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Ratios, Proportions, and Percents


Solving Proportions


To solve proportions, follow these simple steps:

1.  Cross Multiply:
 
 



Multiply the denominator of the second fraction by the numerator of the first.
Multiply the denominator of the first fraction by the numerator of the second.
 

EXAMPLE

2  =  x
3      4

8 = 3x


 









2.  Divide BOTH sides by the COEFFICIENT of the VARIABLE

(The variable is the unknown amount, usually represented by a letter or other symbol.  The coefficient is the number that the variable is multiplied by.  In this case, the variable is x and its coefficient is 3)
 
 



8 = 3x
  3     3

2.666... =  x


 




 
 

Percents

A percent is a ratio of a number to 100 [Per = For Each....Cent = 100]

Therefore another way to think about a percent, say 30%,  is 30/100.

In general,

P%  = P /100
 
 

You can also write a percent as a decimal.
 
 

Remember that when you divide a decimal by 100,

the decimal point moves two places to the left.

So, for example:

45%  = 45/100 = 0.45


 




 

Solving Percent Problems

There are many different types of percent problems,

ranging from tax and commission to percent change, discount, interest, and more.

In general, all of these problems follow a basic format:

P = Percent
N = Number
R = Result
of = multiply

Percent of a Number gives a Result

  P   x N = R
100

Which is equivalent to:

  P  =   R
100      N

The important thing to remember is that your percent is always of a base
(Hence Percent of a number)
this base, when you are given a choice is always the ORIGINAL amount.


 


Example:  What is 15% of 40?
What is your percent?  15
What is your base? 40
What is your result?  unknown at this point...use R

Go to your equation:

  P   x N = R        Substitute all values that you know at this point.
100

  15   x 40 = R
100

Solve (any way you wish...use as a fractions, reduce, then use as fractions, or use decimals)
Any way you do this, you should get R = 6.

15% of 40 is 6.