Video Explanation

Text Explanation

The problem gives us the total price, which is the original price plus the sales tax. Column (A) asks us for the original price, so let's call that x. The sales tax is 15% and the total price is 45, so we can write that as:

1.15x = 45

Remember that when we increase a number by a percent, we multiply by 1 and the percent. 1 for the original amount and 0.15 for the percent. If you prefer, you can think of this equation as 1x + 0.15x = 45, and then you combine the x's to get 1.15 :)

Solve for x and we get approximately 39.13. This is greater than 39, so Column (A) is larger and the answer is (A). 

FAQ: Why can't I take 15% of 45 and then subtract that from the original $45?

We know from the question that the price of the item is $45 "including" sales tax, and we know that sales tax is 15%.

Let's say the price of the item is P. Then we know that P + (sales tax) = $45. We also know that sales tax is 15%. So we can ask, 15% of what? Sales tax is always applied to the initial price, so it's 15% of P

We can re-write our equation as P + (15% of P) = $45
P + 0.15P = $45
1.15P = $45
P = $45/1.15 = ~$39.1

Here's another example. Let's say you are buying an item priced at $30. And let's say sales tax is 10%. Then sales tax would be $3 (10% of the price, $30). And you would pay a total of $33.

Now in the same example, lets say you were given that sales tax is 10% and that the final price including sales tax is $33, and you are asked to compute the initial price before tax. If we did 10% of $33 and subtracted it from $33, we would get $33 - $3.30 = $29.70. But as we can see above the answer is actually $30, because sales tax is applied to the price of the item before tax.

This is the infamous Increase-Decrease Percent trap. See this blog article.

I still don't understand why we multiply by 1.15 and not 0.15

Now that we have the answer, we can demonstrate why :) The original price is approximately $39.13. 15% of that is .15 * 39.13 = approximately 5.87. That's not 45! But, if we add the values, 39.13 + 5.87 = 45. That's why we multiply by 1.15. We include the 1 so we have the original price as well, not just the sales tax. 

Is there a way to do this problem using mental math, or is a calculator necessary? 

This is a good example of a time when a calculator is not only appropriate, but also the fastest and most efficient way to solve a question. While we generally advise against the overuse of or over reliance on calculators, it's important to recognize that the calculator is an essential tool in certain circumstances. See this blog post for more details: Calculator Strategies for the Revised GRE :-) 

One of the absolute BEST places to use a calculator is when you're dealing with decimals, especially those that aren't easy (like 0.22 or 

0.18). That's the case here, with 1.15. We might try converting  to  and then start reducing that: . 

But that doesn't really give us a fraction that we can easily work with. Very few people could do this division in their head, and that's when we realize that it's a perfect time to pull up the calculator. 

However, we can use a different approach than the one given in the explanation video and use mental math. We do this by focusing on the value in Column B.  We can say "what would the taxed price of the item be if it were $39". Finding 15% of $39 can be done in our heads.

10% of 39 is 3.90
Half of that is 1.95

3.90 + 1.95 = 5.85

39 + 5.85 = 44.85

That means that the price of the item must be greater than $39 pre-tax because the total after tax of the item is $45, since $45 > $44.85.