A company that manufactures and sells guitars made changes in their product range. From the start they had 20 models. Then they reduced the number of models to 15. As a result, the company enjoyed a 10% increase in turnover.
a.How much did the stock level change %?
Inspired by the success, suppose the company aim for another 20% increase in turnover and with the same change in stock level as the last time they reduced the number of models. b.How many models should the company go for this time?


To solve this problem, we need to analyze the relationship between the number of guitar models and the company's turnover.

### a. Change in Stock Level (%)

Initially, the company had 20 models and reduced this to 15 models. We need to calculate the percentage change in the number of models.

The percentage change in stock level is calculated as follows:

\[
\text{Percentage change} = \left( \frac{\text{Final number of models} - \text{Initial number of models}}{\text{Initial number of models}} \right) \times 100\%
\]

Substituting the given values:

\[
\text{Percentage change} = \left( \frac{15 - 20}{20} \right) \times 100\% = \left( \frac{-5}{20} \right) \times 100\% = -25\%
\]

Therefore, the stock level changed by \(-25\%\).

### b. Models Needed for a 20% Increase in Turnover

We know that a 25% reduction in models led to a 10% increase in turnover. We want to find the reduction in models needed for a 20% increase in turnover, assuming a linear relationship between the change in the number of models and the change in turnover.

If \(-25\%\) change in stock results in a \(10\%\) increase in turnover, we want to find the stock level change (\(x\)) that would result in a \(20\%\) increase in turnover. 

Using a proportion:

\[
\frac{10\%}{-25\%} = \frac{20\%}{x}
\]

Solving for \(x\):

\[
x = \frac{20\% \times -25\%}{10\%} = -50\%
\]

This means the company needs to reduce the number of models by \(50\%\) from the previous count to achieve a \(20\%\) increase in turnover.

The company currently has 15 models. To achieve a \(50\%\) reduction:

\[
\text{Number of models to keep} = 15 \times \left(1 - \frac{50}{100}\right) = 15 \times 0.5 = 7.5
\]

Since the number of models must be a whole number, the company should aim for either 7 or 8 models. 

### Conclusion

a. The stock level changed by \(-25\%\).

b. The company should aim for either 7 or 8 models to achieve another 20% increase in turnover.

---

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do changes in product variety generally impact consumer demand?
2. Can reducing the number of product models always lead to increased turnover?
3. What factors might influence the relationship between model reduction and turnover increase?
4. Are there risks associated with reducing the number of product models?
5. How might a company assess which models to discontinue?
6. What strategies can be employed to maintain customer satisfaction with fewer models?
7. How do other companies in different industries approach model reduction?
8. Can the linear relationship between model reduction and turnover increase be considered reliable in all cases?

**Tip:** When interpreting percentage changes, consider the context and possible nonlinear effects that might arise.

Understand how changes in stock levels and model reductions affect business turnover. Learn to calculate percentage changes and determine optimal model counts for maximizing revenue. Suitable for business analysts and students studying business mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation