https://file.aiquickdraw.com/mathbot/file/1726605283719b4rrgymr.jpg

Let's break down each question in the image and solve them:

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### **Question 1**:
*Your health clinic increased total sales (output) by 26%, and decreased total costs (input) by 59%. What was your percent change in total productivity?*

The formula for total productivity is:

\[
\text{Productivity} = \frac{\text{Output}}{\text{Input}}
\]

We are given the following percentage changes:
- **Output** increased by 26%, so the new output is $ 1 + 0.26 = 1.26 $.
- **Input** decreased by 59%, so the new input is $ 1 - 0.59 = 0.41 $.

Thus, the new productivity is:

\[
\text{New Productivity} = \frac{1.26}{0.41}
\]

Let's calculate that:

\[
\text{New Productivity} \approx 3.07317
\]

This represents a 207.3% increase in productivity (subtracting 1 and multiplying by 100). Therefore, the change in productivity is:

\[
207.3\%
\]

The answer would be **207**.

---

### **Question 2**:
*You learn their labor productivity last year was 5 and total productivity was 1. You know from public filings that their total output was $10M, overhead was $3M, and the firm's only costs were labor, materials, and overhead. How much did competitor pay for materials last year?*

We know:
- Total output = $10M
- Overhead = $3M
- Productivity = 1, which means:

\[
\text{Total input} = \text{Total output} = 10M
\]

We need to find the amount paid for materials. Let $ L $ be the labor cost and $ M $ be the material cost. Since the only costs are labor, materials, and overhead, the total input is:

\[
L + M + \text{Overhead} = 10M
\]

Substitute overhead:

\[
L + M + 3M = 10M
\]

Thus:

\[
L + M = 7M
\]

From the given labor productivity (output per labor cost) of 5, we know:

\[
\frac{\text{Output}}{L} = 5 \quad \text{so} \quad L = \frac{10M}{5} = 2M
\]

Substitute $ L = 2M $ into the equation:

\[
2M + M = 7M
\]

Thus, the material cost $ M = 5M $. The answer is **5** million dollars.

---

### **Question 3**:
*What is the probability that BOTH of the elevators are functioning?*

- The reliability of the first elevator is 91%, or 0.91.
- The reliability of the second elevator is 82%, or 0.82.

Since the elevators are independent, the probability that both are functioning is the product of the two probabilities:

\[
P(\text{Both Functioning}) = 0.91 \times 0.82 = 0.7462
\]

Converting this to a percentage:

\[
0.7462 \times 100 \approx 74.62\%
\]

Rounding to the nearest percentage point gives **75**.

---

Would you like any further details or explanations on these solutions?

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Here are five related questions to deepen your understanding:
1. What is the formula for calculating productivity change in general terms?
2. How does the decrease in input affect overall productivity in real-world scenarios?
3. If labor productivity changes, how does that impact a company's efficiency?
4. What is the difference between dependent and independent probabilities in statistics?
5. How would the reliability of systems change if there were more than two elevators?

**Tip:** Always check if percentage increases and decreases are applied to different starting values, as it can significantly affect results.

Your health clinic increased total sales (output) by 26%, and decreased total costs (input) by 59%. What was your percent change in total productivity? You are trying to understand the finances of one of your competitors. You learn their labor productivity last year was 5 and total productivity was 1. You know from public filings that their total output was $10M, overhead was $3M, and the firm’s only costs were labor, materials, and overhead. How much did competitor pay for materials last year? I'm the manager of the SOB building with 2 elevators. The first elevator has a reliability of 91%. The second elevator (which is independent from the first) has reliability of 82%. What is the probability that BOTH of the elevators are functioning?

Solve problems involving a clinic's productivity change by calculating the percent increase based on changes in output and input, material costs for a competitor using labor productivity data, and probability of elevator reliability with independent events.

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