•  A discount is the amount of money reduced from the regular price.
• Regular Price – Discount = Sale Price
• Rate of discount is expressed as a percent.
• You can solve discount problems using ratios.
• Be sure the percents go on one side of the ratio.
• Put the dollar values on the other side of the ratio.
• All regular prices = 100%.

Set up the ratio,:                      $200 = 100%
(using discount)                        $25        x

Then cross-multiply:  200x = 2500

Divide each side      200x = 2500
              by 200:            200      200

Solution:  x = 12.5%  The rate of discount is 12.5%.
  (sometimes this is known as Percent Decrease.)

•  Some salespeople earn a percentage of the value of the item they sell (most often cars, real estate, insurance and other large-ticket items).
• The rate of commission is the percentage of the value of the article sold.
• The cost of the sale = 100%

Example #1:
Harry’s commission is $450 when his total sales are $6000.  What is his rate of commission?

$6000 = 100%
 $450        x

So 6000x = 45 000
      6000         6000

x = 7.4979166, which rounds off to a rate of commission of 7.5%.

• The amount of money a person borrows from a bank or credit union is called the principal.
• A person who takes out a loan has to pay the financial institution interest.  This is a percentage of the principal.
• The amount (or amount due) is the total of principal and interest.
• When a person puts money into a savings account or otherwise invests it, the company that is using the money pays out interest.
• Interest is usually paid out over a certain period of time.  You can express years as whole numbers.  Remember, for months, that 1 month = 1/12 of a year.
• Remember that when calculating money, you always round up. (eg:  $10.5333 would round up to $10.54)

Example #1:  A man borrowed $2000 at a rate of 11% for a period of two years.  How much interest did he owe, and what was the amount due?

Remember:  i = prt
           i =  $2000 x 11 x 2
                       1      100    1
                     i = $44 000
                              100
  i = $440
a) The man owed $440 in interest.
b) $2000 + 440 = $2440  (principal + interest = amount due)
The amount due was $2440.
 
 

•  A ratio is a way of comparing two or more quantities with the same unit.
• It can be written in several different ways.  3:5;  three out of five;  three for every five;  three is to five;  3/5; etc.
• Like fractions, ratios are usually most convenient when written in lowest terms.
• A rate is a certain quantity or amount of one thing considered in relation to a unit of another thing.  (eg)  km/hour.  Watch for the “per” sign (/).
• Percent is a special type of ratio or rate:  how much per hundred
• In a proportional situation, two quantities are related by multiplication or division, so each quantity will increase or decrease by a set factor (twice, half, ten times…)
• All of these are practical, everyday applications of algebraic calculations.
• Many types of mathematical problems can be solved using ratio, rate, proportion or percent.
• USE A CALCULATOR FOR THIS WORK.

Important words:

Write the TOTAL opposite 100%.

1. If Ron correctly answers 47 out of 75 questions on an exam, what is his percent?  75 correct = 100%
47 correct =     x     (x = Ron’s percent, which we don’t know yet.)

Then cross-multiply.
75x = 4700
Then divide both sides by 75 to isolate the x.
75x = 4700
75          75
x = 62.67% (rounded)

2.  Jean earned $40 babysitting one weekend and spent 15% of her earnings.  How much did she spend?  $40 = 100%  ($40 is 100% of what she earned.)
                            x      15%   (Now you can work the ratio out.)

3.  The attendance at school on Friday was 551 students.  This was 95% of the total enrolment.  What is the total enrolment of the school?
We don’t know the total enrolment, so     x  = 100%
                                                                                 551     95%
 
 

Remember that “is” means “=” and that “of” means “x” (multiply).

Rewrite the percent as a fraction (/100).  Rewrite the other number as a fraction by putting a 1 under it.

1. 50% of 30 is x  so   50 x 30 = x
                                        100    1
Then multiply.  (50)(30) = x
                            (100)(1)

You now have   1500 = x
                               100

Your solution is x = 15.

The same steps work for questions where the terms are in different orders.

21.  z is 8% of 64     so z =   8   x 64        so z   = (8)(64)     so z = 512 = 5.12
                                                 100      1                    (100)(1)                  100
 
 
 

Note:  for #7, be sure to convert ¼% correctly.  This is actually ¼/100, or 0.25/100, which MUST be written as 25/10 000 (or, better, in its reduced form of 1/ 400).

It’s a good idea NOT to put a decimal point in a numerator.  Write an equivalent fraction instead.  For example, in question 19, 7.5/100 would be better written as its equivalent, 75/1000.

In question 8, the 10.5 is the solution, so leave it alone.  You won’t be rewriting it as a fraction. 10.5 =  x    x  210  so  10.5 = 210x  so 10.5 = 2.1x,
                                         100         1                       100

Where you are using an amount greater than 100%, just use the usual denominator of 100.

For # 9,  use 240/100 x 3500/1 = x