Problem 27 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{2}{3}+\frac{5}{11} $$

Step-by-Step Solution

Answer

\(\frac{37}{33}\)

Step 1: Find a Common Denominator

To add fractions with different denominators, first find the least common multiple of the denominators. Here, the denominators are 3 and 11. The least common multiple of 3 and 11 is 33, so the common denominator is 33.

Step 2: Adjust the Fractions

Change each fraction to have the common denominator of 33. Multiply the numerator and the denominator of \(\frac{2}{3}\) by 11 to get \(\frac{22}{33}\). Multiply the numerator and the denominator of \(\frac{5}{11}\) by 3 to get \(\frac{15}{33}\).

Step 3: Add the Fractions

With the fractions \(\frac{22}{33}\) and \(\frac{15}{33}\) now having the same denominator, you can add them directly. The sum is \(\frac{22+15}{33} = \frac{37}{33}\).

Step 4: Simplify If Necessary

Check if the fraction \(\frac{37}{33}\) can be simplified. Since 37 is a prime number and does not divide evenly into 33, the fraction is already in its simplest form.

Key Concepts

Understanding a Common Denominator

When adding fractions, it's important to have a common denominator — a shared, uniform bottom number for each fraction. This makes the fractions comparable and easy to combine. Imagine you have two different kinds of pies: one cut into three pieces and another into eleven. To fairly combine slices from each pie, you need to re-slice them into the same number of pieces. This re-slicing is done by finding a common denominator. It ensures every fraction represents parts of the same whole.

• Having a common denominator helps make fractions "speak the same language".
• It is crucial for accurate calculation and comparison of fractions.

To combine \( \frac{2}{3} \) and \( \frac{5}{11} \), their denominators needed adjustment to a common ground: 33. This was achieved by converting each fraction to an equivalent fraction with 33 as the denominator. Each equivalent fraction represents the same value in different form.

Finding the Least Common Multiple

Finding the least common multiple (LCM) is a strategy to determine the smallest number that is a multiple of two or more numbers. When dealing with denominators, the LCM is essentially the smallest shared multiple, which becomes the common denominator for fractions. For example, with denominators 3 and 11, you'd list the multiples of each:

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ...
• Multiples of 11: 11, 22, 33, 44, ...

The first common number is 33, which is the LCM. This number forms the basis of converting each original fraction so they can be added seamlessly.
It's like finding a team meeting time that fits into everyone's schedule. LCM simplifies the process, just as a common scheduling slot simplifies planning group activities.

Simplifying Fractions

Once fractions are added, the resulting fraction should be checked to see if it can be simplified. Simplifying is about reducing the fraction to its simplest form, where the greatest common divisor (GCD) of the numerator and the denominator is 1. This process reduces the numerator and the denominator while maintaining the equivalent value of the fraction.Even when a fraction might look complex at first glance, simplifying it reflects its true, simple form with the least, necessary components.

• In our example, the sum was \( \frac{37}{33} \).
• Since 37 is a prime number and does not divide evenly with 33, no further simplification is possible.

Thus, the fraction \( \frac{37}{33} \) remains as it is, fully reduced and accurately representing the sum of the original fractions.