Sara goes to a bookstore and wants to buy a book that is originally priced at $\$ 100$.

Which of the following options gives her the best discounted price?

(A) A discount of $20 \%$

(B) A discount of $10 \%$, then a discount of $10 \%$ off the new price

(C) A discount of $15 \%$, then a discount of $5 \%$ off the new price

(D) A discount of $5 \%$, then a discount of $15 \%$ off the new price

(E) All four options above give the same price
A discount of $20 \%$ on a book priced at $\$ 100$ is a $0.20 \times \$ 100=\$ 20$ discount.

Thus for option (A), Sara's discounted price is $\$ 100-\$ 20=\$ 80$.

A discount of $10 \%$ on a book priced at $\$ 100$ is a $0.10 \times \$ 100=\$ 10$ discount, giving a discounted price of $\$ 100-\$ 10=\$ 90$.

A second discount of $10 \%$ on the new price of $\$ 90$ is a $0.10 \times \$ 90=\$ 9$ discount.

Thus for option (B), Sara's discounted price is $\$ 90-\$ 9=\$ 81$.

A discount of $15 \%$ on a book priced at $\$ 100$ is a $0.15 \times \$ 100=\$ 15$ discount, giving a discounted price of $\$ 100-\$ 15=\$ 85$.

A further discount of $5 \%$ on the new price of $\$ 85$ is a $0.05 \times \$ 85=\$ 4.25$ discount.

Thus for option (C), Sara's discounted price is $\$ 85-\$ 4.25=\$ 80.75$.

A discount of $5 \%$ on a book priced at $\$ 100$ is a $0.05 \times \$ 100=\$ 5$ discount, giving a discounted price of $\$ 100-\$ 5=\$ 95$.

A further discount of $15 \%$ on the new price of $\$ 95$ is a $0.15 \times \$ 95=\$ 14.25$ discount.

Thus for option (D), Sara's discounted price is $\$ 95-\$ 14.25=\$ 80.75$.

Therefore, the four options do not give the same price and option (A) gives Sara the best discounted price.