Verified Answer :

  • Identify the common denominator of 8, 4, 2, and 3, which is 24.
  • Convert each fraction to have the denominator 24: \(\frac{9}{24}\), \(\frac{6}{24}\), \(\frac{12}{24}\), and \(\frac{16}{24}\).
  • Perform the arithmetic on the numerators: \(9 + 6 + 12 - 16 = 11\).
  • State the final fraction: \(\boxed{\frac{11}{24}}\).

Explanation

  1. Analyzing the Problem

To solve the expression \(\frac{3}{8} + \frac{1}{4} + \frac{1}{2} - \frac{2}{3}\), we need to perform addition and subtraction of fractions. The first step in adding or subtracting fractions is to find a common denominator for all the fractions involved. The denominators here are 8, 4, 2, and 3.

  1. Finding a Common Denominator

To find the least common multiple (LCM) of the denominators 8, 4, 2, and 3:

  • Prime factorization of 8 is \(2^3\).
  • Prime factorization of 4 is \(2^2\).
  • Prime factorization of 2 is \(2^1\).
  • Prime factorization of 3 is \(3^1\).

The LCM is the product of the highest power of each prime factor, which is \(2^3 \times 3^1 = 8 \times 3 = 24\). Thus, 24 is our common denominator.

  1. Converting Fractions

Now, we convert each fraction to an equivalent one with 24 as the denominator:

  • \(\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\)
  • \(\frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}\)
  • \(\frac{1}{2} = \frac{1 \times 12}{2 \times 12} = \frac{12}{24}\)
  • \(\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\)
  1. Final Calculation

Now combine the fractions:
\(\frac{9}{24} + \frac{6}{24} + \frac{12}{24} - \frac{16}{24} = \frac{9 + 6 + 12 - 16}{24}\)
Performing the numerator calculation: \(9 + 6 = 15\), \(15 + 12 = 27\), and \(27 - 16 = 11\).
The final result is \(\frac{11}{24}\). Comparing this to the options provided, it matches the first option. The final answer is \(\boxed{\frac{11}{24}}\).

Examples

Knowing how to add and subtract fractions with different denominators is essential in everyday tasks, such as cooking or construction. For instance, if a recipe calls for \(\frac{3}{4}\) cup of flour and you decide to use \(\frac{1}{2}\) cup instead, or if you need to combine several partial measurements of wood for a shelf, you must use a common denominator to accurately calculate the total or the remaining amount needed. This simple arithmetic operation ensures precision in your projects and prevents mistakes when dealing with measurements.

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