anonymous
  • anonymous
Crane A can unload a ship in 10 hours and crane B can unload it in 14 hours. How long will it take the two cranes to unload the ship working together? _______________________________________________________________________________ I just need to know how to set this up into an equation.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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whpalmer4
  • whpalmer4
The key concept here is "find the unit rate". A can unload 1 ship in 10 hours, so its unit rate is 1 ship/10 hours or 0.1 ships/hr. B can unload 1 ship in 14 hours, so its unit rate is 1 ship/14 hours or 0.07142857 ships/hr. A and B working together in perfect harmony ought to be able to unload (1/10 + 1/14) ships per hour.
whpalmer4
  • whpalmer4
Here's a similar example. Bob can mow the lawn in 2 hours, Sam can mow the lawn in 3 hours. How long for them to mow it together? Bob's rate = 1 lawn/2 hours or 1/2 Sam's rate = 1 lawn/3 hours or 1/3 Bob and Sam together mow (1/2 + 1/3) lawns per hour. Make a common denominator to add the fractions:\[\frac{1}{2}*\frac{3}{3} + \frac{1}{3}*\frac{2}{2} = \frac{3}{6}+\frac{2}{6} = \frac{5}{6} \text{ lawns/hr}\] \(1\text{ lawn}/ (5/6\text{ lawns/hr}) = 1\text{ lawn} * \frac{6\text{ hr}}{5\text{ lawns}} = \frac{6}{5} \text{ hours}\) to mow the 1 lawn working together.
anonymous
  • anonymous
so what's the equation I don't really get anything you just said...

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amistre64
  • amistre64
this one?
anonymous
  • anonymous
yeah
anonymous
  • anonymous
I don't really understand what wpalmer was saying
amistre64
  • amistre64
Crane A can unload a ship in 10 hours crane B can unload it in 14 hours How long will it take the two cranes to unload the ship working together? spose a crane works at an even pace; if it takes 10 hours to complete 1 job, how much of the job is completed in 1 hour? in otherwords, what is the hourly rate?
whpalmer4
  • whpalmer4
Every hour, crane A unloads 1/10 of the ship. Every hour, crane B unloads 1/14 of the ship. Together, every hour they unload 1/10 + 1/14 of the ship. Add 1/10 + 1/14. Now, divide 1 by that number to give you the number of hours it takes to unload the ship.
anonymous
  • anonymous
so (1/10 + 1/14) ______________ ? 1
whpalmer4
  • whpalmer4
To divide by a fraction, invert it and multiply instead. For example, 1 / (2/3) can be found by multiplying 1 by (3/2).
whpalmer4
  • whpalmer4
Close, you want: \[\frac{ 1} {\frac{1}{10} + \frac{1}{14}}\]
anonymous
  • anonymous
oh but wouldn't that just equal 1/10 + 1/14
whpalmer4
  • whpalmer4
Certainly not!
whpalmer4
  • whpalmer4
1/10 = 0.1. 1/14 = 0.0714285 Add the two together and you get about 0.17. 1/0.17 does not equal 0.17...
whpalmer4
  • whpalmer4
For the purposes of getting a handle on this, say both cranes work at the same speed as crane A. Crane A does 1/10 of the ship each hour. Crane B does 1/10 of the ship each hour. Together, they do 1/10 + 1/10 = 2/10 of the ship per hour. How long would they take to do the entire ship?
whpalmer4
  • whpalmer4
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whpalmer4
  • whpalmer4
Let's go through adding \[\frac{1}{10} + \frac{1}{14}\]We need to make a common denominator. What is the smallest number that is a multiple of both 10 and 14? That's 140, right? 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 14 28 42 56 70 84 98 112 126 140 154 140 is the first number to appear on both lists. We'll use it as our common denominator. So, to convert each fraction to the common denominator, we multiply it by 1, written as a fraction: \[\frac{1}{10}*\frac{14}{14} + \frac{1}{14}*\frac{10}{10} = \frac{14}{140} + \frac{10}{140} = \frac{24}{140}\]14/14 = 1, and 10/10 = 1, so we haven't changed the value of anything, just the way we write it. We could observe that 24 and 140 both have a factor of 4 in common, and reduce the fraction to 6/35. Therefore, we know that A+B working together will unload 6/35 of a ship each hour. To find the total number of hours to unload, we divide the number of ships to be unloaded (1) by the rate (6/35): \[\frac{1}{(\frac{6}{35})}\]Recalling that dividing by a fraction is equivalent to multiplying by its reciprocal \[\frac{1}{(\frac{6}{35})} = 1*\frac{35}{6} = \frac{35}{6} = 5\frac{5}{6}\] It will take cranes A and B that many hours to unload the ship.
whpalmer4
  • whpalmer4
If you check your work (and you should), crane A working for 35/6 hours does \[\frac{35}{6}*\frac{1}{10} = \frac{35}{60} = \frac{7}{12}\] Crane B does \[\frac{35}{6}*\frac{1}{14} = \frac{35}{84} = \frac{5}{12}\] \[A + B = 1\]\[\frac{7}{12} + \frac{5}{12} = \frac{12}{12} = 1\checkmark\]
whpalmer4
  • whpalmer4
You might want to brush up on your fractions manipulation skills, as most of these problems come down to an exercise in fractions once you've grasped the fundamental concept. Knowing how to set up the problem is only part of the battle :-)

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