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Karen hosts a weekly book club and buys cappuccinos and scones for each person in the group. The cappuccinos cost [tex]$c$[/tex] dollars each, and the scones cost [tex]$s$[/tex] dollars each. There were 12 people who came to the book club last week and 15 people who came to the book club this week.

Pick all the expressions that represent how much Karen spent the past two weeks.

A. [tex]27c + 27s[/tex]

B. [tex]12(c + s) + 15(c + s)[/tex]

C. [tex]54(c + s)[/tex]

D. [tex]12c + 12s + 15c + 15s[/tex]

Answer :

Sure, let's break down the problem step by step:

Karen buys cappuccinos and scones for her book club members. We need to calculate how much she spent over the past two weeks using the given expressions.

1. Understand the Variables:
- [tex]\( c \)[/tex]: Cost of each cappuccino
- [tex]\( s \)[/tex]: Cost of each scone

2. Calculate for Each Week:
- Last Week: 12 people attended
- Cost per person for cappuccino and scone: [tex]\( c + s \)[/tex]
- Total cost for last week: [tex]\( 12 \times (c + s) \)[/tex]
- This Week: 15 people attended
- Cost per person for cappuccino and scone: [tex]\( c + s \)[/tex]
- Total cost for this week: [tex]\( 15 \times (c + s) \)[/tex]

3. Total Cost for Both Weeks:
- Combine the costs of both weeks:
[tex]\[
Total = 12 \times (c + s) + 15 \times (c + s)
\][/tex]

4. Simplify the Expression:
- Factor out the common term, [tex]\( c + s \)[/tex]:
[tex]\[
Total = (12 + 15) \times (c + s) = 27 \times (c + s)
\][/tex]

Now, we will match this result with the given expressions:

- Expression 1: [tex]\( 27c + 27s \)[/tex]
- This matches our simplified expression: [tex]\( 27 \times (c + s) \)[/tex].

- Expression 2: [tex]\( 12(c+s) + 15(c+s) \)[/tex]
- This is the expanded version before simplification and also matches.

- Expression 3: [tex]\( 54(c+s) \)[/tex]
- This does not match. If simplified correctly, it should be [tex]\( 27(c+s) \)[/tex].

- Expression 4: [tex]\( 12c + 12s + 15c + 15s \)[/tex]
- This matches because when you distribute and combine like terms, it equals [tex]\( 27c + 27s \)[/tex].

Correct Expressions are:
- [tex]\( 27c + 27s \)[/tex]
- [tex]\( 12(c+s) + 15(c+s) \)[/tex]
- [tex]\( 12c + 12s + 15c + 15s \)[/tex]

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