Problem 85 - Prealgebra

Question

Conference Attendees At a recent mathematics conference \(\frac{1}{3}\) of the attendees were teachers, \(\frac{1}{4}\) were software salespersons, and \(\frac{1}{12}\) were representatives from various book publishing companies. The remainder of the people in the conference center were employees of the center. What fraction represents the employees of the conference center?

Step-by-Step Solution

Answer

The fraction representing the employees is \(\frac{1}{3}\).

Step 1: Calculate Total Fraction of Attendees Counted

Start by adding up the fractions of attendees who are teachers, software salespersons, and book publishing representatives. \[\frac{1}{3} + \frac{1}{4} + \frac{1}{12}\]

Step 2: Find Common Denominator

The denominators are 3, 4, and 12. The least common multiple of these numbers is 12. Convert each fraction to have a denominator of 12:- Teachers: \(\frac{1}{3} = \frac{4}{12}\)- Software Salespersons: \(\frac{1}{4} = \frac{3}{12}\)- Book Publishing Representatives: \(\frac{1}{12} = \frac{1}{12}\)

Step 3: Add the Fractions

Now, add the fractions together:\[\frac{4}{12} + \frac{3}{12} + \frac{1}{12} = \frac{8}{12}\]Simplify this fraction:\[\frac{8}{12} = \frac{2}{3}\]

Step 4: Calculate Fraction of Employees

The total fraction of attendees at the conference is 1, as this constitutes the whole group. Subtract the combined fraction of the other attendees from 1 to find the fraction representing employees:\[1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}\]

Key Concepts

Adding Fractions

Adding fractions involves combining fractions to see what they amount to altogether. However, one of the main challenges with adding fractions is dealing with different denominators, the bottom numbers of the fractions. To add fractions, they must first have a common denominator. A common denominator is a shared multiple of the denominators of the given fractions.

For example, let's look at the exercise's fractions: \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{1}{12} \). They have different denominators of 3, 4, and 12, respectively. To add these fractions, you would need to convert them into fractions with the same denominator. Once they share a common denominator, you can add the numerators (the top numbers) directly.

In the original exercise, the fractions, once converted, became \( \frac{4}{12} \), \( \frac{3}{12} \), and \( \frac{1}{12} \). You then add the numerators to get \( \frac{8}{12} \). This process is crucial in ensuring that we correctly combine all parts of the fractions involved in the problem.

Least Common Multiple

The least common multiple (LCM) is helpful when you need to find a common denominator for fractions that do not share the same denominator. It is the smallest number that is a multiple of two or more numbers.

In our case, the denominators to find an LCM for are 3, 4, and 12. Think of the LCM as the smallest number that all these denominators can divide into evenly. Working through this problem, we find that the LCM of 3, 4, and 12 is 12 itself because:

• 12 is a multiple of 3 (3 times 4 equals 12)
• 12 is a multiple of 4 (4 times 3 equals 12)
• 12 is also a multiple of itself (1 times 12 equals 12)

Once you have the LCM, you use it as the new common denominator for the fractions you are adding. This simplifies the process, making it easier to add the fractions. Without finding the least common multiple, rewriting fractions with different denominators would be much more complicated.

Fraction Simplification

Fraction simplification makes fractions easier to understand by reducing them to their smallest possible numerator and denominator while maintaining the same value.

This process involves dividing the numerator and the denominator by their greatest common divisor (GCD), which is the largest number that divides both evenly.

Returning to our exercise's result, after adding the fractions we had: \( \frac{8}{12} \). To simplify this fraction, identify the GCD of 8 and 12, which is 4. Divide both the numerator and the denominator by this number:

• Numerator: \( 8 \div 4 = 2 \)
• Denominator: \( 12 \div 4 = 3 \)

The simplified fraction is \( \frac{2}{3} \). Simplifying fractions is important because it presents the fraction in its most basic form, making it easier to understand, compare, or use in other calculations.