How would you set this problem up?

Question

Together, M, B, and K, can do a job in 80 minutes. To do the job alone, M, needs twice the time that B needs and two hours more than K. How long would it take each to complete the job alone?

Answer 1

Okay letting x be how long it would take b to do job.
Then it would take m 2x hours to do jo, b would take x hours, and k 2x-2 hours.
Or, m would do 1/2 job in x hours, b would do 1 job in x hours, and k would do (x/(2x-2)) job in x hours.

So, setting up equation, ((1/2)/x)+(1/x)+((x/(2x-2))/x) = 1/(4/3).
(Note:80 minutes equals 4/3 hour)

This ends up being 6x^2-22x+12=0.
Divide by 2 and factor, and you get (3x-2)(x-3)=0.

Now 3x-2 wouldn't make sense, because it would give k negative hours.

SO, x-3=0, so x=3.
SO, m did job in 3*2=6 hours, b did job in 3 hours, and k did job in 6-2=4 hours.

Answer 2

The key to setting this up is to understand the difference between Rate (how fast I do something) vs Time (how long it takes to do something).

Also, keep a close eye on units - minutes vs hours.

We are told that M, B, and K together accomplish a job in 80 minutes. Each of them work all 80 minutes so T (time) is 80 for all of them but each works at a different rate.

Lets have M be the rate that M works at. B is B's rate and K is K's rate.

So we are told that 80*M + 80*B + 80*K = J (for job) to complete the job

Now we are told that the same job will take an unknown amount of time for M to complete:

M * T = J

We know that B does it in half the time

T/2 * B = J

and K takes 2 hours less (120 minutes)

(T-120) * K = J

Restate these last 3 equations in terms of just T

M = J/T
B = 2*J/T
K = J/(T-120)

So:
80(M+B+K) = J
80(J/T + 2J/T + J/(T-120)) = J
divide each side by 80J to get:
1/T + 2/T + 1/(T-120) = 1/80
((T-120) + 2(T-120) + 1T) / T(T-120) = 1/80
(T - 120 + 2T - 240 + T) / (T^2 - 120T) = 1/80
T + 2T + T - 120 - 240 = (T^2 - 120T)/80
80(4T - 360) = T^2 - 120T
320T - 28800 = T^2 - 120T
T^2 -440T + 28800 = 0

Solve the quadratic:

T = 80 or 360

The answer must be 360 for the time for M to complete the task alone since 80 would be equal to the time it takes for all 3

Therefore:

Time for M = 360 minutes = 6 hours
Time for B = 180 minutes = 3 hours
Time for K = 240 minutes = 4 hours

Answer 3

MBK = 80 mins
M= B x 2 / K + 2 hrs
B= unsure
K= M - 2 hrs

sorry its not the best setup, i dont know how to really explain B

Answer 4

MBK=80 minutes
M amount of time=x
B=x/2
K=x-2 hours

~queenpan1295