Priam bought a jacket that was on sale for 40% off. The...

Priam bought a jacket that was on sale for 40$\%$ off. The original price of the jacket was $\$ 150$ . While the sales clerk figured the price by calculating the amount of discount and then subtracting that amount from $\$ 150$ , Priam found the price faster by calculating 60$\%$ of $\$ 150$ . (a) Explain why Priam was correct. (b) Will Priam's method work for any original price?

Priam bought a jacket that was on sale for 40$\%$ off. The original price of the jacket was $\$ 150$ . While the sales clerk figured the price by calculating the amount of discount and then subtracting that amount from $\$ 150$ , Priam found the price faster by calculating 60$\%$ of $\$ 150$ .
(a) Explain why Priam was correct.
(b) Will Priam's method work for any original price?

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Key Concepts

Percentage Calculation

This concept involves expressing parts of a whole in terms of 100. When calculating discounts, percentages are used to represent the proportion of the price that is discounted or paid. In problems like this, converting a percentage into its decimal equivalent allows for straightforward multiplication to find a portion of the original price.

Complement Percentage

When an item is offered at an X% discount, the amount paid is the complement of the discount, which is (100-X)%. This method is effective because subtracting the discounted percentage from 100% gives the percentage of the original price that remains. In this case, a 40% discount results in a payment of 60% of the original price.

Generalization of Discount Methods

The method of calculating the sale price by multiplying the original price by the complement percentage works for any original price, regardless of its value. The principle behind this is that discounts are applied uniformly as a percentage of the price, ensuring that the complementary calculation is always valid for determining the final sale price.

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