Question

Working alone at a constant rate, machine \(\mathrm{P}\) produces \(a\) widgets in 3 hours. Working alone at a constant rate, machine \(\mathrm{Q}\) produces \(b\) widgets in 4 hours. If machines \(P\) and \(Q\) work together for \(c\) hours, then in terms of \(a, b,\) and \(c,\) how many widgets will machines \(\mathrm{P}\) and \(\mathrm{Q}\) produce? a. \(\frac{3 a c+4 b c}{12}\) b. \(\frac{4 a c+3 b c}{12}\) c. \(\frac{4 a c+3 b c}{6}\) d. \(4 a c+3 b c\) e. \(\frac{a c+2 b c}{4}\)

Short answer

b. \(\frac{4 a c+3 b c}{12}\)

Step-by-step solution

- Calculate the rates of machines

Calculate the rate at which machine P produces widgets. Since machine P produces a widgets in 3 hours, its rate is \(\frac{a}{3}\) widgets per hour. Similarly, calculate the rate at which machine Q produces widgets. Since machine Q produces \(b\) widgets in 4 hours, its rate is \(\frac{b}{4}\) widgets per hour.

- Determine the combined rate

To find the combined rate at which machines P and Q produce widgets when working together, add their individual rates: \[ \text{Combined rate} = \frac{a}{3} + \frac{b}{4} \]

- Find a common denominator

Combine the rates into one fraction by finding a common denominator. The common denominator for 3 and 4 is 12. Thus,\[ \frac{a}{3} + \frac{b}{4} = \frac{4a}{12} + \frac{3b}{12} = \frac{4a + 3b}{12} \]

- Calculate the production for c hours

Multiply the combined rate by the number of hours \(c\) to find the total number of widgets produced in c hours:\[ \text{Total production} = \left( \frac{4a + 3b}{12} \right) c = \frac{(4a + 3b) c}{12} \]

- Identify the correct answer choice

Match the expression \(\frac{(4a + 3b)c}{12}\) with the given choices. The correct answer is option (b).

Key concepts

Machine Productivity

Machine productivity refers to the efficiency with which a machine can produce goods. It's essentially the output rate of the machine per unit time. In our exercise, machine P produces \(a\) widgets in 3 hours, while machine Q produces \(b\) widgets in 4 hours.
To determine the productivity rate, we divide the total output by the time taken:
For machine P: \(\frac{a}{3}\) widgets per hour
For machine Q: \(\frac{b}{4}\) widgets per hour.
This calculation is crucial for solving problems related to combined machine work.

Rate Calculations

Rate calculations involve determining the speed at which work is done or goods are produced by comparing the volume of work or goods to the time taken. In our case, we calculate the rates as follows:
\begin{lowercase}(1). Divide the number of widgets by the hours required by each machine to find their rates:
Rate of machine P: \(\frac{a}{3}\) widgets per hour
Rate of machine Q: \(\frac{b}{4}\) widgets per hour.
These rates are foundational for combining the productivity of the machines.

Combined Production Rates

When machines work together, their combined production rate is the sum of their individual rates. For our exercise:
Combined rate = \(\frac{a}{3}\) + \(\frac{b}{4}\)
To add these, we find a common denominator. The least common multiple of 3 and 4 is 12. Hence:
\(\frac{a}{3} = \frac{4a}{12}\)
\(\frac{b}{4} = \frac{3b}{12}\)
Adding these fractions gives:
Combined rate: \(\frac{4a + 3b}{12}\) widgets per hour.
This rate shows how quickly both machines can produce widgets together.

Fraction Operations

Understanding fraction operations is crucial in solving this type of problem. When adding fractions, we must have a common denominator. For our combined rate:
\(\frac{a}{3} + \frac{b}{4}\)
We found the common denominator (12) and converted each fraction:
\(\frac{a}{3} = \frac{4a}{12}\) and \(\frac{b}{4} = \frac{3b}{12}\)
Then we added them to get:
\(\frac{4a + 3b}{12}\).
This step ensures we represent the combined rate correctly for further calculations.

Algebraic Expressions

In our final calculation, we use algebraic expressions to find the total production over a given time. Multiply the combined production rate by the number of hours (c):
\(\text{Total production} = \frac{(4a + 3b)c}{12}\)
This expression represents the total widgets produced when both machines work together for \(c\) hours.
It ties together rates, fractions, and algebra to help solve the problem efficiently. Recognizing and using such expressions is key for solving complex GMAT problems.