<h3>
Answer: B) 2</h3>
=============================================================
Explanation:
List out the multiples of each number until we find the LCM
- multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, ...
- multiples of 14 are: 14, 28, 42, 56, 70, 84, 98, ...
- multiples of 21 are: 21, 42, 63, 84, 105, ...
We see that 84 shows up in the list and it is the smallest such common item. The LCM of the numbers {12,14,21} is 84.
-------------
Let's say there are 84 items to be moved in the warehouse.
- If person A works alone, then their unit rate is 84/12 = 7 items per hour
- If person B works alone, then their unit rate is 84/14 = 6 items per hour
- If person C works alone, then their unit rate is 84/21 = 4 items per hour
We divide 84 over the respective time values given by your teacher, to find each unit rate.
Let x be the the length of time (in hours) that person A worked. They work alongside persons B and C to get a combined unit rate of 7+6+4 = 17 items per hour. This assumes no worker gets in the way of any others. They work the most efficiently together.
After x hours, they manage to move 17x items in the warehouse.
After x hours are up, there are 84-17x items left to move. Person A leaves B and C to finish up, so B and C's combined rate is 6+4 = 10 items per hour. These two remaining people move the 84-17x items at a rate of 10 items per hour. The time span they have left is 7-x hours.
This means we can form the following equation
(84-17x)/(10) = 7-x
Divide the number of items over the unit rate to find the time duration in hours.
-------------
Solve for x
(84-17x)/(10) = 7-x
84-17x = 10(7-x)
84-17x = 70-10x
84-70 = -10x+17x
14 = 7x
7x = 14
x = 14/7
x = 2
Person A works for 2 hours
-------------
If person A worked for 2 hours, then persons A,B,C moved 17*2 = 34 items. This leaves 84-34 = 50 items left. Person A takes off to leave persons B and C to handle the rest of the job. These two remaining people have a combined rate of 10 items per hour. So they take 50/10 = 5 hours when person A leaves. That gives the 2+5 = 7 hours total. This confirms the correct answer.
