M47-202

Question

Maya bought a framed print at an art gallery and paid a total of $96. If the $96 total consisted of the pre-tax price of the framed print plus sales tax on that pre-tax price, was the sales tax rate less than 20%?

(1) The pre-tax price of the framed print was less than $80.
 
(2) The sales tax Maya paid on the framed print was less than $19.20.

Bunuel · Answer

Official Solution:

Maya bought a framed print at an art gallery and paid a total of $96. If the $96 total consisted of the pre-tax price of the framed print plus sales tax on that pre-tax price, was the sales tax rate less than $20\%$?

Let $P$ be the pre-tax price of the framed print, and let $T$ be the sales tax.

Since Maya paid $96 total:

$P + T = 96$

The question asks whether $\frac{T}{P} < 0.20$.

(1) The pre-tax price of the framed print was less than $80.

This gives $P < 80$. Since $P + T = 96$, we have $T > 16$. So:

$\frac{T}{P} > \frac{16}{80} = 0.20$

Therefore, the sales tax rate was not less than $20\%$.

Sufficient.

(2) The sales tax Maya paid on the framed print was less than $19.20.

This is not enough.

For example, if $T = 1$, then $P = 95$, and the tax rate is $\frac{1}{95}$, which is less than $20\%$.

But if $T = 18$, then $P = 78$, and the tax rate is $\frac{18}{78}$, which is greater than $20\%$ because $20\%$ of 78 is 15.6, and $18 > 15.6$.

Not sufficient.

Answer: A

M47-202

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