Motivation Problem

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Percent Off

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Increased By

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Solve for Original after % Increase or Decrease

# Solving for the Original Amount after an Increase or Decrease

Frequently, we know the new amount and the percent change and need to know the original amount. We reconsider the television problem from the beginning of the lesson. We found that a \$600 TV on sale for 30% off, cost \$420 on sale. We might think, that to find the original cost of the TV that has a sale price of \$420 after a sale of 30% off, we could just take 30% of the \$420 cost and add it back to the \$420 to find the original cost. NOTE THAT THIS DOES NOT WORK:

30% of \$420 is 0.30(\$420) = \$126 and \$420 + \$126 = \$546.

Or, 130% of \$420 is 1.3(\$420) = \$546.

Notice that this is not the \$600 original list price with which we started.

Why doesn't this work?

The method does not work because 30% of \$420 is not the same amount as the 30% of \$600. It may seem obvious when stated this way, but it is a very common error made on problems of this type. We cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

To work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

Example: Since (100% – 30%)(original price) = (sale price)

70% of the  (original price) is \$420.

0.7x  = 420

x = 600

The original cost was \$600.

We may also solve this problem using the two-step method. Working the problem in two steps requires us to use more algebra skills. We have to know that x = 1x (Identity Property of Multiplication) and that 1x – 0.3x = 0.7x (subtracting like terms). Then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = \$420

x = 600

The original cost was \$600.

Note that with both of the above methods we do obtain the correct original amount.

Example: After a 5% pay raise, Hermione is earning \$22,680 per year. What was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

x = 21,600

Hermione earned \$21,600 per year before the pay raise.

## Self Check Problem

A VEDM, Inc. states that it make as 30% profit over its expenses for producing its new product the hMat. VEDM, Inc. sells each hMat for \$377. How much does it cost VEDM, Inc. to make each hMat?

### Joke or Quote

Three traffic engineers were boasting about improvements in their respective city traffic flows, which improved 911 emergency vehicle response times.

First Engineer: "Since we installed our new satellite navigation system, we've cut our emergency response time by ten percent."

Second Engineer: "Not bad,but by using a computer model of traffic patterns, we we cut our average time by 20 percent."

Third Engineer: "That's nothing, since our city manager passed the bar exam, we've cut our emergency response time in half!"