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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?
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?
- a)1/2
- b)2
- c)3
- d)5
- e)6
Correct answer is option 'E'. Can you explain this answer?
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Verified Answer
Machine A and Machine B can produce 1 widget in 3 hours working togeth...
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 1/3 of a widget. In 1 hour, Machine A can produce 1/a widgets and Machine B can produce 1/b widgets. Together in 1 hour, they produce 1/a + 1/b = 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 a/2. Under these new conditions, in 1 hour Machine A and B could produce 1/(a/2) + 1/b = 1/2 widgets. We now have two unknowns and two different equations. We can solve for a.
The two equations:
2/a + 1/b = 1/2 (Remember, 1/(a/2) = 2/a)
1/a + 1/b = 1/3
2/a + 1/b = 1/2 (Remember, 1/(a/2) = 2/a)
1/a + 1/b = 1/3
Subtract the bottom equation from the top:
2/a – 1/a = 1/2 – 1/3
1/a = 3/6 – 2/6
1/a = 1/6
Therefore, a = 6.
2/a – 1/a = 1/2 – 1/3
1/a = 3/6 – 2/6
1/a = 1/6
Therefore, a = 6.
The correct answer is E.
Most Upvoted Answer
Machine A and Machine B can produce 1 widget in 3 hours working togeth...
To solve this problem, we need to set up a system of equations based on the given information and then solve for the unknown variable.
Let's assume that Machine A's rate is A widgets per hour and Machine B's rate is B widgets per hour.
1) Set up the equation based on the first scenario:
Together, Machine A and Machine B can produce 1 widget in 3 hours. This means that in 1 hour, they can produce 1/3 of a widget. Therefore, the equation is:
A + B = 1/3
2) Set up the equation based on the second scenario:
If Machine A's speed were doubled, the two machines could produce 1 widget in 2 hours. This means that in 1 hour, they can produce 1/2 of a widget. Since Machine A's rate is doubled, it becomes 2A. Therefore, the equation is:
2A + B = 1/2
Now we have a system of equations with two unknowns (A and B). We can solve this system to find the values of A and B.
3) Solving the system of equations:
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.
From equation 1, we have A = 1/3 - B.
Substituting this value of A into equation 2, we get:
2(1/3 - B) + B = 1/2
2/3 - 2B + B = 1/2
2/3 - B = 1/2
-B = 1/2 - 2/3
-B = 3/6 - 4/6
-B = -1/6
B = 1/6
Now that we have the value of B, we can substitute it back into equation 1 to find A:
A + 1/6 = 1/3
A = 1/3 - 1/6
A = 2/6 - 1/6
A = 1/6
4) Finding the time it takes for Machine A to produce 1 widget on its own:
Since A represents Machine A's rate, and we know that rate is equal to output per unit time, we can say that it takes Machine A 1/A hours to produce 1 widget on its own. Therefore, it takes Machine A 1/(1/6) = 6 hours to produce 1 widget on its own.
Therefore, the correct answer is option E.
Let's assume that Machine A's rate is A widgets per hour and Machine B's rate is B widgets per hour.
1) Set up the equation based on the first scenario:
Together, Machine A and Machine B can produce 1 widget in 3 hours. This means that in 1 hour, they can produce 1/3 of a widget. Therefore, the equation is:
A + B = 1/3
2) Set up the equation based on the second scenario:
If Machine A's speed were doubled, the two machines could produce 1 widget in 2 hours. This means that in 1 hour, they can produce 1/2 of a widget. Since Machine A's rate is doubled, it becomes 2A. Therefore, the equation is:
2A + B = 1/2
Now we have a system of equations with two unknowns (A and B). We can solve this system to find the values of A and B.
3) Solving the system of equations:
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.
From equation 1, we have A = 1/3 - B.
Substituting this value of A into equation 2, we get:
2(1/3 - B) + B = 1/2
2/3 - 2B + B = 1/2
2/3 - B = 1/2
-B = 1/2 - 2/3
-B = 3/6 - 4/6
-B = -1/6
B = 1/6
Now that we have the value of B, we can substitute it back into equation 1 to find A:
A + 1/6 = 1/3
A = 1/3 - 1/6
A = 2/6 - 1/6
A = 1/6
4) Finding the time it takes for Machine A to produce 1 widget on its own:
Since A represents Machine A's rate, and we know that rate is equal to output per unit time, we can say that it takes Machine A 1/A hours to produce 1 widget on its own. Therefore, it takes Machine A 1/(1/6) = 6 hours to produce 1 widget on its own.
Therefore, the correct answer is option E.
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