot \frac{3}{10}\right)$

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).

$2\frac{9}{10}-\left(\frac{30}{70}\right)$

$2\frac{9}{10}-\left(\frac{3}{7}\right)$

Now convert $2\frac{9}{10}$ to a non-mixed fraction. It will become $\frac{29}{10}$ .

$\frac{29}{10}-\frac{3}{7}$

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

$\frac{203}{70}-\frac{30}{70}$

Now subtract, and find the answer!

$\frac{173}{70}$ is the answer

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Question

Solve:

$\frac{\frac{2}{3}}{\frac{1}{5}} + \frac{9}{\frac{3}{2}} =$

Answer

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

$\frac{\frac{2}{3}}{\frac{1}{5}} + \frac{9}{\frac{3}{2}} =$

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

$\frac{2}{3} \cdot\frac{5}{1} + \frac{9}{1}\cdot \frac{2}{3} =$

$\frac{10}{3} + \frac{18}{3} =$

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

$\frac{28}{3}$

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Question

Simplify:

$\frac{\frac{1}{2}}{8}+\frac{\frac{x}{4}}{4}$

Answer

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:

$\frac{\frac{1}{2}}{\frac{8}{1}}+\frac{\frac{x}{4}}{\frac{4}{1}}$

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

$\frac{1}{2} \cdot \frac{1}{8} + \frac{x}{4} \cdot \frac{1}{4}$

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

$\frac{1}{16} + \frac{x}{16}$

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

$\frac{1+x}{16}$

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Question

Simplify:

$\frac{\frac{3}{5}}{\frac{3}{10}}+\frac{12}{5}$

Answer

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

$\frac{\frac{3}{5}}{\frac{3}{10}} = \frac{3}{5} \cdot \frac{10}{3}$

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

$\frac{1}{1} * \frac{2}{1}$ , which is merely . Thus, your problem is:

$2+\frac{12}{5}$

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

$\frac{10}{5}+\frac{12}{5} = \frac{22}{5}$

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Question

Solve for :

Answer

To add decimals, simply treat them like you would any other number. Any time two of the digits in a particular place (i.e. tenths, hundredths, thousandths) add up to more than ten, you have to carry the one to the next greatest column. Therefore:

So .

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Question

Solve for :

Answer

To solve this problem, subtract from both sides of the eqution,

Therefore, .

If you're having trouble subtracting the decimal, mutliply both numbers by followed by a number of zeroes equal to the number of decimal places. Then subtract, then divide both numbers by the number you multiplied them by.

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Question

Solve for :

Answer

To solve, you need to do some algebra:

Isolate x by adding the 4.150 to both sides of the equation.

Then add the decimals. If you have trouble adding decimals, an effective method is to place one decimal over the other, and add the digits one at a time. Remember to carry every time the digits in a given place add up to more than .

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Question

Solve for :

Answer

To solve for , first add to both sides of the equation, so that you isolate the variable:

Then, add your decimals, and remember that .

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Question

Solve for :

Answer

To solve, first add to both sides of your equation, so you isolate the variable:

Then add the decimals together:

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Question

Solve for :

Answer

To solve, first add to both sides of the equation:

Then add the decimals together:

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Question

The sum of seven consecutive even integers is 0.

Column A: The product of the seven integers

Column B: 2

Answer

For the sum of 7 consecutive even integers to be zero, the only sequence possible is –6, –4, –2, 0, 2, 4, 6. This can be determined algebraically by assigning the lowest number in the sequence to be “y” and adding 2 for each consecutive even integer, and then setting this equal to zero.

y, y + 2, y + 4, y + 6 . . .

The product of any number and 0 is 0.

Therefore, column B must be greater.

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Question

and are both even whole numbers.

What is a possible solution for ?

Answer

If and are both even whole numbers, then their addition must be an even whole number as well. Although is an even number, it is not a whole number and could therefore not be a solution. This means the only possible solution would be .

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Question

If John buys twenty-two apples on Monday, and thirty-four bananas on Tuesday, how many pieces of fruit does John have?

Answer

Plugging in the values given we arrive at the total fruit John has:

A good note about adding even numbers--any even numbers--is that if you add two even numbers, their sum will ALWAYS be an even number.

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Question

Choose the answer below which best solves the following problem:

Answer

To deal with a problem with this many digits, often the best strategy is to line up one number over the other, then add the places one at a time. Don't forget to carry a one every time the addition goes over ten. Also, note that any time you add two even numbers, their sum will ALWAYS be an even number.

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Question

If $5^2 \times 5^n = 5^{12}$ , what is the value of ?

Answer

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

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Question

Simplify: y3x4(yx3 + y2x2 + y15 + x22)

Answer

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

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Question

Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.

$\dpi{100} \small n>0$

Quantity A: $\dpi{100} \small 16^{n+2}$

Quantity B: $\dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

Answer

By using exponent rules, we can simplify Quantity B.

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{n+1})^{2}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times (8^{2n+2})\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{3(2n+2)}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{4}\times 2^{6n+6}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 4^{n}$

$\dpi{100} \small \dpi{100} \small 2^{6n+10}\div 2^{2n}$

$\dpi{100} \small 2^{4n+10}$

Also, we can simplify Quantity A.

$\dpi{100} \small 16^{n+2}$

$\dpi{100} \small =2^{4(n+2)}$

$\dpi{100} \small =2^{4n+8}$

Since n is positive, $\dpi{100} \small 4n+10>4n+8$

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Question

If $\frac{3^{y - 1}}{3^{-2}} = 27^{y}3^{y}$ , what is the value of ?

Answer

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

$3^{y-1}3^2=27^y3^y$

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

$3^{y-1}3^2=(3^3)^y3^y$

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

$3^{(y-1)+2}=(3)^{3y}3^y$

$3^{y+1}=3^{3y+y}=3^{4y}$

We now know that the exponents must be equal, and can solve for .

$\frac{1}{3}=y$

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Question

Simplify $x^{^{3}}x^{^{5}} - x^{^{2}} + (y^{^{2}})^{^{2}}$ .

Answer

First, simplify $x^{3}x^{5}$ by adding the exponents to get $x^{8}$ .

Then simplify $(y^{2})^{2}$ by multiplying the exponents to get $y^{4}$ .

This gives us $x^{8} - x^{^{2}} + y^{4}$ . We cannot simplify any further.

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Question

Simplify $x^{3}x^{6} + (x^{3})^{3} + x^{3}x^{2}x^{2}x^{2}$ .

Answer

Start by simplifying each individual term between the plus signs. We can add the exponents in and so each of those terms becomes . Then multiply the exponents in so that term also becomes . Thus, we have simplified the expression to which is $3x^{9}$ .

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