Percent word problems: tax and discount

- 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. And 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%, is going to be equal to 6% times my original price and that's what they're asking for, the original price. So times, I'm just going to call it P for short, original price. And so you have it. We have a fairly straightforward equation now. And now I'm just going to rewrite everything as decimal, so we could write 0.72 is equal to 6% is the same thing as six per 100, which is the same thing as six hundredths, 0.06, six hundredths times p. And now we can just divide both sides by six hundredths to solve for the original price, so let's do that. Alright. So on the right-hand side I have a P. On the left-hand side if I have 72 hundredths divided by six hundredths, well that's going to be equal to 12, 12 times six is 72. Six hundredths times 12 would be 72 hundredths. And so this is going to be equal to 12. So the original price of the bracelet, before tax is $12. And you can verify that. 6% of $12, 0.06 times 12 is indeed 72 hundredths. Let's do another example. A store has a 25% 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, they're actually quite useful because you will encounter this all of 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. 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 because that the price with the discount. So can we solve for the original price? So one way to think about it, this is 100% of the original price or one times the original price minus 25% of the original price is going to equal this. Well, 100% minus 25% of something is going to be 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 75%. And 75%, that's the same thing as 75 per 100 or 75 hundredths. So I could divide both sides by 75 hundredths. On the left-hand side these two are equivalent, so I'll be left with my original price. And now I just have to figure out what is this going to be. So if I take 75 hundredths and let me actually multiply both of these times 100. So this is going to be the same thing as 3,450 divided by 75. I just did that to get rid of the decimals. I moved, essentially, both of the decimals to the right two places. Multiplying the numerator and the denominator by 100. So let's see how many times 75 goes into 3,450. Let's see, 75 doesn't go into three. Doesn't go into 34. It does go into 345. What is that, four times? Four times five is 20. Four times seven is 28 plus two, 300. And then you subtract. Get 45. Bring down the zero. 450. 75 goes into 450 six times. Six times five is 30. Six times seven is 42 plus three is indeed 45 and we luckily have no remainder. So this right over here is going to be equal to $46. Our original price is $46. You take off one-fourth or 25%, you're left with $34.50. And that makes intuitive sense as well. Always good to just take whatever answer you have and put it back in and see if it makes sense. In terms of if you somehow got $460 and you said taking 25% off you get to $34.50, you'd say "Well that sounds off." Or if you got $4 and you said you take 25% off and you got $34.50 that would have also felt off, but $46 feels right. And you can even calculate that. What's 25% of $46, subtract that from $46 and you should get $34.50.