PERCENTS GREATER THAN 100
WARNING: There are two prerequisites for learning about percents: knowledge of
decimals and fractions.
There is a series of questions listed below. In order for you to continue,
you should answer yes to all of them. If you are unsure of any of these points,
click on the appropriate question for a review.
INTRODUCTION TO PERCENTS GREATER THAN 100
So far all of the percents we have seen were less than 100. (percents < 100)
But there is no law that says percents can't be greater than 100
(at least not in most states)
On this page we'll explore percents greater than 100 (percents > 100)
Remember the rectangle diagram from our first page? Well here it is again.
Only now it brought its identical twin brother.
In the diagram below you will see two rectangles. Each of these rectangles
is identical.
Consider only one of them to be the base. It is important for you to remember that we
did not double the base, we just printed it twice.
The idea of a percent larger than 100 means we're going to have the entirety of one
of the rectangles (100%) plus a portion of the second one.
In this second diagram you'll again see the first rectangle
and in the second rectangle 90 of the squares are white and 10 of them are colored red.
Imagine if you will that we are interested in the entire first rectangle (100% of the rectangle)
plus only the white squares in its duplicate (90% of the rectangle).
Altogether we care about 190% of the rectangle.
What will soon become clear is that whenever we are taking a percent of a base:
- If the percent is smaller than 100, the result will be smaller than the base.
- If the percent is greater than 100, the result will be larger than the base.
PERCENTS, DECIMALS, & FRACTIONS
Starting with a percent
First of all percents greater than 100 are treated just like any other percent
You still are going to change them into fractions by
placing them over a denominator of 100.
To change a percent into a decimal you will still drop the "%" sign and
move the decimal point two places to the left.
The only change will be the fraction will now be an improper fraction which you'll
change into a mixed fraction,
and the decimal will no longer have "0" as its whole number part.
Below you'll see a few examples of changing percents into
fractions and decimals
- 125%
125% = 125/100 = 1 25/100 = 1 1/4 (notice that this is a mixed fraction)
After moving the decimal two places to the left 125 becomes 1.25 thus 125% = 1.25
(Notice that there is now a "1" before the decimal part)
- 130%
130% = 130/100 = 1 30/100 = 1 3/10 (Notice that htis is a mixed fraction)
After moving the decimal two places to the left 130 becomes 1.30 thus 130% = 1.30
(Notice there is now a "1" before the decimal part)
Starting with a fraction
If we started with a mixed fraction we could separate its whole number and fraction parts, then
change the fraction part into a decimal by dividing the numerator by the denominator.
Then simply add the whole number part to the decimal.
Now we have two different ways to change our fraction to a percent.
We could multiply its equivalent decimal by 100 (move the decimal point two places to the
right)
or
We could change the improper form of the fraction into a ratio of some number over 100.
(i.e. a/b = ?/100). To solve this problem multiply the fraction by 100 (i.e. ? = 100 * a/b).
That number will be our percent.
Below you'll sere a few examples of changing a fraction into a decimal and a percent.
- 1 1/5
1 1/5 = 1 + 1/5
Now 1.000 divided by 5 equals 0.20 so 1/5 = 0.2
Thus 1 1/5 = 1 + 1/5 = 1 + .2 = 1.20
To get a percent we could take 1.2 and move the decimal point two places to the right
1.20 = 120%
or
1 1/5 = ?/100
Well 1 1/5 = 6/5 (the whole number part 1 multiplied by the denominator 5 and
then added to the numerator 1 gives us 6, our new numerator)
6/5 = ?/100
100 * 6/5 = 20 * 6 = 120
1 1/5 = 120%
- 2 3/4
2 3/4 = 2 + 3/4
Now 3.000 divided by 4 equals 0.75 so 3/4 = 0.75
Thus 2 3/4 = 2 + 3/4 = 2 + .75 = 2.75
To get a percent we could take 2.75 and move the decimal point two places to the right
2.75 = 275%
or
2 3/4 = ?/100
Well 2 3/4 = 11/4 (the whole number part 2 multiplied by the denominator 4 and
then added to the numerator 3 gives us 11, our new numerator)
11/4 = ?/100
100 * 11/4 = 25 * 11 = 275
2 3/4 = 275%
Starting with a decimal
To change the decimal into a fraction, first separate the whole number and decimal
parts (just as you did above). Change the decimal part into a fraction by putting its
corresponding whole number over the appropriate power of ten
(If the decimal has one decimal place, put it over 10 ; if it has two decimal places,
put it over 100, if it has three decimal places, put it over 1000, etc.).
