In 6 hours,an experienced cook prepares enough pies to supply a local restaurant's daily order. Another cook prepares the same number of pies in 7 hours. Together with a third cook, they prepare the pies in 2 hours. Find how long it takes the third cook to prepare the pies alone.
Well, it seems like these cooks are quite efficient under pressure! Let's do some math with a dash of humor, shall we?
Let's say the number of pies they prepare together in 2 hours is "x".
Now, if the first cook can prepare enough pies in 6 hours alone, then in 2 hours, they can whip up x/6 pies. Similarly, the second cook can make x/7 pies in 2 hours.
So, when they work together, their combined pie-making skills add up to x/6 + x/7 pies in 2 hours. Now, let's call the third cook "Mystery Chef", because we don't know how fast they are yet.
Together, they can make x/6 + x/7 pies in 2 hours, but we want to find out how long it would take the Mystery Chef to prepare x pies alone.
If all three of them work together for 2 hours and prepare x pies, then the Mystery Chef's pie-making speed would be x/2 pies per hour.
Now, we need to set up an equation to solve for the Mystery Chef's time. We can say that x/2 = 1/T, where T represents the time taken by the Mystery Chef, which is what we're trying to find.
Multiplying both sides by 2, we get x = 2/T.
Now, we need to substitute the values we found earlier: x/6 + x/7 = 2/T.
Getting a common denominator, we have (7x + 6x) / 42 = 2/T.
Combining like terms, we get 13x / 42 = 2/T.
To get rid of those pesky fractions, we can cross-multiply: 13xT = 84.
Finally, we divide both sides by 13 to solve for T: T = 84/13.
So, based on my hilarious calculations, it would take the Mystery Chef approximately 6.461538461538462 hours to prepare the pies alone.
Let's assume that the number of pies the experienced cook can prepare in 1 hour is x.
Therefore, the number of pies the other cook can prepare in 1 hour is also x.
Since they prepare enough pies to supply a local restaurant's daily order in 6 hours, it means that in 1 hour, the two of them together will prepare 1/6 of the pies.
Similarly, since they prepare enough pies to supply a local restaurant's daily order in 7 hours, it means that in 1 hour, the two of them together will prepare 1/7 of the pies.
And when they work together with a third cook, they are able to prepare the pies in 2 hours, which means that in 1 hour, all three of them can prepare 1/2 of the pies.
Let's set up an equation to solve for x:
x + x = 1/6 (equation 1)
x + x = 1/7 (equation 2)
x + x + x = 1/2 (equation 3)
Simplifying equations 1 and 2:
2x = 1/6
2x = 1/7
Now, we can solve for x:
2x = 1/6
x = (1/6) / 2
x = 1/12
2x = 1/7
x = (1/7) / 2
x = 1/14
Substituting x into equation 3:
(1/12) + (1/12) + x = 1/2
(1/6) + x = 1/2
x = 1/2 - 1/6
x = 1/3 - 1/6
x = 2/6 - 1/6
x = 1/6
Therefore, the third cook can prepare the pies alone in 6 hours.
So, it takes the third cook 6 hours to prepare the pies alone.
To solve this problem, we can use the concept of work done. The work done by each cook is inversely proportional to the time it takes them to complete the work.
Let's assume that the number of pies that each cook can prepare per hour is x.
So, the first cook can prepare x pies per hour and takes 6 hours to complete the work. Therefore, the work done by the first cook is (x pies/hour) x (6 hours) = 6x pies.
Similarly, the second cook can prepare x pies per hour and takes 7 hours to complete the work. So, the work done by the second cook is (x pies/hour) x (7 hours) = 7x pies.
Together, the first and second cooks can complete the work in 2 hours, so their combined work is (x pies/hour) x (2 hours) = 2x pies.
To find the work done by the third cook in 2 hours, we subtract the combined work of the first and second cooks from the total work required (which is x pies): x pies - 2x pies = -x pies.
However, since work cannot be negative, we need to make the total work positive. Therefore, we take the absolute value of -x: |-x| = x.
Hence, the third cook can prepare x pies in 2 hours.
To find how long it takes the third cook to prepare the pies alone, we need to divide the total work (x pies) by the work done by the third cook per hour (x pies/hour): x pies / (x pies/hour) = 1 hour.
Therefore, the third cook can prepare the pies alone in 1 hour.
if the 3rd cook takes x hours,
1/2 = 1/6 + 1/7 + 1/x
x = 21/4 = 5 1/4