Arithmetic - GRE Quantitative Reasoning

(2_x_ + 4)/(x + 2)

To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.

a prime number is divisible by itself and 1 only

list the factors of each number:

6: 1,2,3,6

9: 1,3,9

71: 1,71

51: 1, 3,17,51

15: 1,3,5,15

The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.

The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.

Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.

But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.

Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.

Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.

Therefore the answer is "Cannot be determined".

Let's first represent the total costs per month:

C = 25000 + 45n, where n is the number of widgets manufactured.

The profit can be represented as 60n – C or 60n – 25000 – 45n = 15n – 25000. Now, we merely have to solve this for 0 in order to find the "break even" line.

15n – 25000 = 0 → 15n = 25000 → n = 1666.67. We must sell a whole number of widgets, so it must be 1667.

For this problem, pay attention only to the values between 2002 and 2006 - each of which is 17 (thousand) and 22 (thousand). Note that the decrease within this range does not need to be accounted for, as the question asks only for values relative to 2002 and 2006.

To calculate percent change, take it as a fraction: 22/17 = 1.294; this represents approximately 129% of the actual value or a 29% increase in sales across these 4 years.

In the problem, 150% of 6 represents the change in the value, not the new value. Percent "greater than" is a key phrase that refers to percent change. Therefore, 9, or 150% of 6, is how much 6 increases. The answer is 6 + 9 = 15.

Formula: Original * (1 + percent increase/100) = New

6(1 + 1.50) = 15

First we need to find out how many possible numbers there are. The number of possible four-digit numbers with four different digits is simply 4 * 4 * 4 * 4 = 256.

To find the sum, the formula we must remember is sum = average * number of values. The last piece that's missing in this formula is the average. To find this, we can average the first and last number, since the numbers are consecutive. The smallest number that can be created from 1, 2, 3, and 4 is 1111, and the largest number possible is 4444. Then the average is (1111 + 4444)/2.

So sum = 5555/2 * 256 = 711,040.

For each quantity, only count the integers that aren't in the other quantity. Both quantities include the numbers 51 to 98, so those numbers won't affect which is greater. Only Quantity A has 49 and 50 (for a total of 99) and only Quantity B has 99. Since the excluded numbers from both quantities equal 99, you can conclude that the 2 quantities are equal.

The sum of all integers from 1 to 30 can be found using the formula

, where is 30. In this case, the sum equals 465.

This is a straightforward percentage problem. After the first discount, the price of the weights is $0.8X. This price is then further reduced by 10% for the employee discount. 90% of 0.8X = 0.9(0.8X) = 0.72X.

The easiest way to solve this is to note that a 30% markdown reduces the cost to 70% of the original cost. Likewise, a 40% markup is the same as making the price 140% of its current value. Using this, we can solve our problem in one string of multiplications:

New cost = Original * 0.7 * 1.4 * 0.9

New cost = 20 * 0.7 * 1.4 * 0.9 = 17.64

To find 25% of $50 you multiply 50 and 0.25. This gives you 12.50. This means the dress is $12.50 cheaper, so it now costs 50 – 12.50 = $37.50. You have an additional 10% off, so you multiply 37.50 by 0.10 to get 3.75. 37.50 – 3.75 = $33.75. This is the cost of the dress. NOTE: You CANNOT add 25% and 10% and take 35% off of the $50 dress. You will get the wrong answer.

There are 6 ways to choose the first rose bush, 5 ways to choose the second, 4 ways to choose the third, and 3 ways to choose the fourth. In total there are 6 * 5 * 4 * 3 = 360 ways to arrange the rose bushes.

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.

With this problem the first thing to do is cancel out variables. The x2 can all be divided by each other because they are present in each system. The equation will now look like this:

Now we can see that the equation can all be divided by y, leaving the answer to be:

_x_2 – _y_2 can be also expressed as (x + y)(x – y).

Therefore, the fraction now can be re-written as (x + y)(x – y)/(x + y).

This simplifies to (x – y).

Following this equation, you divide 4 by 8 to get 1/2. When a variable is raised to an exponent, and you are dividing, you subtract the exponents, so 6 – 3 = 3.

125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.