Now reduce this fraction to lowest terms. Once you're finished reintroduce
the whole number part to complete your mixed fraction
To change the decimal to a percent, multiply it by 100
(move the decimal two places to the right)
Below you'll see a few examples of changing a decimal into a fraction and a percent.
- 1.375
1.375 = 1 + 0.375
Now .375 = 375/1000 (there are three decimal places so place 375 over 1000)
375/1000 = 3/8
So 1.375 = 1 + 0.375 = 1 + 3/8 = 1 3/8
1.375 = 1 3/8
If we move the decimal place two places to the right 1.375 becomes 137.5
1.375 = 137.5%
- 2.5
2.5 = 2 + 0.5
Now 0.5 = 5/10 (there is only one decimal place so place the 5 over 10)
5/10 = 1/2
2.5 = 2 + .5 = 2 + 1/2 = 2 1/2
2.5 = 2 1/2
2.5 = 2.50
By moving the decimal point two places to the right 2.50 becomes 250
2.5 = 250%
Below you'll see a table.
In each row only one figure is supplied, either a percentage, a decimal, or a fraction.
On each row, fill in the missing data, and then check your answers.
Make sure your fractions are mixed fractions and in lowest terms.
| PERCENT |
DECIMAL |
FRACTION |
CHECK THE ANSWERS ON THIS ROW |
RESET THE BOXES IN THIS ROW |
|
|
|
|
|
|
|
|
|
PERCENTS OF A NUMBER
Let's start with our trusted friend, the percent formula
BASE * PERCENT = AMOUNT
The BASE is again the number (or object) that we are working on. It constitutes 100%
Each PERCENT is 1/100 of the base.
The percent can be expressed as either a fraction or a decimal.
Nothing has changed in how we go about calculating the amount.
The only difference will be that now the amount at the end will be larger than the base.
To see this more clearly, let's look at a few examples.
Example 1
125% of 400
400 is our base
125 is our percent. 125% = 125/100 = 1.25
(or if you prefer 125% = 125/100 = 1 25/100 = 1 1/4)
1.25 * 400 = 500 (or if you prefer 1 1/4 * 400 = 5/4 * 400 = 5 * 100 = 500)
125% of 400 is 500
Notice that our answer 500 is greater than our base 400
Notice the following: 100% of 400 is 400, and 25% of 400 is 100 (Assignment 1: verify that 25% of 400 is 100)
Our solution 500 = 400 + 100 = (100% of 400) + (25% of 400)
Example 2
110% of 120
120 is our base and 110 is our percent
110% = 110/100 = 1 10/100 = 1 1/10 (or if you prefer 110% = 1.10)
120 * 1 1/10 = 120 * 11/10 = 12 * 11 = 132 (or if you prefer 120 * 1.1 = 132)
110% of 120 is 132
Notice that our answer 132 is greater than the base 120
Notice the following: 100% of 120 is 120, and 10% of 120 is 12 (Assignment 2: verify that 10% of 120 is 12)
Our solution 132 = 120 + 12 = (100% of 120) + (10% of 120)
In general whenever you take p percent of a number, where p > 100,
you are going to arrive at
(100% of the number) + (p-100)% of the number.
So if we took 130% of 45 we would get
(100% of 45) + (130-100)% of 45 =
(100% of 45) + (30% of 45) =
45 + 13.5 = 58.5
There you have it: two ways to calculate p percent of a number
when p > 100
Let's compare percents greater than 100 and percents less than 100.
Example 3
What is 115% of 35.00?
