Question

Machine A can process 6,000 envelopes in 3 hours. Machines \(\mathrm{B}\) and \(\mathrm{C}\) working together but independently can process the same number of envelopes in \(2 \frac{2}{5}\) hours. If machines \(A\) and \(C\) working together but independently process 3,000 envelopes in 1 hour, then how many hours would it take machine \(\mathrm{B}\) to process 12,000 envelopes?

Short answer

Machine B would take 8 hours to process 12,000 envelopes.

Step-by-step solution

- Determine Machine A's Rate

Machine A can process 6,000 envelopes in 3 hours. This gives a processing rate of \( \frac{6000 \text{ envelopes}}{3 \text{ hours}} = 2000 \text{ envelopes/hour} \)

- Determine Combined Rate of Machines B and C

Machines B and C together can process 6,000 envelopes in 2 \frac{2}{5} hours. Converting the mixed number to an improper fraction gives \frac{12}{5} hours. Thus, the combined rate of B and C is \( \frac{6000 \text{ envelopes}}{\frac{12}{5} \text{ hours}} = 2500 \text{ envelopes/hour} \)

- Determine Combined Rate of Machines A and C

Machines A and C together can process 3,000 envelopes in 1 hour. Therefore, their combined rate is \( 3000 \text{ envelopes/hour} \)

- Determine Machine C's Rate

From Step 1 and Step 3, subtract Machine A's rate from the combined rate of A and C to find Machine C's rate: \( 3000 \text{ envelopes/hour} - 2000 \text{ envelopes/hour} = 1000 \text{ envelopes/hour} \)

- Determine Machine B's Rate

From Step 2, subtract Machine C's rate from the combined rate of B and C to find Machine B's rate: \( 2500 \text{ envelopes/hour} - 1000 \text{ envelopes/hour} = 1500 \text{ envelopes/hour} \)

- Calculate Time for Machine B to Process 12,000 Envelopes

Using Machine B's rate, calculate the time required to process 12,000 envelopes: \( \frac{12000 \text{ envelopes}}{1500 \text{ envelopes/hour}} = 8 \text{ hours} \)

Key concepts

Rate Calculations

Understanding how to calculate rates is crucial when dealing with work and time problems. A rate is essentially the amount of work done per unit of time. In our exercise, we determined the rate of Machine A by dividing the number of envelopes it processes by the time it takes. Given that Machine A processes 6,000 envelopes in 3 hours, we calculate its rate as follows:

• Rate of Machine A = \[ \frac{6000 \text{ envelopes}}{3 \text{ hours}} = 2000 \text{ envelopes/hour} \]

This means Machine A can process 2,000 envelopes every hour. Calculating rates for other machines or combined efforts will follow a similar approach, as seen in following sections.

Work and Time Problems

Work and time problems involve calculating how long it takes multiple workers or machines working together, or individually, to complete a task. The key is to find the rate at which each worker or machine completes the work. This problem requires us to determine such rates for machines A, B, and C. For example, machines B and C together process 6,000 envelopes in \[2\frac{2}{5}\text{ hours}\]. First, convert the mixed number into an improper fraction \[\frac{12}{5}\]. Then, calculate the combined rate of B and C:

• Combined Rate of B and C = \[\frac{6000 \text{ envelopes}}{\frac{12}{5} \text{ hours}} = 2500 \text{ envelopes/hour} \]

This combined rate shows that B and C together can process 2,500 envelopes per hour.

Algebraic Problem-Solving

Algebra is a powerful tool for solving rate and time problems. We often need to set up equations with unknowns (variables) and solve for these unknowns. In the exercise, we determined the rates of individual machines by subtracting and solving for their rates algebraically. We know Machine A's rate and the combined rate of A and C. Therefore, to find Machine C's rate, we subtract Machine A's rate from their combined rate:

• Rate of Machine C = \[\text{Combined Rate of A and C} - \text{Rate of Machine A} = 3000 \text{ envelopes/hour} - 2000 \text{ envelopes/hour} = 1000 \text{ envelopes/hour} \]

This means Machine C processes 1,000 envelopes per hour.

Machine Rates

Determining the rate at which machines work independently or together is key to solving problems involving machines. Once we know the rates of two machines working together, we can find out the rate of a single machine by subtracting. To find Machine B's rate, we subtract Machine C's rate from the combined rate of B and C:

• Rate of Machine B = \[\text{Combined Rate of B and C} - \text{Rate of Machine C} = 2500 \text{ envelopes/hour} - 1000 \text{ envelopes/hour} = 1500 \text{ envelopes/hour} \]

Lastly, to find out how long it takes Machine B to process 12,000 envelopes, we divide the total number of envelopes by Machine B's rate:

• Time for Machine B = \[\frac{12000 \text{ envelopes}}{1500 \text{ envelopes/hour}} = 8 \text{ hours} \]

Here, Machine B takes 8 hours to process 12,000 envelopes.