Percentages
A percent is
just a shorthand way of expressing a fraction whose denominator is 100. Percent
means "per 100", "out of 100", or "divided by 100". For example, 25% =
25/100 = 0.25 and
0.3% = 0.3/100 = 0.003. In terms of money, 50 cents out of a dollar is 50 cents out of 100,
which is 50/100 of a dollar or 50% of a dollar.
To find a percentage
of something, the percents must be converted to decimals and then
multiplied by some number. Never directly add and/or subtract percents; you must first multiply
them by something. When finding percentages, convert the sentence into a mathematical
equation. For example, "5 is what percent of 100" can be converted into "5 = x(100)"
where x
is the percent you are looking for.
When working with
percents, it is usually necessary to convert percents to decimals before
performing computations with them. Since a percent is just a fraction with denominator 100,
you convert a percent to a decimal by moving the decimal point two places from right to left.
For example, 6% is equivalent to (.06). In the following example, it is necessary to convert the
percent to a decimal.
Example 1: What
is 30 percent of 200?
To find 30% of
200, convert 30% to .30. Then multiply 200 by .30, which results in 60.
Hence, 60 is 30% of 200.
To convert a
decimal to a percent, move the decimal point two places from left to right and
add a % sign. For example, 0.8 = 80% and 0.02 = 2%.
To convert a
percent to a fraction, just make the percent the numerator of a fraction with
denominator 100 and reduce the fraction. For example, 40% = 40/100 = 4/10 = 2/5.
To convert a
fraction to a percent, divide the numerator by the denominator and move the
decimal point two places to the right.
Example 2: Express
4/5 as a percent.
To do so, use long
division and move the decimal point. 5 divided into 4 is 0.8 which is
equivalent to 80%. Hence, 4/5 = 0.8 = 80%.
The following is
a list of percents and their fraction and decimal equivalents which should
be committed to memory:
|
0.01
=
|
1/100
=
|
1%
|
|
0.1
=
|
1/10
=
|
10%
|
|
0.2
=
|
1/5
=
|
20%
|
|
0.25
=
|
1/4
=
|
25%
|
|
0.5
=
|
1/2
=
|
50%
|
|
0.75
=
|
3/4
=
|
75%
|
Ratios
Ratios,
like fractions, decimals, and percents, are just another way of expressing division.
Every fraction is a ratio and every ratio is a fraction. A fraction is just the ratio of the
numerator
to the denominator. The ratio 1: 2 (read "1 to 2") is equivalent to the fraction 1/2 or the
decimal
0.5 or 50% or just 1 divided by 2. On the GRE, ratios may be expressed in any of the
following ways:
x/y
the ratio of x to y
x is to y
x : y
Anytime you see
a ratio, treat it just like a fraction. Anything you can do to a fraction, you
can also do to a ratio, including cross-multiplying, reducing, finding common denominators, etc.
Here are some examples
of how to work ratio problems on the GRE.
Example 3:
If you have 3 coins and the ratio of pennies to nickels is 2 : 1, how many pennies
and how many nickels are there?
Clearly, there
are 2 pennies and 1 nickel since you only have 3 coins altogether and there are 3
parts in the ratio (2 and 1 = 3).
Example 4: If
you have 24 coins and the ratio of pennies to nickels is 2 : 1, how many of each
type of coin do you have?
Since the ratio
again contains 3 parts (2 and 1), divide 24 by 3 to get the number of coins in
each part; then multiply each part of the ratio by the result . Dividing 24 by 3 yields 8 which
means each of the 3 parts in the ratio consists of 8 coins. Two of the parts are pennies at 8
coins per part, so there are 16 pennies. One part is nickels which makes 8 nickels.
Example 5: At
a camp for boys and girls, the ratio of girls to boys is 5 : 3. If the camp’s
enrollment is 160, how many of the children are boys?
(a) 20 (b) 36 (c)
45 (d) 60 (e) 100
A ratio of 5 :
3 means 8 total parts. To find out how many children are in each part, divide the
total enrollment by the total number of parts. (160 divided by 8 is 20); that means each part is
20 children. Three parts are boys and 3 multiplied by 20 is 60 which is the answer (d). Note
that there is no need to find out how many girls there are. Also, be careful and use the correct
part of the ratio that answers the question.
