SOLUTION: If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines

Algebra ->  Rate-of-work-word-problems -> SOLUTION: If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines      Log On


   

Question 1208171: If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines
Found 5 solutions by ikleyn, josgarithmetic, greenestamps, Edwin McCravy, mccravyedwin:
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
If 8 men take 12 days to assemble 16 machines,
how many days will it take 15 men to assemble 50 machines
~~~~~~~~~~~~~~~~~~~~~~~ 

Let x be the number of days in the second scenario.


The rate of work in the first scenario is  16%2F%288%2A12%29 = 2%2F12 = 1%2F6 of the machine per man per day.  

The rate of work in the second scenario is  50%2F%2815%2Ax%29 = 10%2F%283x%29 of the machine per man per day.  


The rate of work is assumed to be the same, so we write this proportion

    1%2F6 = 10%2F%283x%29.


From this proportion, find x

    x = %2810%2A6%29%2F3 = 10*2 = 20 days.


ANSWER.  20 days for 15 men to assemble 50 machines.

Solved.

----------------

To see many other similar  (and different)  problems,  solved by the same method,  look into the lesson
    - Rate of work problems
in this site.

This lesson is written specially for you,  to make your horizon  WIDER.



Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
Simple formula without explanation, %28NR%29T=W

Quick setup
system%288%2Ar%2A12=16%2C15%2Ar%2Ax=50%29

%2815rx%29%2F%288%2Ar%2A12%29=50%2F16

%285x%29%2F%288%2A4%29=25%2F8

x=%2825%2F8%29%28%288%2A4%29%2F5%29

x=%285%2A5%2A8%2A4%29%2F%288%2A5%29

x=20---------twenty days

Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


You have, so far, two responses showing very different methods for solving the problem.

There are numerous other methods; below is a solution by the method I find easiest.

The problem asks for how many days it will take. So start with the given number of days, 12, and multiply it by factors based on how the values of the other parameters change.

The given scenario has 8 men; the new scenario has 15 men. More men means fewer days, so the number of days changes by a factor of 8/15.

The given scenario has 16 machines; the new scenario has 50. More machines to make means more days, so the number of days changes by a factor of 50/16.

The calculation is then

%2812%29%288%2F15%29%2850%2F16%29=%2812%2A8%2A50%29%2F%2815%2A16%29

Cancel some common factors before performing the multiplication:

%2812%29%288%2F16%29%2850%2F15%29=%2812%29%281%2F2%29%2810%2F3%29=%286%29%2810%2F3%29=20

ANSWER: 20 days

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
You can also use the worker-time-job formula, which is:

%28W%5B1%5DT%5B1%5D%29%2FJ%5B1%5D%22%22=%22%22%28W%5B2%5DT%5B2%5D%29%2FJ%5B2%5D

where

W1 = the number of workers in the first situation.
T1 = the number of time units (days in this case) in the first situation.
J1 = the number of jobs in the first situation.

W2 = the number of workers in the second situation.
T2 = the number of time units (days in this case) in the second situation.
J2 = the number of jobs in the second situation.

W1 =  8             W2 = 15     
T1 = 12             T2 = the unknown quantity 
J1 = 16             J2 = 50

%28%288%29%2812%29%29%2F16%22%22=%22%22%28%2815%29%28T%5B2%5D%29%29%2F50

96%2F16%22%22=%22%22%28%283%29%28T%5B2%5D%29%29%2F10

6%22%22=%22%22%28%283%29%28T%5B2%5D%29%29%2F10

60%22%22=%22%223T%5B2%5D

20%22%22=%22%22T%5B2%5D

Answer: 20 days

Edwin

Answer by mccravyedwin(405) About Me  (Show Source):
You can put this solution on YOUR website!

Here is what I now believe is the best way to do this kind of problem, and how
to think it out,

I believe it should be taught this way:

I will use COMBINED VARIATION, defined as follows:

Combined variation describes a situation where a variable depends on two (or
more) other variables, and varies directly with some of them and varies
inversely with others (when the rest of the variables are held constant).

We are asked for TIME REQUIRED, so let's see how TIME varies with each of the
other two variables, the number of workers and the number of jobs.

Time required varies DIRECTLY with the number of jobs IF the number of workers
remains constant. 
(The more jobs, the more time required. The less jobs, the less time required.
Obvious!) 

Time required varies INVERSELY with the number of workers IF the number of jobs
remains constant. (The more workers, the less time required.  The less workers,
the more time required. Obvious!) 

Therefore, when we let everything vary, we have a case of combined variation
So to state the combined variation involved:

Time required varies directly with the number of jobs
and inversely with the number of workers.

Let T = time required, J = number of jobs, W = number of workers.

T=k%2Aexpr%28J%2FW%29
8 men take 12 days to assemble 16 machines

12=k%2Aexpr%2816%2F8%29

Solve for k:

12=k%2A2

6=k

Substitute 6 for k:

T=6%2Aexpr%28J%2FW%29
how many days will it take 15 men to assemble 50 machines?

T=6%2Aexpr%2850%2F15%29
  
T=20

answer: 20 days.

Edwin