Question

Jane makes toy bears. When she works with an assistant, she makes 80 percent more bears per week and works 10 percent fewer hours each week. Having an assistant increases Jane's output of toy bears per hour by what percent? A. $$20 \%$$ B. $$80 \%$$ C. $$100 \%$$ D. $$180 \%$$ E. $$200 \%$$

Short answer

The percent increase in output per hour is 100%. Answer: C

Step-by-step solution

- Define variables

Let's denote Jane's original weekly output of toy bears as \( B \) and her original weekly working hours as \( H \).

- Calculate output with an assistant

When Jane works with an assistant, she makes 80 percent more bears. Therefore, her new weekly output is \( 1.8B \).

- Calculate reduced working hours

With an assistant, Jane works 10 percent fewer hours. So, her new weekly working hours are \( 0.9H \).

- Calculate output per hour without assistant

Jane's original output of toy bears per hour is \( \frac{B}{H} \).

- Calculate output per hour with assistant

With the assistant, her new output per hour is \( \frac{1.8B}{0.9H} \). Simplifying this fraction, we get \( 2 \frac{B}{H} \).

- Calculate percentage increase

The new output per hour \( 2 \frac{B}{H} \) is twice the original output per hour \( \frac{B}{H} \). This is an increase of 100 percent because \( 2 \frac{B}{H} = \frac{B}{H} + 100\% \frac{B}{H} \).

Key concepts

percentage increase in productivity

Understanding the percentage increase in productivity is essential for solving word problems like the given exercise. In this context, we calculate the increase in Jane's output of toy bears when she works with an assistant. Jane originally produces a certain number of toy bears, denoted as $$B$$, in a specified number of hours, denoted as $$H$$. Working with an assistant increases her productivity by 80%, which mathematically converts to multiplying her output by 1.8. This means her new weekly output is $$1.8B$$. Additionally, her working hours decrease by 10%, so her new weekly working hours are $$0.9H$$. Combining these factors gives her new productivity in terms of output per hour as: $$\frac{1.8B}{0.9H}$$. Simplifying this to $$2\frac{B}{H}$$ indicates that she now produces twice the number of bears per hour as she did before, which translates to a 100% increase in productivity, as $$2 \frac{B}{H} = \frac{B}{H} + 100 \frac{B}{H}$$.

collaborative work benefits

Collaborative work provides numerous benefits, and Jane's situation is a perfect example. By working with an assistant, Jane can produce 80% more toy bears and reduce her working hours by 10%. The assistant helps by managing tasks, allowing Jane to focus on her core activity—making toy bears. This leads to significant advantages:

• Increased output: Jane can produce more bears, boosting potential revenue.
• Reduced fatigue: Working fewer hours reduces stress and fatigue, potentially enhancing the quality of her work.
• Efficiency: Collaboration allows for better task management, ensuring higher productivity per hour worked.

These benefits emphasize the importance of teamwork and efficient resource utilization in work settings.

calculating efficiency rates

Efficiency rate calculations help us understand how productive a person or process is. In Jane's case, her original efficiency without an assistant is given by her output per hour: $$\frac{B}{H}$$. When an assistant joins her, the productivity increases. To calculate her new efficiency rate, we use the provided details: an 80% increase in output and a 10% reduction in working hours. This gives a new efficiency of $$\frac{1.8B}{0.9H}$$. Simplifying this, we find: $$2\frac{B}{H}$$. The comparison between her old efficiency, $$\frac{B}{H}$$, and her new efficiency, $$2\frac{B}{H}$$, demonstrates a doubling of productivity, showing a 100% increase. These kinds of efficiency rate calculations are crucial in resource planning and management.

mathematical reasoning

Developing strong mathematical reasoning is vital for solving word problems. This involves identifying the key information, forming relevant equations, and performing accurate calculations. Let's break down Jane's problem:

• Identify the base quantities: original weekly output $$B$$ and original weekly hours $$H$$.
• Calculate new outputs based on given percentages: 80% more bears leading to $$1.8B$$ and 10% fewer hours resulting in $$0.9H$$.
• Determine productivity per hour both before and after the change: $$\frac{B}{H}$$ initially and $$\frac{1.8B}{0.9H}$$ after the change.
• Simplify and interpret the results: $$\frac{1.8B}{0.9H} = 2\frac{B}{H}$$ which means a 100% increase in productivity.

This logical sequence showcases how we can use mathematical principles to draw meaningful conclusions from given data, underscoring the importance of step-by-step problem solving.