Problem 32 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{6}{11}-\frac{1}{2}+\frac{3}{8} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{37}{88}\).

Step 1: Identify the Least Common Denominator (LCD)

The denominators in the expression are 11, 2, and 8. To combine these fractions, we need a common denominator. The least common denominator of 11, 2, and 8 is 88.

Step 2: Convert Each Fraction to the LCD

Convert each fraction to have a denominator of 88 by finding equivalent fractions:- \(\frac{6}{11} = \frac{6 \times 8}{11 \times 8} = \frac{48}{88}\)- \(\frac{1}{2} = \frac{1 \times 44}{2 \times 44} = \frac{44}{88}\)- \(\frac{3}{8} = \frac{3 \times 11}{8 \times 11} = \frac{33}{88}\)

Step 3: Perform the Operations

Now, perform the operations using the fractions with the common denominator:\[\frac{48}{88} - \frac{44}{88} + \frac{33}{88} = \frac{48 - 44 + 33}{88} = \frac{37}{88}\]

Step 4: Simplify the Result

Verify that the fraction \(\frac{37}{88}\) is in its simplest form. Since 37 is a prime number and does not divide 88, \(\frac{37}{88}\) is the simplest form.

Key Concepts

Least Common Denominator

How do you handle fractions with different denominators? The answer is by finding the Least Common Denominator (LCD). The LCD is the smallest number that all denominators in a set of fractions can divide evenly. This makes it possible to rewrite each fraction so they can be added or subtracted easily.

In our example, we have fractions with denominators of 11, 2, and 8. To combine them, we need a shared bottom number: our LCD. Through calculation or by listing multiples, we find that 88 is the smallest number that 11, 2, and 8 can all divide into equally.

The process of finding a common denominator ensures that the fractions are speaking the same language, so to speak. With the LCD, adding or subtracting fractions becomes a straightforward task.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. By changing the numerator and denominator proportionally, you can make different-looking fractions comparable. This technique is very effective when working with the Least Common Denominator.

In our example, each original fraction needs to be converted to this common denominator. We multiply both the numerator and the denominator of each fraction to get them all with the same bottom number, which is 88 in this case.

• For \(\frac{6}{11}\), multiplying by \(\frac{8}{8}\) gives us \(\frac{48}{88}\).
• For \(\frac{1}{2}\), multiplying by \(\frac{44}{44}\) results in \(\frac{44}{88}\).
• For \(\frac{3}{8}\), multiplying by \(\frac{11}{11}\) leads to \(\frac{33}{88}\).

Each has a denominator of 88, making them easier to add or subtract. This step allows us to compare apples to apples, by converting all fractions to equivalent forms with the same denominator.

Simplest Form

Once the operations of addition or subtraction are completed, we need to ensure the resulting fraction is in its simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

In our example, after performing the operations: \(\frac{48}{88} - \frac{44}{88} + \frac{33}{88}\), we get \(\frac{37}{88}\). To check if this is in the simplest form, we look at the numbers 37 and 88. The number 37 is prime, meaning it only divides evenly by 1 and itself. Since 37 does not divide 88, \(\frac{37}{88}\) is indeed already in its simplest form.

Simplifying fractions makes them easier to read and understand, but also ensures accuracy in further mathematical operations and comparisons.