CALCULATING BASES AND ADDING PERCENTS

Next we'll go back to the sale one last time. But first it's time for practice.

These are the type of problems we'll explore in this section.

First let's consider the discount problem.

You paid $10.50 for a shirt at a 30% off sale. How much was the shirt before it was discounted?

First of all we are going from a higher number (which we don't currently know yet) to a lower one, so this is a discount problem.
We know from our discussion on the previous page that if we took (100 - 30)% of the whatever the original price was, we would arrive at the sale price.
So this problem becomes "70% of what number is $10.50?"
Dividing 10.5 by .7 (70% = 0.70 = 0.7) we get 10.5 / .7 = 15.00
Thus $15.00 was the original price.
$15 dicounted by 30% = (100-30)% of 15 = 70% of 15 = 0.7 * 15 = 10.50

Let's look at one more discount problem before going on to our raise (increase) problem.

A baseball player has fallen on hard times. Over the course of the season his batting average fell 5% to a final average of 285. What was his average?

First of all we are going from a higher number (which we don't currently know yet) to a lower one, so this is a discount problem.
We know from our discussion on the previous page that if we took (100 - 5)% of whatever his average was, we would arrive at the current average of 285.
Therefore this problem becomes "285 is 95% of what number?"
Well 95% = 95/100 = 19/20
285 ¸ 19/20 = 285 * 20/19 = 5700/19 = 300
His batting average was 300.
300 discounted by 5% = (100-5)% of 300 = 95% of 300 = 0.95 * 300 = 285

The general rule of thumb for the discount problem "some number discounted by p% is amount."
To calculate the missing number divide the amount by (100-p)%

Now let's start increasing.

Let's start with your raise.

Congratulations, you just got a 5% raise. Your paycheck currently reads $861. How much were you making before the raise?

First of all we know this is an increase problem, since we intend to go from a smaller number (which we do not currently know) to a larger one.
We know from our discussion on the previous page that (100+5)% of whater your old salary was will be your new salary of 861.
So this problem becomes "105% of what number is 861?"
105% = 1.05
861/1.05 = 820
Your old paycheck was for $820
820 increased by 5% = (100 + 5)% of 820 = 105% of 820 = 1.05 * 820 = 861

I'll show you this once more and then give you the general rule of thumb (although by that time you might have guessed what it's going to be)

A product cost $48.15 once the 7% sales tax is factored in. What would it have cost without the tax?

First of all we know this is an increase problem, since we intend to go from a smaller number (which we do not currently know) to a larger one.
We know from our discussion on the previous page that (100+7)% of whater the price was will be the final cost of 48.15.
So this problem becomes "107% of what number is 48.15?"
107% = 1.07
48.15/1.07 = 45
The price of that item without tax is $45.00
450 increased by 7% = (100 + 7)% of 450 = 107% of 450 = 1.07 * 450 = 481.15

Have you figured out what the general rule is? I bet you have. But just in case you haven't here it comes
The general rule of thumb for the increase problem "some number is being increased by p% yeilds amount."
To calculate the missing number divide the amount by (100+p)%

Occasionally you might come across a situation where there is a limited number of options.
And you might want some type of barometer measuring which of the options was the most favorable.
Well percents would be that barometer.
However it is important to remember that if we added the percents for each option we must reach 100.

What are some real examples of this you might ask?

Well here are three examples.

• An election where you only have two candidates.
  If the winning candidate gets 78% of the vote
  then the other candidate must have received (100 - 78)% = 22% of the vote.
  
  We could write this as follows: The total vote = The vote for the winning candidate + The vote for the losing candidate
  or the equivalent statement
  (100% of the vote) = (78% of the vote) + (22% of the vote)
  
  Now let's say the total vote (our base) was 1,000,000 people, and you knew from some previous calculation that 78% of 1,000,000 is 780,000. Now I know you could calculate 22% of 1,000,000 just as easilly, but's let's say you're feeling a little lazy. By subtracting 780,000 from 1,000,000 you'll get your answer.
  (i.e. (22% of the vote) =(100% of the vote) -(78% of the vote))
  Thus you could conclude 22% of the vote = 220,000.
  
  
• A library noted that on a given day 56% of the books circulated were fiction. What percent of the books circulated on that day were non-fiction?
  
  Well either a book is fiction or non-fiction. There is no third alternative. So the two percents would have to add up to be 100.
  100 - 56 = 44
  Thus 44% of the books circulated that day were non-fiction.
  
  We could write this as follows:
  The total amount of books circulated = The fiction books circulated + The non-fiction books circulated
  or the equivalent statement
  (100% of the books circulated) = (56% of the books circulated) + (44% of the books circulated)
  
  Now let's say the total number of books circulated (our base) was 150,000, and you knew from some previous calculation that 56% of 150,000 is 84,000. Now I know you could calculate 44% of 150,000 just as easilly, but's let's say you're feeling a little lazy. By subtracting 84,000 from 150,000 you'll get your answer.
  (i.e. (44% of the books circulated) =(100% of the books circulated) - (56% of the books circulated))
  Thus you could conclude 44% of the books circulated is 66,000.
  
  
• This last example is a little harder, so pay careful attention
  
  A multiplex movie theater showing three different films.
  If 35% of the people seeing movies at that theater saw film 1 and 32% of the people seeing movies at that theater saw film 2, what percent of people seeing movies at that theater saw film 3?
  
  Well the percents of all three films have to add up to be 100
  The percents that we know are 35 and 32.
  35 + 32 = 67.
  100 - 67 = 33.
  Thus we can conclude 33% of the people seeing movies at that theater saw film 3.
  
  We could write this as follows:
  The total amount of people seeing movies at that theater = The amount of people seeing movies at that theater seeing film 1 + The amount of people seeing movies at that theater seeing film 2 + The amount of people seeing movies at that theater seeing film 3
  or the equivalent statement
  (100% of the people seeing movies at that theater) = (35% of the people seeing movies at that theater) + (32% of the people seeing movies at that theater) + (33% of the people seeing movies at that theater)
  
  Now let's say the total number of people seeing movies at that theater is 250,000. And let's say you know that 35% of 250,000 is 87,500, and 32% of 250,000 is 80,000.
  To calculate the amount of people seeing the third film you could compute 33% of 250,000
  or
  You could subtract (87,500 + 80,000) from 250,000, getting a result of 82,500.
  (i.e. (33% of the people seeing movies at that theater) = (100% of the people seeing movies at that theater) - (35% of the people seeing movies at that theater) - (32% of the people seeing movies at that theater))