35 is the base and 115 is the percent
115% = 1.15 (or if you prefer 1 3/20)
35 * 1.15 = 40.25 (or 35 * 1 3/20 = 40 1/4)
Back on page 1 we found out that 15% of 35 is 5.25
So 15% of 35 yields 5.25 (5.25 being less than 35)
And 115% of 35 gives us 40.25 (40.25 being larger than 35)
Example 4
What is 105% of 2?
2 is the base and 105 is the percent.
105% = 105/100 = 1 5/100 = 1 1/20 (or if you prefer 105% = 1.05)
2 * 1 1/20 = 2 * 21/20 = 1 * 21/10 = 2 1/10 (or 2 * 1.05 = 2.1)
Back on page 1 we found out that took 5% of 2 is 0.10 (0.10 is less than 2)
105% of 2 is 2.1 (2.1 is greater than 2)
Now is you chance to practice taking percents
Make sure you give your answer as a decimal.
The computer cannot check answers with commas in them, so please omit any commas
Thus, if your answer came to 23.456 then enter it as 23456.
Exercise 1
What is 115% of 34?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 2
What is 185% of 210?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 3
Ehat is 199% of 300?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 4
What is 202% of 10?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 5
What is 106% of 13?
Go to the top of this page
Go to the percent home page
Go to the next section
CALCULATING PERCENTS LARGER THAN 100
On the second page of this chapter you were forced to calculate percents.
There I presented problems like "What percent of 46 is 34.5?"
You calculated a decimal by dividing the end figure by the base, and then by
multiplying that decimal by 100 you reached the correct percent.
In the problem that I just quoted you found
34.5 / 46 = 0.75 = 75%
However since I was always going from a higher number to a lower one,
you were assured that the percent would be less than 100.
If I changed that by going from a lower number to a higher one, then we would
reach a percent larger than 100.
I'll illustrate this point with the examples below
What percent of 45 is 90.9?
Since 45 is less than 90.9 (45 < 90.9), we are going from a smaller number to a larger one. Therefore
we're assured that our answer should be greater than 100.
90.9 / 45 = 2.02 = 202%
202% of 45 = 90.9
What percent of 45 is 0.9?
Since 45 is greater than 0.9 (45 > 0.9), we're going from a larger number to a smaller one.
Therefore the percent should be less than 100.
0.9 / 45 = 0.02 = 2%
2% of 45 is 0.9
What percent of 125 is 225?
Since 125 < 225, we're going from a smaller number to a larger one. Therefore the percent
will be greater than 100.
225/125 = 1.80 = 180%
180% of 125 is 225.
What percent of 125 is 100?
Since 125 > 100, we're going from a larger number to a smaller one. Therefore the percent
will be less than 100.
100/125 = 0.80 = 80%
80% of 125 is 100
At the end of this set of exercises we'll go back to increasing and decreasing.
As usual, state your answer as a decimal.
Exercise 1
What percent of 34 is 36.72?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 2
What percent of 34 is 23.8?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 3
What percent of 145 is 65.25?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 4
What percent of 145 is 233.45?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 5
What percent of 75 is 120?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 6
What percent of 75 is 45?
Go to the top of this page
Go to the percent home page
Go to the next section
INCREASES AND DISCOUNTS
So far this discussion is quite theoretical. Who uses percents like 110 anyway?
To give you a real instance when these type of percents are used, let's go back and
re-examine the increase and decrease examples used on page 1 of this unit.
Example 1
Suppose you earn $35,460 annually. And you're getting 7% raise.
How much will you be earning?
What we have to do is take 7% of 35,460 and then add this to the original
base figure of 35,460.
7% of 35,460 = 0.07 * 35,460 = 2482.20
35460 + 2482.20 = 37942.20
You will be earning $37,942.20
Now let's look at this same problem under a different light.
You basically went from 35460 to 37942.20
Now let's say we wanted to know what percent of 35460 is 37942.20?
Since we're going from a lower figure (our base) to a larger one (our amount),
the percent should be greater than 100.
37942.20 / 35460 = 1.07 = 107%
What's interesting about this is 7% was the original percent set forth as
the percent increase and
107 = 100 + 7
Example 2
A $2 item with a 5% sale tax.
