The word percent is a fraction whose denominator is 100. For example 26% is equivalent to the fraction 26/100.
To change a decimal number to a percent, we simply multiply by 100%:
0.32 × 100% = 32%
If a percentage is given, move the decimal two places to the left to express its equivalent decimal form. The number 0.321 is equivalent to 32.1%.
You can convert a fraction to a decimal by dividing the fraction then moving the decimal point two spaces over and adding a percent sign:
34/100 = .34 = 34%
Correspondingly, when you multiply something by a percentage, move the decimal point two spaces to the left.
500 × 20% = 500 × 20 = 10,000, then move the decimal two to left to 100%
(of course you could have quickly realized that 20% = 1/5 and solved in your head to get 100.)
10% multiplication trick:
To get 10% of anything, simply slide the decimal point over by 1.
3456 × 10% = 345.6
Example
Convert 4% into a decimal and a fraction in lowest terms.
Solution
To convert 4% into a decimal, we move the decimal point two places to the left:
4% = 0.04
To express 4% as a fraction, we divide by 100:
4/100 = 1/25
Fraction / Percent Issues
To convert from a percent to a decimal (and vice versa)
You must take a decimal point and insert it into the percent number two spaces from the right:
80% = 0.80, while 225% equals 2.25.
To convert back, move the decimal two places over and add a percent sign. 0.8 becomes 80%
Percent to a fraction
Take the percentage number and place it as a numerator with 100 as the denominator. If you have 70% of something, then you can convert it into the fraction as 70/100.
Fraction to a percent or decimal
Divide the numerator by the denominator. (See the chart below of common conversions). 1/5 equals 5 divided into 1, or the decimal 0.2.
To make this a percentage, multiply by 100% to get 20%
Common Decimal/Fraction Conversions
1 = 100%
1/3 = 33.33
1/4 = 25%
1/5 = 20%
1/6 = 16.66%
1/7 = 14.28%
1/2 =50%
2/3 = 66.66
3/4 = 75%
2/5 = 40%
5/6 = 83.66 %
2/7 = 28.56%
3/5 = 60%
3/7 = 42.85%
4/5 = 80%
4/7 = 57.14%
5/7 = 71.42%
6/7 = 85.71%
1/8 = 12.5%
1/9 = 11.1%
1/10 = 10%
1/11 = 9.09%
1/12 = 8.3%
3/8 = 37.5%
2/9 = 22.2%
3/10 = 30%
2/11 = 18.18%
5/12 = 41.7%
5/8 = 62.5%
4/9 = 44.4%
7/10 = 70%
3/11 = 27.27%
7/12 = 58.3%
7/8 = 87.5%
5/9 = 55.6%
9/10 = 90%
4/11 = 36.36%
11/12= 91.7%
7/9 = 77.7%
5/11 = 45.45%
8/9 = 88.8%
6/11 = 54.54%
7/11 = 36.36%
8/11 = 72.72%
9/11 = 81.81%
10/11 = 90.90%
Cross Multiplication with Fractions
Use cross-multiplication to solve for percentages. Multiply the upper left by the lower right and set it equal the the upper right multiplied by the lower left.
Numbers
Percentage
Part Original/Whole
=
Part
100%
What is 30% of 210?
_N_
210
=
30%
100%
N (100%) = (30%)(210)
n = 63
Increase and Decrease of Percentages
Percent increase: If the price of a $10 item increases by 10%, the new price is the original $10 plus 10% of the $10 original (or 110% or the original). "Increase" may go by other names such as the "markup" from "wholesale" (cost from factory) to "retail" (cost to public).
Percent decrease: If the price of an item decreases by 10%, the new price is the original $10 minus 10% of the $10 original (or 90% of the original). "Decrease" may go by other names such as "sale".
How good is a 100% increase?
An increase of 100% is the same as doubling something. So 10 increased by 100% is 20 because you are adding 100% of 10 to 10 to get 20. Re-read this if you didn't get it the first time.
Example
If the price of a stock falls from $50 to $40, what is the percentage of decrease?
