Professors Lambert and Stassen teamteach 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?

Options

A. `x+(3)/(2)x=40`
B. `(1)/(x)+(2)/(3x)=(1)/(40)`
C. `(1)/(x)+(50)/(x)=(1)/(40)`
D. `(2)/(x)+(3)/(x)=(1)/(40)`

Explanation

To solve the problem systematically, we will follow these steps:

### Step 1: Define Variables
Let \( t \) be the time (in hours) that Professor Lambert takes to grade all the exams. Since Professor Stassen takes 50% more time than Professor Lambert, her time will be \( 1.5t \).

### Step 2: Write the Rates of Work
The rate of work for each professor can be expressed as:
- For Professor Lambert:
\[ \text{Rate of Lambert} = \frac{W}{t} \]
- For Professor Stassen:
\[ \text{Rate of Stassen} = \frac{W}{1.5t} = \frac{W}{\frac{3}{2}t} = \frac{2W}{3t} \]
- When they work together, they can complete the grading in 40 hours:
\[ \text{Rate together} = \frac{W}{40} \]

### Step 3: Set Up the Equation
Since the rates of work add up when they work together, we can write the equation:
\[ \frac{W}{40} = \frac{W}{t} + \frac{2W}{3t} \]

### Step 4: Simplify the Equation
We can cancel \( W \) from all terms (assuming \( W \neq 0 \)):
\[ \frac{1}{40} = \frac{1}{t} + \frac{2}{3t} \]

### Step 5: Combine the Right Side
To combine the fractions on the right side, we need a common denominator, which is \( 3t \):
\[ \frac{1}{t} + \frac{2}{3t} = \frac{3}{3t} + \frac2{3t} = \frac{5}{3t} \]
Thus, we have:
\[ \frac{1}{40} = \frac{5}{3t} \]

### Step 6: Cross Multiply
Cross-multiplying gives us:
\[ 3t = 40 \times 5 \]
\[ 3t = 200 \]

### Step 7: Solve for \( t \)
Now, divide both sides by 3:
\[ t = \frac{200}{3} \approx 66.67 \text{ hours} \]

### Final Equation
The equation that represents the situation described is:
\[ \frac{1}{40} = \frac{1}{t} + \frac{2}{3t} \]