Problem 221 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{3}{8}-\frac{1}{6}+\frac{3}{4} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{23}{24}\).

Step 1: Find a common denominator

To simplify the expression \(\frac{3}{8}-\frac{1}{6}+\frac{3}{4}\), first find a common denominator for the fractions. The least common multiple (LCM) of 8, 6, and 4 is 24.

Step 2: Convert the fractions

Convert each fraction to an equivalent fraction with the common denominator of 24: \(\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\), \(\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}\), and \(\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}\).

Step 3: Combine the fractions

Now, substitute the equivalent fractions back into the expression: \(\frac{9}{24} - \frac{4}{24} + \frac{18}{24}\).

Step 4: Perform the addition and subtraction

Add and subtract the fractions: \(\frac{9}{24} - \frac{4}{24} = \frac{5}{24}\) and then \(\frac{5}{24} + \frac{18}{24} = \frac{23}{24}\).

Step 5: Simplify the final result

The final simplified result of the expression \(\frac{3}{8} - \frac{1}{6} + \frac{3}{4}\) is \(\frac{23}{24}\).

Key Concepts

Common Denominator

When simplifying fractions in expressions, finding a common denominator is essential. A common denominator allows you to combine fractions by giving them the same bottom number. This simplifies both the addition and subtraction of fractions.

To find a common denominator:

• List the denominators of all the fractions involved.
• Determine the least common multiple (LCM) of these numbers. This will be your common denominator.

For example, in the exercise \(\frac{3}{8}-\frac{1}{6}+\frac{3}{4}\), the denominators are 8, 6, and 4. To find the common denominator, calculate the least common multiple of these numbers.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of all the given numbers. To find the LCM, you can use several methods such as prime factorization or listing the multiples.

Here are the steps for finding the LCM using listing multiples:

• List a few multiples of each number.
• Identify the smallest number that appears in all lists.

For the numbers 8, 6, and 4:

• Multiples of 8: 8, 16, 24, 32, 40, ...
• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, ...

The smallest multiple common to all three lists is 24. Thus, 24 is the LCM.

Combining Fractions

Once you have a common denominator, you can combine the fractions easily. This involves converting each fraction to an equivalent fraction with the new common denominator and then adding or subtracting the numerators.

Here are the steps to combine fractions:

• Convert each fraction to have the common denominator by multiplying the numerator and denominator by the same number.
• Write each fraction with the common denominator, then add or subtract the numerators and place the result over the common denominator.

For example, in the exercise \(\frac{3}{8}-\frac{1}{6}+\frac{3}{4}\), once we convert fractions:

• \(\frac{3}{8} = \frac{9}{24}\)
• \(\frac{1}{6} = \frac{4}{24}\)
• \(\frac{3}{4} = \frac{18}{24}\)

Combine these fractions:

• \(\frac{9}{24} - \frac{4}{24} + \frac{18}{24} = \frac{23}{24}\)

Therefore, \(\frac{3}{8} - \frac{1}{6} + \frac{3}{4} = \frac{23}{24}\), which is the simplified result.