Question

A given university has an average professor pay of 40,000 a year and an average administrator pay of 45,000 per year. If the ratio of professors to administrators is 4 to 3, and the total pay for professors and administrators in a year is $40,415,000, how many professors does the college have? 
 

• A. 500
• B. 375
• C. 411
• D. 548
• E. 475

Hint:

For problems involving ratios, using a common multiplier (like 'k') is a very powerful technique. It simplifies the setup by reducing the number of variables in your main equation, making it easier to solve.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves setting up and solving a system of equations based on a given ratio and a total amount. We need to find the number of professors.
Step 2: Key Formula or Approach:
Let P be the number of professors and A be the number of administrators.
From the ratio, we can say \(\frac{P}{A} = \frac{4}{3}\). A common way to handle this is to use a multiplier, k.
Let \(P = 4k\) and \(A = 3k\), where k is a positive integer.
The total pay is given by the equation: \[ (\text{Pay per professor} \times P) + (\text{Pay per administrator} \times A) = \text{Total Pay} \] Step 3: Detailed Explanation:
Substitute the given values into the total pay equation: \[ 40,000 \times P + 45,000 \times A = 40,415,000 \] Now substitute our expressions for P and A in terms of k: \[ 40,000(4k) + 45,000(3k) = 40,415,000 \] Calculate the products: \[ 160,000k + 135,000k = 40,415,000 \] Combine the terms with k: \[ 295,000k = 40,415,000 \] Now, solve for k by dividing both sides by 295,000: \[ k = \frac{40,415,000}{295,000} = \frac{40,415}{295} \] To simplify the fraction, we can divide the numerator and denominator by 5: \[ k = \frac{8083}{59} \] Now perform the long division: \[ k = 137 \] The question asks for the number of professors, which is P. \[ P = 4k \] Substitute the value of k we found: \[ P = 4 \times 137 = 548 \] Step 4: Final Answer:
The college has 548 professors.