Tax, discount and tip examples

We're told that Casey buys a bracelet. She pays for the bracelet and pays 72 cents in sales tax. The sales tax rate is 6%. What is the original price of the bracelet before tax?

So pause this video and see if you can figure this out.

Well, let's think about how your sales tax is calculated. Your sales tax would be equal to the sales tax rate—I'll just call that the rate—so that's going to be equal to the rate times your original price.

What do we know here? Well, they tell us what our sales tax is. Our sales tax is 72 cents, so I'll write that here. So 72 cents is going to be equal to my sales tax rate. They also tell us that that is 6 percent. Is going to be equal to 6 percent times my original price—and that's what they're asking for: the original price.

So, times—I'll just call it—let's just call it p for short: original price.

And so, you have it. We have a fairly straightforward equation now. I'm just going to rewrite everything as a decimal. We could write 0.72 is equal to 6 percent, which is the same thing as 6 per 100, which is the same thing as 0.06 times p.

Now we can just divide both sides by 0.06 to solve for the original price, so let's do that.

On the right-hand side, I have a p. On the left-hand side, if I have 0.72 divided by 0.06, well that's going to be equal to 12. Twelve times 0.06 is 0.72; and so this is going to be equal to 12.

So, the original price of the bracelet before tax is 12 dollars. You can verify that 6 percent of 12, which is 0.72 times 12, is indeed 0.72.

Let's do another example. A store has a 25 percent off sale on coats. With this discount, the price of one coat is $34.50. What is the original price of the coat?

So, once again, pause this video and see if you can figure it out.

These examples are actually quite useful because you will encounter this all the time when you are shopping, or you're trying to calculate tax, or you're buying something. So think about this: can you figure out the original price of the coat?

Alright, well let's just write it out the same way we did last time. So let's say our original price, and then there's a 25% discount. So minus 25% of the original price is going to be equal to $34.50.

Can we solve for the original price? One way to think about it: this is 100% of the original price, or 1 times the original price minus 25% of the original price is going to be equal to this.

Well, 1 - 0.25 of something is going to be 0.75 of our original price. This is why people use letters for variables so they don't have to keep writing this over and over again. Is going to be equal to $34.50.

Now, to solve for our original price, we just divide both sides by 0.75. And 0.75, that's the same thing as 75 per 100, or 0.75. So I could divide both sides by 0.75.

On the left-hand side, these two are equivalent, so I'll be left with my original price. Now I just have to figure out what this is going to be. So if I take $34.50 divided by 0.75, this is going to be the same thing as $34.50 divided by 75.

I just did that to get rid of the decimals. I moved essentially the decimal both decimals to the right two places, multiplying the numerator and the denominator by 100.

So let's see how many times 75 goes into $34.50.

See, 75 doesn't go into 3; it doesn't go into 34. But it does go into 345. What is that? Four times. Four times 5 is 20. Four times 7 is 28 plus 2, which is 30. Then you subtract; you have 45.

Bring down the zero: 450. 75 goes into 450, 6 times. Six times 5 is 30. Six times 7 is 42 plus 3 is indeed 45, and we luckily have no remainder.

So this, right over here, is going to be equal to 46 dollars. Our original price is 46. You take off one fourth, or 25 percent, you're left with $34.50.

That makes intuitive sense as well. It's always good to just take whatever answer you have and put it back in and see if it makes sense. If you got $460 dollars instead, taking 25 percent off, you'd get to $34.50—that sounds off. If you got $4, you said you take 25 percent off and you got $34.50—that would have also felt off.

But 46 feels right, and you can even calculate that: what is 25 percent of 46 dollars? Subtract that from 46, and you should get $34.50.