Gre math Review 1 Arithmetic

part that is a certain percent of a whole, you can either multiply the whole by the decimal equivalent of the percent or set up a proportion to find the part.

Example 1.7.3: To find 30% of 350, multiply 350 by the decimal equivalent of 30%, or 0.3, as follows.

  x = 350 times 0.3, which is equal to 105

To use a proportion, you need to find the number of parts of 350 that yields the same ratio as 30 out of 100 parts. You want a number x that satisfies the proportion

 

part over whole = 30 over 100, or x over 350 = 30 over 100

Solving for x yields   x = the fraction with numerator 30 times 350, and denominator 100, which is equal to 105, so 30% of 350 is 105.

Given the percent and the part, you can calculate the whole. To do this you can either use the decimal equivalent of the percent or you can set up a proportion and solve it.

Example 1.7.4: 15 is 60% of what number?

Solution: Use the decimal equivalent of 60%. Because 60% of some number z is 15, multiply z by the decimal equivalent of 60%, or 0.6.
0.6z = 15
Now solve for z by dividing both sides of the equation by 0.6 as follows.

z = 15 over 0.6, which is equal to 25
Using a proportion, look for a number z such that

part over whole = 60 over 100, or, 15 over z = 60 over 100

Hence,  60z = 15 times 100

and therefore,
 
z = the fraction with numerator 15 times 100 and denominator 60, which is equal to 1,500 over 60, or 25.
That is, 15 is 60% of 25.

Although the discussion about percent so far assumes a context of a part and a whole, it is not necessary that the part be less than the whole. In general, the whole is called the base of the percent. When the numerator of a percent is greater than the base, the percent is greater than 100%. For example, 15 is 300% of 5, since

  15 over 5 = 300 over 100,
and 250% of 16 is
  open parenthesis 250 over 100, close parenthesis, times 16, which is equal to 2.5 times 16, or 40.
Note that the decimal equivalent of 250% is 2.5.

It is also not necessary for the part be related to the whole at all, as in the question, “a teacher’s salary is what percent of a banker’s salary?”

When a quantity changes from an initial positive amount to another positive amount, for example, an employee’s salary that is raised, you can compute the amount of change as a percent of the initial amount. This is called percent change. If a quantity increases from 600 to 750, then the percent increase is found by dividing the amount of increase, 150, by the base, 600, which is the initial number given:
 
amount of increase over base = the fraction with numerator 750 minus 600, and denominator 600, which is equal to 150 over 600, which is equal to 25 over 100, which is equal to 0.25, or 25%.
We say the percent increase is 25%. Sometimes this computation is written as
 
open parenthesis, the fraction with numerator 750 minus 600, and denominator 600, close parenthesis, times 100% = open parenthesis, 150 over 600, close parenthesis, times 100%, which is equal to 25%.
If a quantity doubles in size, then the percent increase is 100%. For example, if a quantity changes from 150 to 300, then the percent increase is
 
change over base = the fraction with numerator 300 minus 150, and denominator 150, which is equal to 150 over 150, or 100%.
If a quantity decreases from 500 to 400, calculate the percent decrease as follows.
 
change over base = the fraction with numerator 500 minus 400 and denominator 500, which is equal to 100 over 500, which is equal to 20 over 100, which is equal to 0.20, or 20%
The quantity decreased by 20%.

When computing a percent increase, the base is the smaller number. When computing a percent decrease, the base is the larger number. In either case, the base is the initial number, before the change.

Example 1.7.5: An investment in a mutual fund increased by 12% in a single day. If the value of the investment before the increase was $1,300, what was the value after the increase?

Solution: The percent increase is 12%. Therefore, the value of the increase is 12% of $1,300, or, using the decimal equivalent, the increase is  0.12 times $1,300 = $156. Thus, the value of the investment after the change is

$1,300 + $156 = $1,456.

Because the final result is the sum of the initial investment, 100% of $1,300, and the increase, 12% of $1,300, the final result is 100% + 12% = 112% of $1,300. Thus, another way to get the final result is to multiply the value of the investment by the decimal equivalent of 112%, which is 1.12:

  $1,300 times 1.12 = $1,456.

A quantity may have several successive percent changes. The base of each successive percent change is the result of the preceding percent change.

Example 1.7.6: The monthly enrollment at a preschool decreased by 8% during one month and increased by 6% during the next month. What was the cumulative percent change for the two months?

Solution: If E is the enrollment before the first month, then the enrollment as a result of the 8% decrease can be found by multiplying the base E by the decimal equivalent of   100% minus 8% = 92%, which is 0.92:

0.92E.

The enrollment as a result of the second percent change, the 6% increase, can be found by multiplying the new base 0.92E by the decimal equivalent of 100% + 6% = 106%, which is 1.06:

 1.06 times 0.92 times E = 0.9752E.
The percent equivalent of 0.9752 is 97.52%, which is 2.48% less than 100%. Thus, the cumulative percent change in the enrollment for the two months is a 2.48% decrease.