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NY Regents Exam - Integrated Algebra: Help and Review24 chapters | 260 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When you go shopping, you don't want to be surprised by the amount of tax on an item. You also want to fully understand discounts to know how much you can save. In this lesson, we'll practice calculating both tax and discounts.

Let's go shopping and talk tax and discounts. We're going to take a trip to Crazy Larry's Discount Barn, where you can buy pretty much anything, as long as you don't mind shopping in a barn. We're talking money - money saved and costs added to your purchase. Algebra doesn't get much more useful than this.

Crazy Larry loves to offer wacky discounts. A **discount** is a deduction from the original cost. You might find a shovel on sale for 61% off or a box of cereal on sale for 18.5% off. Crazy Larry's nothing if not random in his specials and choices of merchandise. But these unusual discounts can be confusing. How do you know how much an item costs if there's going to be a percentage discount?

Plus, most items come with a tax. A **tax**, in this context, is an added fee on many purchases that usually goes to the city or state. Maybe you planned to buy a pack of gum, and you know it costs one dollar. You only brought one dollar with you, but then the cashier says it's $1.08. Where'd those eight cents come from, and why is it standing in the way of your minty fresh breath?

Let's start with discounts. Crazy Larry is selling a kid's bike for $97. It's on sale for 22% off. How can we figure out the sale price?

There are a few different ways. Let's start by setting up an equation that compares two ratios. We're trying to find the sale price, so let's call that *x*. The original price is $97, so the ratio of the sale price to the original price is *x*/97. That's equal to the discounted percentage over 100%. The sale is 22% off. That means the sale price is 78% of the original, or 100 - 22. So, *x*/97 = 78/100. If we cross multiply, we get 100*x* = 97 * 78. 97 * 78 = 7566. Divide that by 100 to get *x* = 75.66. So, the sale price is $75.66.

That's one perfectly good way to find the discounted price. But here's another. We know the discounted price is 78% of the original. If we convert 78% to a decimal, .78, we can set up this equation: *x* = .78 * 97, where *x* is, again, the discounted price. Guess what .78 * 97 is. Yep, $75.66. Crazy Larry can't fool us. We have two methods to figure out the sale price.

Let's try another. Shelley Shopper is a coupon wizard. She goes to Crazy Larry's and gets great deals by combining sale prices and coupons. There's a painting she wants to buy.

The original price is $224. It's on sale for 34% off. Shelley has a coupon good for an additional 16% off the sale price. What will the painting cost her?

Can we just add the two discounts? 34 + 16 is 50, so is it just 50% off? That'd be $112.

No, that won't work. The coupon is for 16% off the sale price. So, we have to complete two steps. First, let's find 34% off the original price. 34% off means the sale price is 100 - 34, or 66%, of the original price. If we use our second method, we know *x* = .66 * 224. That's $147.84.

Now let's find out what 16% off that is. 100 - 16 = 84. So, *y* = .84 * 147.84. So, Shelley Shopper pays $124.19.

Notice the diminishing effect of the discounts. If we just took 50% off from the start, the painting would cost $112. But since we did 34% off, then 16% off the adjusted price, the 16% ended up giving us a smaller discount. Still, Shelley's happy. She still got a great deal on a painting!

Crazy Larry offers great deals on everything he can stuff into his discount barn, but his customers still need to pay sales tax. Fortunately, if you know what the tax percentage is, you can calculate how much it will be.

There's a couch that Crazy Larry sells for $988. Normally, there'd be a discount, but we're here on a Tuesday, and Tuesdays are no discounts on couches or potatoes days. The tax is 7.5%. So, what will the couch cost with tax?

What we need to figure out is what 7.5% of 988 is. Again, there are a few paths we could follow. Let's try two, and see if they lead to the same destination. First, let's convert 7.5% to a decimal. It becomes .075. Now, we can say that if *x* is the cost of the tax, then *x* = .075 * 988. That's 74.1. In dollars, that's $74.10. If we add that to 988, we get $1,062.10.

There's a shortcut, though. Instead of multiplying 988 by .075, let's multiply it by 1.075. Why? Because then we'll find out the cost of the couch and the tax together, all at once. 988 * 1.075 = 1062.10. So, we got the same answer!

Let's take a trip to the snack bar at Crazy Larry's. Their food selection is just as esoteric as the merchandise options. Here's Bob. Bob orders a $6.50 bowl of gumbo, a $3.25 side of Brussels sprouts and a $4.00 slice of cherry pie. Bob has a coupon for 11% off. The tax on food is 5%. What is Bob's total cost?

Oh, Bob. I'm not sure you should mix those foods. But, hey, why not? Let's start by adding up the costs. $6.50 + $3.25 + $4.00 = $13.75.

Now we have to figure out both the discount and the tax. Which goes first? Well, Bob is only going to be taxed on the amount he spends, so we should figure out the discount first - then the tax on the discounted total. His coupon is for 11% off. So, he's paying 89% of $13.75. Let's do *x* = .89 * 13.75. That gets us $12.24.

Now let's figure out the tax with our shortcut method. Be careful when solving problems involving tax, though. If the question just asks for the cost of the tax, you wouldn't use this method. But this time we want the total cost. So, let's do 12.24 * 1.05. That's 12.85. So, Bob's total cost, after the discount, but including tax, is $12.85. Enjoy the meal, Bob!

To summarize, we looked at discounts and taxes. A discount is the amount a price is reduced. A tax, in this context, is an extra cost that's added on to purchases to benefit the state or local government.

To calculate a discount, you can set up two ratios that compare the sale price to the original price and the discounted percentage to 100%. You can also convert the discounted percentage to a decimal and multiply that by the original price.

To calculate a tax, you can convert the percentage to a decimal, then multiply it by the price. If you want to know the total cost, including the tax, you can multiply the original price by one plus the decimal.

Happy shopping!

After this lesson, you'll be able to:

- Define discounts and taxes as they relate to purchases
- Explain how to calculate discounts and taxes

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NY Regents Exam - Integrated Algebra: Help and Review24 chapters | 260 lessons

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