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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

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

Step by step solution

01

- 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.
02

- Determine the combined rate

To find the combined rate at which machines P and Q produce widgets when working together, add their individual rates: \[ \text{Combined rate} = \frac{a}{3} + \frac{b}{4} \]
03

- 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,\[ \frac{a}{3} + \frac{b}{4} = \frac{4a}{12} + \frac{3b}{12} = \frac{4a + 3b}{12} \]
04

- 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:\[ \text{Total production} = \left( \frac{4a + 3b}{12} \right) c = \frac{(4a + 3b) c}{12} \]
05

- Identify the correct answer choice

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

One letter is selected at random from the 5 letters \(\mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y},\) and \(\mathrm{Z}\), and event \(A\) is the event that the letter \(\mathrm{V}\) is selected. A fair six-sided die with sides numbered \(1,2,3,4,5,\) and 6 is to be rolled, and event \(B\) is the event that a 5 or a 6 shows. A fair coin is to be tossed, and event \(C\) is the event that a head shows. What is the probability that event \(A\) occurs and at least one of the events \(B\) and \(C\) occurs? a. \(\frac{1}{30}\) b. \(\frac{1}{15}\) c. \(\frac{1}{10}\) d. \(\frac{2}{15}\) e. \(\frac{1}{5}\)

An ornithologist has studied a particular population of starlings and discovered that their population has increased by \(400 \%\) every ten years starting in \(1890 .\) If the initial population in 1890 was 256 birds, how large was the population of starlings in \(1970 ?\) a. 102,400 b. 10,000,000 c. 16,777,216 d. 20,000,000 e. 100,000,000

If \(x>0, y>0,\) and \(\frac{7 x^{2}+72 x y+4 y^{2}}{4 x^{2}+12 x y+5 y^{2}}=4,\) what is the value of \(\frac{x+y}{y} ?\) a. \(\frac{3}{4}\) b. \(\frac{4}{3}\) c. \(\frac{10}{7}\) d. \(\frac{7}{4}\) e. \(\frac{7}{3}\)

Two workers have different pay scales. Worker A receives \(\$ 50\) for any day worked plus \(\$ 15\) per hour. Worker \(B\) receives \(\$ 27\) per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9: 21 a.m. and receives the \(\$ 50\) upon arrival and Worker \(\mathrm{B}\) arrives at 10: 09 a.m. assuming both work continuously, at what time would their earnings be identical? a. \(11: 09 \mathrm{a.m}.\) b. \(12: 01 \mathrm{p.m}.\) c. \(2: 31 \mathrm{p.m}.\) d. \(3: 19 \mathrm{p.m}.\) e. \(3: 36 \mathrm{p.m}.\)

A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10 Which of the following could NOT be the sum of their numbers? a. 1,254 b. 1,428 c. 3,972 d. 4,316 e. 8,010

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