The expression $1.32(0.65 p)$ represents the amount Cai paid for dinner at a restaurant after he applied a discount coupon and added tip. Which changes to the original price could have resulted in the expression? $\newline$ A discount of $0.65 \%$ and then $32 \%$ tip $\newline$ $32 \%$ tip and then a discount of $35 \%$ $\newline$ A discount of $65 \%$ and then $0.32 \%$ tip $\newline$ $32 \%$ tip and then a discount of $65 \%$

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Q. The expression

1.32(0.65 p)

represents the amount Cai paid for dinner at a restaurant after he applied a discount coupon and added tip. Which changes to the original price could have resulted in the expression?

\newline

A discount of

0.65 \%

and then

32 \%

tip

\newline

32 \%

tip and then a discount of

35 \%

\newline

A discount of

65 \%

and then

0.32 \%

tip

\newline

32 \%

tip and then a discount of

65 \%

Understand the expression: Understand the expression $1.32(0.65p)$ . This expression represents two operations applied to the original price $p$ . The first operation is multiplication by $0.65$ , which suggests a discount, and the second operation is multiplication by $1.32$ , which suggests adding a tip to the discounted price.
Interpret the first operation: Interpret the first operation, $0.65p$ . Since $0.65$ is less than $1$ , it represents a percentage of the original price $p$ . To find the discount percentage, subtract $0.65$ from $1$ and then multiply by $100$ . $\newline$ Discount percentage = $(1 - 0.65) \times 100 = 0.35 \times 100 = 35\%$ $\newline$ This means a discount of $35\%$ was applied to the original price.
Interpret the second operation: Interpret the second operation, $1.32$ . Since $1.32$ is more than $1$ , it represents the original price plus an additional percentage, which is the tip. To find the tip percentage, subtract $1$ from $1.32$ and then multiply by $100$ . $\newline$ Tip percentage = $(1.32 - 1) \times 100 = 0.32 \times 100 = 32\%$ $\newline$ This means a tip of $32\%$ was added after applying the discount.
Compare the calculated discount and tip percentages: Compare the calculated discount and tip percentages with the given options to determine which changes to the original price could have resulted in the expression $1.32(0.65p)$ . Option A suggests a discount of $0.65\%$ followed by a $32\%$ tip, which is incorrect because the discount calculated is $35\%$ , not $0.65\%$ . Option B suggests a $32\%$ tip followed by a discount of $35\%$ , which is incorrect because the order of operations in the expression indicates the discount is applied before the tip. Option C suggests a discount of $65\%$ followed by a $0.32\%$ tip, which is incorrect because the discount is $35\%$ , not $65\%$ , and the tip is $32\%$ , not $0.32\%$ . Option D suggests a $32\%$ tip followed by a discount of $65\%$ , which is incorrect for the same reasons as Option C. The correct changes are a discount of $35\%$ followed by a $32\%$ tip, which is not explicitly listed among the options but is implied by the correct interpretation of the expression.