The total cost equaled 2 + (5% of 2) = 2 + .10 = 2.10
Now let's say just for the sake of argument that we wanted to know what percent of 2 is 2.10?
We know that the percent will be larger than 100 since we're going from a smaller
number to a larger one.
2.1 / 2 = 1.05 = 105%
Now 5 was the original percent set forth in the example and our
answer equals 100 + 5.
Actually what you just saw in the two previous examples is no coincidence.
Whenever you are increasing a number by percent (which we will call p), you can achieve the
final figure by
taking (100 + p) percent of that number.
Let's look at a new problem, before we examine decreasing.
Example 3
Your rent is going up by 6%. It is currently $450 a month. What will the rent be.
Well we are looking for a number higher than the base figure so we know the
percent needed will be higher than 100.
Using what we just discovered, we can achieve our result by taking 106% (100 + 6) of
$450 (our base)
106% of 450 = 1.06 * 450 = 477
Your new rent will be $477 a month.
Let's see if discounting works the same way.
Example 4
Back on page one I asked if an item that usually sells for $25 and is
on sale at 40% off, what will its sale price be?
Here the base number is 25.
40% of $25.00 = 25 * 40/100 = 25 * 2/5 = 5 * 2/1 = 10
So the price is being discounted by $10.00
25 - 10 = 15.
Thus the sale price is $15.00
Now let's look at these figures again
We reduced 25 to 15.
So if I wanted to know what percent of 25 is 15,
I would expect that percent to be less than 100.
15/25 = 0.6 = 60%
What's interesting about this is 60 = 100 - 40
(40 was the original percent that we were discounting with)
Let's see if the same thing happens when we re-examine the second discounting problem
Example 5
The second discounting problem was as follows:
The popularity of a presidential candidate is going down.
According to a poll, after the last debate the number of people who said
they would vote for him dropped by 12%. Before the debate 110,250 (out of 200,00)
supported him. How many people
support him now?
Basically we have to discount 110,250 by 12%.
110,250 * 0.12 = 13230
110,250 - 13,230 = 97,020
97,020 people currently support the candidate.
Now what percent of 110250 is 97020?
Since we're going down the percent should be less than 100.
97020 / 110250 = 0.88 = 88%
And 88 = 100 - 12 (12 being the percent we were discounting with)
The general rule that we just realized is as follows:
Whenever you are decreasing a number by percent (which we will call p), you can achieve the final
figure by taking (100-p) % of the base figure.
Example 6
So if your rent of $450 was being decreased by 6% (like that would ever happen) how much
would your rent be.
According to the rule all we have to do is take (100 - 6)% or 450.
450 * 94% = 423
Your new rent will be $423
In each of the exercises listed below, you'll be asked to increase or descrease a number
using percentages.
To help you along, there will be two steps to each problem.
First you will be asked what percent you need to take of base figure in
order to achieve the desired answer.(Step 1)
And then you will be asked for the answer.(Step 2).
Make sure you give your answer as a decimal and omit any commas (they confuse the computer)
Exercise 1
A baseball player had a 250 batting average in the year 1999.
In the year 2000 his average went up by 30%
What was his average in the year 2000?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 2
You are a stamp collector. The number of stamps in your collection is 1450.
For your birthday several of your friends gave you stamps, and now your collection
has grown by 2%. How many stamps do you now own.
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 3
You just saw a newspaper advertisement proclaiming that
a $450 refrigerator is on sale at 20% off. How much will its sale price be?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 4
In the last contract, your union negotiated a 6.5% raise.
You currently make $800.00 per week.
What will your paycheck look like after the raise?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 5
According to the ratings, on the week of May 13, 2002 a certain television series
received a 5.5 share.
(each share represents a million households)
The very next week its populatity fell by 12%.
(Entertainment is such a fickle busines-isn't it)
What was its share the week of May 20th?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 6
According to opinion polls the popularity of a senator fell by 3%.
Last week 36,000 (out of 40,000) people approved of the job he was doing.
How many people approve of him now?
Go to the top of this page
Go to the percent home page
Go to the next page
