Question

carol is planning a trip to south africa. she plans to spend ten nights there, so she is investigating her options for lodging. she has made arrangements to stay at least part of the time at a hostel, which is her cheapest option. however, she would like to spend as many nights as she can in a private hotel. the hostel costs r 34 (34 rands) per night, and the hotel costs r 55. if carol has a budget of 430 rands for lodging, how many nights can she afford to stay in the private hotel?
a. six
b. five
c. four
d. three

Explanation:

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<reasoning>

Formulate the system of constraints

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &x + y = 10 \\ &55x + 34y \le 430 \\ &x, y \in \mathbb{Z}_{\ge 0} \end{aligned}$$

\]

Substitute and solve for the maximum nights in the hotel

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &y = 10 - x \\ &55x + 34(10 - x) \le 430 \\ &21x + 340 \le 430 \\ &21x \le 90 \\ &x \le \frac{90}{21} \approx 4.28 \end{aligned}$$

\]

Determine the maximum integer value

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &x_{\max} = 4 \\ &y = 10 - 4 = 6 \\ &\text{Cost} = 55(4) + 34(6) = 220 + 204 = 424 \le 430 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. six</mcq-option>
<mcq-option>b. five</mcq-option>
<mcq-correct>c. four</mcq-correct>
<mcq-option>d. three</mcq-option>
</answer>

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Answer:

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<reasoning>

Formulate the system of constraints

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &x + y = 10 \\ &55x + 34y \le 430 \\ &x, y \in \mathbb{Z}_{\ge 0} \end{aligned}$$

\]

Substitute and solve for the maximum nights in the hotel

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &y = 10 - x \\ &55x + 34(10 - x) \le 430 \\ &21x + 340 \le 430 \\ &21x \le 90 \\ &x \le \frac{90}{21} \approx 4.28 \end{aligned}$$

\]

Determine the maximum integer value

Using the Budget Calculation knowledge point
\[

$$\begin{aligned} &x_{\max} = 4 \\ &y = 10 - 4 = 6 \\ &\text{Cost} = 55(4) + 34(6) = 220 + 204 = 424 \le 430 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. six</mcq-option>
<mcq-option>b. five</mcq-option>
<mcq-correct>c. four</mcq-correct>
<mcq-option>d. three</mcq-option>
</answer>

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