Loading...
Of the faculty members at a certain university, \(\frac{4}{9}\) are in the School of Liberal Arts and Sciences, \(\frac{1}{5}\) are in the School of Business, \(\frac{1}{6}\) are in the School of Education, and the remaining \(51\) are in the School of Nursing. If no faculty member is in more than one school, how many faculty members are there at the university?
Let's break down what we know in everyday language:
Think of the university faculty like a pie that's divided into four pieces. We know the size of three pieces as fractions, and we know that the fourth piece contains exactly 51 people.
Here's what we have:
• Liberal Arts and Sciences: \(\frac{4}{9}\) of all faculty
• Business: \(\frac{1}{5}\) of all faculty
• Education: \(\frac{1}{6}\) of all faculty
• Nursing: exactly 51 faculty members
The key insight is that these four groups make up ALL the faculty at the university - no one belongs to multiple schools.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Now we need to figure out what fraction of the total faculty the 51 nursing faculty represent.
First, let's find what fraction is already accounted for by the three schools with known fractions. To add fractions, we need a common denominator.
Looking at \(\frac{4}{9}\), \(\frac{1}{5}\), and \(\frac{1}{6}\), let's find the least common multiple of 9, 5, and 6:
• \(9 = 3^2\)
• \(5 = 5\)
• \(6 = 2 \times 3\)
• \(\mathrm{LCM} = 2 \times 3^2 \times 5 = 90\)
Converting each fraction:
• \(\frac{4}{9} = \frac{4 \times 10}{9 \times 10} = \frac{40}{90}\)
• \(\frac{1}{5} = \frac{1 \times 18}{5 \times 18} = \frac{18}{90}\)
• \(\frac{1}{6} = \frac{1 \times 15}{6 \times 15} = \frac{15}{90}\)
Total fraction in the three specified schools: \(\frac{40}{90} + \frac{18}{90} + \frac{15}{90} = \frac{73}{90}\)
This means the nursing school represents: \(1 - \frac{73}{90} = \frac{90}{90} - \frac{73}{90} = \frac{17}{90}\) of all faculty
Now we can set up our equation. We know that:
• The nursing school has 51 faculty members
• The nursing school represents \(\frac{17}{90}\) of the total faculty
If we call the total number of faculty 'T', then:
\(\frac{17}{90}\) of T = 51
In mathematical notation: \(\frac{17}{90} \times T = 51\)
To solve for T, we multiply both sides by \(\frac{90}{17}\):
\(T = 51 \times \frac{90}{17}\)
\(T = 51 \times 90 \div 17\)
\(T = 4590 \div 17\)
\(T = 270\)
Let's verify this makes sense:
• Liberal Arts: \(\frac{4}{9} \times 270 = 120\) faculty
• Business: \(\frac{1}{5} \times 270 = 54\) faculty
• Education: \(\frac{1}{6} \times 270 = 45\) faculty
• Nursing: 51 faculty
• Total: \(120 + 54 + 45 + 51 = 270\) ✓
The total number of faculty members at the university is 270.
This matches answer choice A. 270.
1. Misunderstanding the constraint 'no faculty member is in more than one school'
Students might overlook this crucial constraint and assume there could be overlapping memberships between schools. This would lead them to incorrectly set up the problem, potentially thinking they need to use inclusion-exclusion principles or that the fractions don't need to add up to exactly 1.
2. Incorrectly identifying what the 51 represents
Students may misread the problem and think that 51 is additional faculty beyond the four schools, or that it represents faculty in multiple schools. The key insight that 51 is the exact count for the fourth school (Nursing) and represents the 'remaining' faculty is critical to the correct approach.
3. Setting up the wrong equation structure
Rather than recognizing this as a 'parts of a whole' problem where fractions must sum to 1, students might try to set up separate equations for each school or attempt to use ratios incorrectly, missing the fundamental relationship that all four schools together equal the total faculty.
1. Arithmetic errors when finding the common denominator
Students frequently make mistakes when finding the LCM of 9, 5, and 6, or when converting fractions to the common denominator of 90. For example, they might incorrectly convert \(\frac{4}{9}\) to something other than \(\frac{40}{90}\), or make errors in the LCM calculation itself.
2. Calculation errors when adding fractions
Even with the correct common denominator, students often make arithmetic mistakes when adding \(\frac{40}{90} + \frac{18}{90} + \frac{15}{90}\), potentially getting a sum other than \(\frac{73}{90}\). This error would cascade through the rest of the solution.
3. Division errors in the final calculation
When solving \(T = 51 \times \frac{90}{17}\), students may make computational errors. They might incorrectly calculate \(51 \times 90 = 4590\) or make mistakes in dividing 4590 by 17, leading to an incorrect total that doesn't match any of the answer choices.
1. Failing to verify the answer
Students might arrive at 270 but fail to check their work by verifying that \(\frac{4}{9} \times 270 + \frac{1}{5} \times 270 + \frac{1}{6} \times 270 + 51\) actually equals 270. Without this verification step, they might second-guess their correct answer and select a different choice.
2. Misreading the answer choices
After correctly calculating 270, students might hastily select a similar-looking number like 315 or misread their own calculation, especially if they're rushing or feeling uncertain about their approach.
This problem can be effectively solved using smart numbers by choosing a convenient total that makes the fractional calculations clean.
Step 1: Choose a smart number for the total faculty
Since we're dealing with fractions \(\frac{4}{9}\), \(\frac{1}{5}\), and \(\frac{1}{6}\), we need a number that's divisible by 9, 5, and 6. The LCM of 9, 5, and 6 is 90. Let's use a multiple of 90 that gives us a reasonable remainder when we subtract the known fractions.
Let's try \(T = 270\) (which is \(3 \times 90\)).
Step 2: Calculate faculty in each school using our smart number
• Liberal Arts and Sciences: \(\frac{4}{9} \times 270 = 4 \times 30 = 120\) faculty
• Business: \(\frac{1}{5} \times 270 = 54\) faculty
• Education: \(\frac{1}{6} \times 270 = 45\) faculty
Step 3: Calculate remaining faculty
Remaining faculty = \(270 - 120 - 54 - 45 = 51\) faculty
Step 4: Verify against the problem statement
The problem states that 51 faculty members are in the School of Nursing, which matches our calculation exactly!
Why this smart number approach works:
Instead of setting up algebraic equations, we strategically chose a total that would make fraction calculations simple and then verified it against the given constraint. The fact that our chosen total of 270 produces exactly 51 remaining faculty members (matching the problem statement) confirms this is the correct answer.