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.

0.7*x *= 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* = 1*x* (Identity Property of Multiplication) and that 1*x* – 0.3*x* = 0.7*x* (subtracting like terms). Then we solve the problem as follows:

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

*x* – 0.3*x* = 420

0.7*x* = $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

1.05*x* = 22,680

*x* = 21,600

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

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?

**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!"**