A store is offering a 20% discount on a certain item. The store’s sale of the item is subject to a 6% sales tax.

Question

Quantity A :

The item’s purchase price, if the discount is applied to the after-tax price

Quantity B :

The item’s purchase price, if the tax is applied to the discounted price

Which of the following is true ?

1) Quantity A is greater than Quantity B

2) Quantity A is less than Quantity B

3) Quantity A is equal Quantity B

I performed my maths and I found that Quantity A is greater than Quantity B, but the answer given is that both are equal.

So, what is the correct answer?

Answer 1

Indeed they are both equal.

Let $P$ be the original price prior to any discounting or tax. Let $F$ be the final price after tax.

Now let us consider the two scenarios A and B.

Scenario A

After applying 6% tax, the price becomes, $1.06P$

Thus after the discount,

$F = 0.8\times1.06P$

Scenario B

After applying an 20% discount, the price becomes $0.8P$

Thus after the tax,

$F = 1.06\times0.8P = 0.8\times1.06P$

Thus it is evident that the order of applying the tax/discount does not change the final price.

Answer 2

They are equal.

If the original price of the item is $P$, then $$A=0.80\times(1.06\times P)$$ while $$B=1.06\times(0.80\times P)$$

Because multiplication of real numbers is associative and commutative, these are both equal to $$0.848\times P$$ and so are the same.

Comment: You may get the wrong answer for these quantities if you forget that the second operation applies to the result of the first operation, not to the original quantity. For example, in computing $A$, you discount the taxed amount, not the original amount; so you have $(1.06\times P) - 0.20\times(1.06\times P)$ which is the $0.80\times (1.06\times P)$ shown above. Don't make the mistake of doing something like $(1.06\times P) - 0.20\times P$ instead, which is wrong.

Answer 3

Let $x$ be the original price of the item. Now let's calculate quantities $A$ and $B$.

For quantity $A$ the discount is applied after the tax. The tax makes the item cost $1.06x$ and so the discount makes the final item cost $1.06x(1-.2)=1.06x(.8)=.848x$.

For quantity $B$ the tax is applied to the discounted price, so we have the discounted price is $.8x$, and applying the tax gives as a final item cost of $.8x(1.06)=.848x$.

You can see that the quantities are indeed the same. The key point used here is that multiplication is commutative - it doesn't matter in which order we choose to multiply.