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Solving Problems involving Percents

Solving Problems involving Percents

Author: Colleen Atakpu

This lesson demonstrates how to use equations with percent discounts and sales tax.

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College Algebra

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Today we're going to talk about solving equations using percents. And one of the most common places that you use percentages in the real world is with sales tax. So we're going to look at three different problems involving sales tax and show you how you can solve them using percentages.

So we can use percentages to find the sales tax that you have to pay on certain items. So for example, let's say I'm buying a $30 shirt and I have to pay 8% sales tax. So there's a couple of things you could do with that information. You could find the amount in dollars of sales tax that you have to pay or you could find the price, so the final price, of what you have to pay, which would include the sales tax.

So let's start by finding the amount of sales tax that we'll have to pay. So we need to first convert our 8% to be a decimal so 8% means 8 out of 100, percent means out of 100. So if I convert 8 divided by 100 into a decimal, I'm going to get 0.08. So now that I have my sales tax percentage in terms of a decimal, I'm going to multiply that by the cost of my shirt. So I'm going to do $30 multiplied by my decimal for my sales tax. Now the product of the price of my shirt and the decimal form of my sales tax is going to be the amount of sales tax that I have to pay. So multiplying these together, I find that the amount of sales tax that I have to pay is going to be $2.40.

So let's look at how you could find the price of your shirt including the sales tax. So again, we need to start by converting our percentage, 8% or 8 out of 100, into a decimal. We already know that's 0.08. So now I'm going to multiply that by the cost of my shirt. However, because I want to include the original amount of my shirt in my overall cost, I'm going to multiply by 1 plus 0.08.

So to simplify this expression, I'm going to start by adding in my parentheses. This gives me 1.08 and I'll multiply that by the $30, which is going to give me a total cost of $32.40. So I found that the price that I have to pay for my shirt and the sales tax on the shirt is $32.40. So if we compare these two answers, we see that it makes sense, because if I know the original price of my shirt is $30 and I know that I have to pay $2.40, overall that's going to give me $32.40, which is what I found using this method.

So here's another example. I've got five boxes of the markers that cost a total amount of $18.55 and that's including tax. And the tax rate is 6%. So I can set up an equation to figure out the amount of money that each box of markers is or the unit price before tax. So I know that the cost that I have to pay, the total cost-- I'll represent that with a c variable-- that's going to be equal to 1 plus the amount that I have to pay in tax, so the percentage rate of my tax converted as a decimal multiplied by the unit price of each box of markers multiplied by the number of markers that I want to buy.

So before I can start substituting values into this equation, I know that I need to convert my sales tax percentage into a decimal. So again, 6% means 6 out of 100. Dividing those two numbers to get a decimal, I get 0.06. So now I can substitute 0.06 into my formula, bring down the 1 plus. I'm trying to find the unit price of each box of markers before tax. And I know that I bought 5 markers. And I also know that I paid a total of $18.55.

I'm going to start to solve for x by simplifying where I can. So 1 plus 0.06 gives me 1.06. Bring down the rest of my equation. Now I can simplify further by multiplying 1.06 times 5 and that is going to give me 5.3. I'll bring down my x and bring down my $18.55. Now all I need to do is divide both sides by 5.3. These will cancel. And I'll see $18.55 divided by 5.3 gives me a value for x of $3.50.

So here's my last example. At a store, the original price of a book was $12 excluding tax. The total price paid for four books was $40 and that was including a 7% tax. So what percent were the books discounted? So before I get started, I want to go ahead and convert my 7% tax into a decimal. So again, to convert 7% into a decimal, 7 out of 100 is 0.07. So I know that my tax is going to be 0.07.

I'm going to start by figuring out what my cost was before tax. So if I bought four books at $12, the original price, then that's going to give me a cost before tax of $48. Now to figure out what the cost would have been after tax, I'm going to use my 0.07 and I'm going to add 1 to that, because I want to know what the price is including the original amount that I paid for the four books.

So 1 plus 0.07, multiply that by the cost of my books-- $48-- and that's going to give me 1 plus 0.07 is 1.07 times $48. And that's going to give me $51.36. Now I only paid $40 so I know that to figure out the discount, I can find the difference between what I should have paid before the discount and what I actually paid. So $51.36 minus $40 that I actually paid means that I saved $11.36.

If I want to find the percent that is though I can use division. So $11.36 out of the original amount that I should have paid, $51.36. Dividing those two numbers will give me a decimal of approximately 0.22, which if I converted that into a percentage is approximately equal to 22%. So I saved a total of 22% on my four book purchase.

So let's go over our key points from today. Make sure you get them in your notes if you don't already so you can refer to them later. Percentages can be used for solving everyday problems such as those involving tax. And when calculating the percentages, you have to convert the percentage to decimal form by dividing by 100. So I hope that these key points and examples helped you understand a little bit more about solving problems using percentages. Keep using your notes and keep practicing and soon you'll be a pro. Thanks for watching.