Math: Work Scenarios - My Take on it

[Kaushik]

Homework Statement

Professors Lambert and Stassen team-teach an advanced psychology class. Their students recently took their final exams, and the professors are preparing to grade them. Professor Stassen predicts that, if she graded all the exams, she would take 50% more time than professor Lambert would if he graded all the exams. They estimate that it will take them 40 hours to grade all the exams if they grade together. Which of the following equations best represents the situation described?

Relevant Equations

$W=R*t$

My take on it:

I used $W = Rt$ where $W$ is work done, $R$ is rate and t is the time taken
let $W = 1$ and $t$ be the time taken by lambert.
for Lambert:
$1 = R(t)$
$R = \frac{1}{t}$ ... (1)
for Stassen:
$1 = \frac{r}{(3t/2)}$
$r = \frac{2}{(3t)}$...(2)
from (1) and (2),
$R' = \frac{5}{(3t)}$ ... net rate
total time taken when both do together = 40
$W = R'(40)$
but here $W = 2$ as they are grading 2 bundles
$2 = \frac{200}{(3t)}$
$t = \frac{100}{3}$ is the answer I am getting.

Ans given is t = 100

 

[PeroK]

but here $W = 2$ as they are grading 2 bundles

There is only one class of students and one bundle of exam papers. You did everything right except this.

 

[Kaushik]

There is only one class of students and one bundle of exam papers. You did everything right except this.

Ahhhhhh ! So is it supposed to be t = 200/3 ?

 

[PeroK]

Ahhhhhh ! So is it supposed to be t = 200/3 ?

What time are you looking for?

 

[Kaushik]

What time are you looking for?

We just want an equation. Here 't' is the time taken by Lambert.

 

[Kaushik]

This is their solution. I don't get the part where they are saying Rare of Professor Lmaber = 1.5/t

Shouldn't it be 1/(1.5t) = 2/(3t) ?

 

[PeroK]

View attachment 258330

This is their solution. I don't get the part where they are saying Rare of Professor Lmaber = 1.5/t

Shouldn't it be 1/(1.5t) = 2/(3t) ?

First, let's take an example. You should really get into the habit of doing this. a) to see what's going on and b) to get an answer in a specific case.

Let's assume there are 200 exam papers. Together they take 40 hours, so they do 5 an hour between them. That's your $R'$.

Suppose Stassen does 2 per hour and Lambert 3 per hour (that represents Lambert doing them 50% faster than Stassen). That adds up to 5. And, to check, Stassen would take 100 hours to do them all and Lambert 200/3 hours. 200/3 hours + 50% is 300/3 = 100.

That all checks out. Note that all you need for an example is any job that takes them 40 hours between them. In one sense, that example is a complete solution. Although, whoever marks your exam may not agree with me, so I would stop short of suggesting you do this!

I don't like the book solution. Why $\frac 1 x$ instead of just $x$? What is $x$ anyway?

In your solution, you need to clarify what $t$ means.

 

[Kaushik]

In your solution, you need to clarify what $t$ means.

In my solution $t$ represents the time taken by Lambert when she is grading all the papers all by herself.

Now let us consider Lambert to take $t = 200/3$ hours to grade the paper herself.
So time taken by Stassen will be $50%$ more than Lambert. This implies that Time taken by Stassen to grade the paper all by herself will be $200/3 + 100/3 = 100$ hours. From the time, we can deduce their rate.
The rate for Lambert = $3/200$
The rate for Stassen = $1/100$
The net rate is the sum of their respective rates.
Hence, $R' = 3/200 + 1/100$
$R' = 5/200 = 1/40$
Now when both do the work together,
$1 = R'(t)$
$t = 1/R'$
$t = 40$ hours and this matches with the data given in the question.

So is $t = 200/3$ correct? And once again t is the time taken by Professor Lamber when she grades all the paper by herself (without Professors Stassen's help)

 

[PeroK]

Here's a different take. Both your solution and the book's solution are lacking in analysis. An alternative approach is to be explicit about everything. If the problem were more complicated, then I would do it this way. The key is to miss nothing out. For example:

Let $W$ be the total work to be done. Let $t_s, t_l$ be the time it takes Stassen and Lambert to complete the work on their own. Let $t$ be the time it takes to complete the work together. Let $r_s, r_l$ be the rates at which Stassen and Lambert work. Let $r$ be the rate at which they work together.

That's all your variables defined. Now, we can write down what we know.

We are given the relationship in times:

$t_s = 1.5 t_l$

Use this to get the relationship in rates

$r_s = \frac W {t_s} = \frac W {1.5 t_l} = \frac 2 3 \frac W {t_l} = \frac 2 3 r_l$

Note that this proves that the rates are in inverse proportion to the times. I think the book actually assumes you can just write this down. But, there's no harm in checking it out.

And so on ...