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Productivity Problems

Problem $1$:

A factory produces $7,500$ widgets. Each widget has a standard selling price of $$45.$ The labor input is $750$ hours for $$\frac{30}{?}$ hour. The cost of materials is $$10,000,$ and the overhead cost is $1\mathrm{.}5$ times the labor cost. Calculate the multifactor productivity.

Problem $2$:

A service company processes $200$ transactions daily. The price per transaction is $$20.$ They use $160$ hours of labor each day, which costs $$\frac{18}{?}$ hour. Other costs include $$800$ daily for materials and twice the daily labor cost for overhead. Calculate the multifactor productivity.

Problem $3$:

In $2023,$ a call center's labor productivity was measured at $90$ calls per hour. By $2024,$ the productivity improved to $100$ calls per hour. Calculate the productivity growth from $2023$ to $2024.$

Problem $4$:

A car repair shop services $120$ cars a month. They charge an average of $$500$ per service. The labor input amounts to $1,000$ hours per month at a rate of $$20$ per hour. The cost of parts $($materials$)$ is $$30,000$ per month, and the overhead is equal to the total labor cost. Determine the multifactor productivity.

Problem $5$:

In the first quarter of $2023,$ a manufacturing plant produced $50,000$ units with labor productivity at $50$ units per hour. In the first quarter of $2024,$ the plant managed to increase its labor productivity to $55$ units per hour while producing the same number of units. What is the percentage of productivity growth?

Problem $6$:

A supervisor reviewed the weekly report and noticed that an employee completed $150$ tasks over $60$ hours in one week. In the following week, the employee managed to complete $130$ tasks but only worked $50$ hours. Determine whether the employee's productivity increased, decreased, or remained unchanged from one week to the next. Provide your reasoning and calculations to support your conclusion.