Proportions
The GRE often contains
questions in which you must compare two ratios which are
proportional. These questions take a given ratio, or relationship, and project it onto a larger or
smaller scale while leaving out one piece of information.
Example 6: If
10 baskets contain a total of 50 eggs, how many eggs would 7 baskets contain?
(a) 10 (b) 17 (c)
35 (d) 40 (e) 50
To solve a problem
like this, set up the two proportional ratios, one of which will have a
missing piece of information. Think of the ratios like this: 10 baskets is to 50 eggs as 7 baskets
is to x eggs. Then set up these ratios as proportional fractions and cross- multiply:
OR
10 : 50
7 : x
THEN
10x = 350
x = 35
Hence, the answer
is (c). Note that we could have reduced the fraction 10/50 to 1/5 and made
the cross-multiplication easier.
Averages
The average, or
arithmetic mean, of a set of numbers is the sum of all the numbers in the set
divided by the total number of numbers in the set. The formula to remember is
average = the
sum of the numbers being averaged
the number of elements
For example, the
average of the numbers 1, 2, 3, 4, 5, 6, 7 is 28 divided by 7 which is 4. The
GRE always refers to an average as an "average (arithmetic mean)". Just ignore the
parenthetical remark so it doesn’t confuse you.
In an averaging
problem, you may be asked to find the total first. For example, suppose a
problem states that the average of 4 test scores is 80 find the sum of the tests. Recall the
formula above. In this case we already know the average and the number of elements; we need
to find the sum of the tests. Hence, just cross-multiply and the total of the test scores is the
product of the two: 80
4 = 320.
Suppose you are
told that two of these scores are 90 and 95 and you want to find the average
of the other two scores. The sum of 90 and 95 is 185. So, the total of the other two scores is
320 – 185 = 135. Hence, the average of the remaining two scores is 135 divided by 2 which is
(67.5).
Do not get confused if
two or more of the elements being averaged are the same. For
example, the average of 5, 5, 5, and 20 is 5 + 5 + 5 + 20 divided by 4 which is 35 divided by
4 which is (8.75). You do not add 5 and 20 and divide by 2, nor do you add 5 and 20 and
divide by 4.
Another situation
which may confuse you is when a new element is added to a set that has
already been averaged. Suppose you take two tests and earn scores of 70 and 80. The
average of your two tests is 75. Now, suppose you take a third test and earn another score of
70. Does your average remain 75? No. Your new average is (70 + 80 + 70)/3 = 73.33. Note
that your new average is NOT (75 + 70)/2 = 72.5. Suppose your third test score was 75.
Then your average over all three tests is still 75 (Why?).
Another common
error occurs when averaging a set of numbers that includes 0. For
example, what is the average of 0, 0, 0, and 4? A careless person would say 4, but the answer
is 1! An easy averaging problem can be made difficult on the GRE if certain information is left
out.
Example 7: The
average test score earned by a group of students is 80. If 40% of the students
have an average score of 70, what is the average score of the remaining 60%?
(a) 70.33 (b) 80
(c) 86.67 (d) 90 (e) 95
An important piece
of information is missing from this question. Do not try to solve this problem
by setting up a complex algebraic equation. Since the problem is dealing with percentages, the
actual number in the group is irrelevant. Just pick a number that’s easy to work with. Since all
of our percents are multiples of 10, choose 10 as the number of students in the group. 40%, or
4, of these students have an average score of 70. We want to know the average score of the
remaining 6 (60% of 10).
Now that we’ve
gotten rid of the percents, it is just an averaging problem. So first find the
totals. If the average score of 10 students is 80, then the total sum of their test scores is 800.
Just remember the averaging formula and cross-multiply the average and the number of
elements. Since 4 of the students have an average score of 70, their total score is 280. Again,
just cross-multiply. To find the total score of the remaining 6 students, just subtract: 800 –
280
= 520. Hence, the average score must be 520 divided by 6 which is 86.67 or choice (c).
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