Solution
First, subtract the numbers resulting in the decrease: 50 - 40 = 10. Then divide by the original amount:
50 - 40
50
=
10
50
= 0.2
Convert to a percentage by moving the decimal point two places to the right and adding a percent sign:
% decrease = 20%
if you cross multiply:
Numbers
Percentage
40
50
=
n
100%
40(100%) = 50n
20% = n
Discounts & Markup
Discount is the percent reduced on the price of an item. Markup is the amount of increase when the cost of an item is increased a certain percent. The following examples will illustrate this concept.
For markups and discounts, calculate:
If the value is negative, that is the amount of the discount. If the number is positive, that is the amount of the markup
Example
A pair of aerobic shoes is marked $120 and is discounted to $90. What is the percent discount?
Solution
The percent discount is based on the initial cost. It is
% discount = ((120 - 90) / 120) × 100
30/120 × 100 = 25%
Example
A pair of running shoes is purchased at wholesale for $90 and is sold to a store for $120. What is the percent markup?
Solution
The percent markup is based on the original cost. It is
% markup ((120 - 90) / 90) × 100
= 30 / 90 × 100 = 33 1/3%
Example
An employee is to mark up a piece of jewelry 120%. If its wholesale cost is $100, what should its selling price be?
Solution
The amount of the markup is 1.2 × 100 = $120
The selling price is then $100 + $120 = $220
Numbers
Percentage
_n_ 100
=
120
100%
n(100%) = 50n
20% = n
Example
A college bookstore purchases trade books on a 40% margin, i.e., it purchases a trade book for 40% less than its retail price. What is the percentage markup based on its wholesale price?
Solution
Since the retail price is not given, the percentage markup that we seek must be the same for all trade books. Therefore, let the retail price of a trade book be $100 (rather than the symbol x). Then the bookstore's purchase price is
100 - 100 × 0.4 = 100 - 40 = $60
If a book sells for $100 and costs $60, its percentage markup is
%markup = (100 - 60) / 60 × 100 = 40 / 60 × 100 = 66%
Example (easy)
Kathy buys a bike for $240 after a 40% markdown. What was the original price?
Solution
Let P be the original price. Then
P - P × 0.4 = 240
P - 0.4P = 240
0.6P = 240
divide both sides by .6
therefore, P = $400
Example (easy)
Find the number of residents of a city if 20% of them, or 6,200 people, ride bicycles.
Solution
Let R be the number of residents. The equation that represents the verbal statement is
0.2R = 6,200. R = 6200/.2 = 62000/2 = 31,000 people.
Example
Kent pays 20% taxes on income between $10,000 and $20,000 and 30% on income over $20,000. The first $10,000 is tax free. If he pays $14,000 in taxes, what was his income?
Solution
Let Kent's income be L. Then the total tax is:
0.2(20,000 - 10,000) + 0.3(L - 20,000) = 14,000
2,000 + 0.3L - 6,000 = 14,000
0.3L = 14,000 + 4,000 = 18,000
L = 18,000/.3 = $60,000
Example (medium)
How many gallons of pure water must be added to 100 gallons of a 4% saline solution to provide a 1% saline solution?
Solution
Let x be the gallons of pure water to be added. There are 100 × 0.04 = 4 gallons of salt and 96 gallons of pure water in a 4% saline solution. The total number of gallons will be x + 100. The amount of salt will remain constant. Hence,
0.01(x + 100) = 4
0.01x + 1 = 4
0.01x = 3
x = 3/.01 = 300 gallons
Taking a Percent of a Percent
What happens if you have to take a percentage of something and then modify it again? You multiply the existing total by the new total.
Example
If a stock valued at $100 increases by 10% during the first year, then increases again by another 20% in the second year, what is the stock's final value?
Solution
100 × 110% = 110
then for the second year:
110 × 120% = $132
Sometimes on these double percent questions, you should just Plug In 100 as a test number to get a handle on the quantities.
Example
Joe's portfolio lost 80% of its value, then it gained 10% of its value back. What was his final loss?
Solution
Here you can just substitute 100 for the original value to help solve the question. It goes from 100 to 20. And then add 10% to 20 to get 22.
