Problem 82 - Prealgebra

Question

Popular Majors Enrollment figures show that the most popular programs at a local college are liberal art studies and business programs. The liberal arts studies program accounts for \(\frac{1}{5}\) of the student enrollment while business programs account for \(\frac{1}{10}\) of the enrollment. What fraction of student enrollment chooses one of these two areas of study?

Step-by-Step Solution

Answer

The fraction of students enrolled in either program is \( \frac{3}{10} \).

Step 1: Understand the Problem

The problem asks us to find out what fraction of students is enrolled in either liberal arts studies or business programs, given that these account for \( \frac{1}{5} \) and \( \frac{1}{10} \) of the enrollment, respectively.

Step 2: Express Fractions in Common Denominator

The fractions \( \frac{1}{5} \) and \( \frac{1}{10} \) must be expressed with a common denominator to be easily added. The least common denominator of 5 and 10 is 10. So, we express \( \frac{1}{5} \) as \( \frac{2}{10} \) (since \( \frac{1}{5} = \frac{2}{10} \)).

Step 3: Add the Fractions

Now, we add the fractions \( \frac{2}{10} \) and \( \frac{1}{10} \). Adding these fractions gives us \( \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \).

Step 4: Simplify the Result (if required)

The sum \( \frac{3}{10} \) cannot be simplified further, so the fraction \( \frac{3}{10} \) represents the total student enrollment in either liberal arts studies or business programs.

Key Concepts

Common Denominator

Before you can add or subtract fractions, you must make sure they share a common denominator. The denominator is the bottom part of the fraction; it tells you how many equal parts the whole is divided into. When fractions have different denominators, like \( \frac{1}{5} \) and \( \frac{1}{10} \), you can’t directly add them together like whole numbers.

• Finding a common denominator: To find a common denominator, you need to identify a number that is a multiple of both denominators. Here, the denominators are 5 and 10. The smallest number that both can divide evenly into is 10, making it our common denominator.
• Equivalent fractions: Once you have the common denominator of 10, convert each fraction so they both have the same bottom number. This means \( \frac{1}{5} = \frac{2}{10} \), since 5 goes into 10 two times.

Once fractions have a common denominator, they’re much more manageable for addition or subtraction.

Adding Fractions

After establishing a common denominator, you can proceed to adding the fractions. When fractions share the same bottom number, you only need to add the numerators, the top numbers, and keep the denominator the same.
Enjoy the simple process:

• Identifying numerators: With fractions \( \frac{2}{10} \) and \( \frac{1}{10} \), we can now proceed to add them by focusing only on the numerators: 2 and 1.
• Adding the numerators: Add these numbers together: \( 2 + 1 = 3 \). The denominator remains the same (10).

So, \( \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \). It's as simple as that! This fraction, \( \frac{3}{10} \), now represents the total fraction of students enrolled in either liberal arts or business programs at the college.

Student Enrollment Analysis

Understanding fractions in the context of student enrollment can be quite insightful for schools and colleges. Analyzing which programs attract more students helps institutions plan better resources and allocate support where needed. By knowing that 30% (or \( \frac{3}{10} \)) of students are enrolled in two popular programs, institutions can make informed decisions.

• Resource allocation: If these fractions suggest large enrollments, colleges might decide to enhance facilities related to liberal arts and business studies, ensuring quality education.
• Trend Analysis: Over time, enrollment fractions can highlight trends. Increasing fractions for certain programs might prompt expansion or maintenance strategies.

Overall, fraction analysis in student enrollment offers a comprehensive overview that enables educational frameworks to be tailored to current student interests and needs.