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The Dreaded Percentage Off Problems

Most people understand how to work a problem like the following:

The normal price of a camera is $150. However, during Saturday's sale, the camera will be discounted by 30%. What will be the price of the camera on Saturday?


The first step is the price times the percentage that will be taken off of the price. Thus, the problem is $150*.30. The answer to that is $45.
The next step is to subtract the answer from the last step from the original price. Thus, the next problem is $150-$45. The answer to that is $105.
This means the answer to the problem is $105.

Using this same data, assuming that somebody bought the camera, the following information might be given.
At a sale, Jeremy bought a camera for $105. This is the price after a 30% discount was taken off of the original price. What was the original price of the camera?
If there are multiple answers, the problem can be worked backwards from each step. However, there is an easy formula to find the answer. The original price is what is unknown, so it is called x. The price of $105 is the original price minus 30% of the original price, thus $105=(x-.30x).
All that is needed is to solve the problem.
$105=(x-.30x)
$105=(.70x)
$105/.70=.70x/.70
$150=x
The original price was $150.
The last type of problem is as follows:
Jeremy bought a camera on sale for $105. It usually sells for $150. What is the percentage discounted from the original price?
First, find the difference of the prices. $150-$105 is $45.
This tells us that the price that was discounted from $150 is $45. We then need to find what percentage 45 is of 150.
We know that to find a percentage of a number, we multiply by that percentage, which is the same as multiplying that number over 100 by the original number.
Thus, we are looking for x% of 150 to equal 45.
Our equation is x/100*150=45
This can be written while keeping the 150/1 as well - x/100*150/1=45.
We multiply what we have first giving us 150x/100=45.
Now that we have a single fraction on one side we may want to write our whole number as a fraction because we want it keep cross multiplying simple.
Our equation is now 150x/100=45/1.
Now we cross multiply to get 150x=4500.
To solve this, we divide both sides by 150 and we get x=30.
Thus, 30% is our answer.