Problem 48 - Algebra A Combined Function

Question

Add or subtract as indicated. See Examples 15 through \(20,22,\) and \(23 .\) $$ \frac{11}{12}-\frac{1}{16} $$

Step-by-Step Solution

Answer

The result is \( \frac{41}{48} \).

Step 1: Identify the Operation

We need to subtract two fractions: \( \frac{11}{12} - \frac{1}{16} \). This requires that both fractions have a common denominator before we can perform the subtraction.

Step 2: Find Common Denominator

Determine the least common denominator (LCD) of the denominators 12 and 16. The prime factors are 12 = 2² × 3 and 16 = 2⁴. The LCD will be the highest power of each prime factor, which is 2⁴ × 3 = 48.

Step 3: Adjust Fractions

Convert each fraction to an equivalent fraction with the common denominator of 48.For \( \frac{11}{12} \): Multiply both the numerator and the denominator by 4 to get \( \frac{44}{48} \).For \( \frac{1}{16} \): Multiply both the numerator and the denominator by 3 to get \( \frac{3}{48} \).

Step 4: Perform Subtraction

Now that both fractions have the same denominator, we can subtract the numerators:\[ \frac{44}{48} - \frac{3}{48} = \frac{44 - 3}{48} = \frac{41}{48} \].

Step 5: Simplify the Result

Check to see if \( \frac{41}{48} \) can be simplified. Since 41 is a prime number and does not divide 48, \( \frac{41}{48} \) is already in its simplest form.

Key Concepts

Least Common Denominator

When you need to subtract fractions, as in the problem \( \frac{11}{12} - \frac{1}{16} \), it's vital to first find a common denominator. This ensures that you are comparing apples to apples, so to speak. The least common denominator (LCD) is the smallest number that both denominators (12 and 16, in this case) can divide into evenly.

To find the LCD, start by determining the prime factorization of each denominator:

• The number 12 breaks down into \(2^2 \times 3\).
• The number 16 is \(2^4\).

The LCD is found by taking the highest power of each prime factor present in any of the numbers:

• For the prime number 2, the highest power is \(2^4\).
• For the prime number 3, it's \(3\).

Multiply these together to get the LCD: \(2^4 \times 3 = 48\).

With the LCD, you're set to convert the fractions to a common denominator, facilitating a smooth subtraction process!

Converting Fractions

Once the least common denominator (LCD) is identified, the next important step is converting each fraction to this common denominator. In our example, after finding the LCD to be 48, both fractions \(\frac{11}{12}\) and \(\frac{1}{16}\) need to be converted.

To convert a fraction:

• Multiply both the numerator and the denominator of \(\frac{11}{12}\) by 4, resulting in \(\frac{44}{48}\).
• Multiply both the numerator and the denominator of \(\frac{1}{16}\) by 3, giving \(\frac{3}{48}\).

Now both fractions have the same denominator, 48, making it possible to add or subtract them directly. Converting fractions like this ensures you're working with equivalent fractions, preserving the original value but allowing for smoother arithmetic operations.

Simplifying Fractions

After performing the subtraction \( \frac{44}{48} - \frac{3}{48} = \frac{41}{48} \), the final step is to simplify the resulting fraction. This means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

For \( \frac{41}{48} \):

• Check to see if there are any common factors between 41 and 48. Since 41 is a prime number, it doesn't divide 48.
  This indicates \( \frac{41}{48} \) is already simplified.
• If the numbers had common factors, you would divide both the numerator and the denominator by their greatest common divisor (GCD) to simplify.

Simplifying fractions makes them easier to understand and use in further calculations. Even if a result doesn’t reduce, checking for simplification is a good habit to develop.