Question

A math teacher finishes a specific homework assignment three times as fast as the quickest student. If it takes the teacher 40 minutes to finish the assignment, which of the following equations can be used to determine how many minutes it would take for both the teacher and the quickest student to finish the assignment if they work together?

Answer 1

hmm... say the speed of the teacher is 3x , and the speed of the student is x . if there is a assignment of work W . then the time taken by the teacher is 40=W/3x [like distance covered by speed] . Then if both combine and solve the assignment , their speed will be 4x ie they will do the work in W/4x time but { 1/3*(W/x) =40 ; W/x = 120 } 1/4*W/x = (1/4)*120 = 30 .

So they will complete the assignment in 30 mins .

Answer 2

This one is a bit tricky. There is an intuitive and non intuitive solution.

Non intuitive.
The rate of teacher is Rt say and the rate of the student is Rs. ****So Rt = 3Rs since the teacher's rate is 3x of the student.

Now they have a common workload to complete, say W.
Now work rate = workload/time, so we can say ****Workload (W) = rate x time = 40 Rt for teacher or 120 Rs for student (it takes 3 times as long)

Now for this task, the student and teacher work in parallel. So the sum of their workloads must add to the complete workload of any one person.
Let the teachers workload in their combined session be Wt for teacher and the student's Ws.

****Now Ws + Wt = W (total workload) = 120Rs
Also workload = rate x time, so Ws = Rs T and Wt = Rt T where T is the time in minutes. Since both complete the task at the same time, we just write T rather than Ts and Tt.

Sub in Ws and Wt into equation and you get
RsT + RtT = 120Rs. Almost done! Replace Rt with 3Rs and you get:

RsT + 3RsT = 120Rs
T(4Rs) = 120Rs so T = 120Rs/(4Rs) = 30 minutes.

Answer 3

The intuitive way is this:

The teacher always does things 3 times as quick as the student. So what a student does in a given time, the teacher must do 3 times that amount of work.

So if effective time of student is t, the teacher does 3t worth of work. Altogether they do 3t + t = 4t worth of work.

Now 4t worth of work is enough for teacher to complete 40 minutes of work.
So t = 40/4 = 10 minutes. But that's the teacher. The student will need to do 30 minutes. As they do it together and finish the same time, together they take 30 minutes.

Alternatively you could say 4t worth of work is 120 minutes of student's work. So t = 120/4 = 30 minutes of student's work. As the longer time (compared to the teacher), it is the answer.