Work Rate Practice Questions

Question 1 of 5 Medium

Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates. If Machine A's speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates. How many hours does it currently take Machine A to produce 1 widget on its own?

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Correct answer: E

Let $a$ be the number of hours it takes Machine A to produce $1$ widget on its own. Let $b$ be the number of hours it takes Machine B to produce $1$ widget on its own.

The question tells us that Machines A and B together can produce $1$ widget in $3$ hours. Therefore, in $1$ hour, the two machines can produce $\frac{1}{3}$ of a widget.

In $1$ hour, Machine A can produce $\frac{1}{a}$ widgets and Machine B can produce $\frac{1}{b}$ widgets. Together in $1$ hour, they produce $\frac{1}{a} + \frac{1}{b} = \frac{1}{3}$ widgets.

If Machine A's speed were doubled it would take the two machines $2$ hours to produce $1$ widget. When one doubles the speed, one cuts the amount of time it takes in half. Therefore, the amount of time it would take Machine A to produce $1$ widget would be $\frac{a}{2}$. Under these new conditions, in $1$ hour Machine A and B could produce $\frac{1}{\frac{a}{2}} + \frac{1}{b} = \frac{1}{2}$ widgets.

We now have two unknowns and two different equations. We can solve for $a$.

The two equations: 
$\frac{2}{a} + \frac{1}{b} = \frac{1}{2}$ (Remember, $\frac{1}{\frac{a}{2}} = \frac{2}{a}$) 
$\frac{1}{a} + \frac{1}{b} = \frac{1}{3}$

Subtract the bottom equation from the top: 
$\frac{2}{a} - \frac{1}{a} = \frac{1}{2} - \frac{1}{3}$ 
$\frac{1}{a} = \frac{3}{6} - \frac{2}{6}$ 
$\frac{1}{a} = \frac{1}{6}$

Therefore, $a = 6$.

The correct answer is E.

Question 2 of 5 Medium

Adam and Brianna plan to install a new tile floor in a classroom. Adam works at a constant rate of 50 tiles per hour, and Brianna works at a constant rate of 55 tiles per hour. If the new floor consists of exactly 1400 tiles, how long will it take Adam and Brianna working together to complete the classroom floor?

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Correct answer: C

Because Adam and Brianna are working together, add their individual rates to find their combined rate: $50 + 55 = 105$ tiles per hour

The question asks how long it will take them to set $1400$ tiles.

$$\text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{1400 \text{ tiles}}{105 \text{ tiles / hour}} = \frac{40}{3} \text{ hours} = 13\frac{1}{3} \text{ hours} = 13 \text{ hours and } 20 \text{ minutes}$$

The correct answer is C.

Question 3 of 5 Medium

A copy machine, working at a constant rate, makes 35 copies per minute. A second copy machine, working at a constant rate, makes 55 copies per minute. Working together at their respective rates, how many copies do the two machines make in half an hour?

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Correct answer: B

This is a straightforward combined rate problem. Machine 1 rate: 35 copies per minute Machine 2 rate: 55 copies per minute Combined rate: $ 35 + 55 = 90 $ copies per minute Time working together: 30 minutes (half an hour) Total copies: $ 90 \times 30 = 2700 $ copies The correct answer is B.

Question 4 of 5 Medium

Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

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Question 5 of 5 Medium

Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A and B work together at their respective rates to complete the job, which of the following represents the fraction of the job that B will not have to complete because of A's help?

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