Problem 106 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{11}{12}-\frac{7}{15}-\frac{9}{20} $$

Step-by-Step Solution

Answer

The result of the subtraction is 0.

Step 1: Identify the fractions

We have three fractions to work with: \( \frac{11}{12}, \frac{7}{15}, \frac{9}{20} \). Each fraction has different denominators, so we cannot subtract them directly.

Step 2: Find the Least Common Denominator (LCD)

To subtract fractions, we first need a common denominator. The denominators are 12, 15, and 20. We find the Least Common Multiple (LCM) of these numbers. \(\text{LCM}(12,15,20) = 60\). So, 60 is our least common denominator.

Step 3: Rewrite each fraction with the LCD

Convert each fraction to an equivalent fraction with the denominator of 60.- Multiply \(\frac{11}{12}\) by \(\frac{5}{5}\) to get \(\frac{55}{60}\).- Multiply \(\frac{7}{15}\) by \(\frac{4}{4}\) to get \(\frac{28}{60}\).- Multiply \(\frac{9}{20}\) by \(\frac{3}{3}\) to get \(\frac{27}{60}\).

Step 4: Perform the subtraction

With a common denominator of 60, subtract the numerators:\[ \frac{55}{60} - \frac{28}{60} - \frac{27}{60} = \frac{55 - 28 - 27}{60} = \frac{0}{60} \].

Step 5: Simplify the result

The resulting fraction is \(\frac{0}{60}\). Since the numerator is 0, the entire fraction evaluates to 0.

Key Concepts

Understanding the Least Common Denominator

When subtracting fractions, our primary goal is to ensure they all share the same denominator. This shared denominator is known as the Least Common Denominator (LCD). This ensures the fractions are of the same "size" in terms of parts, making them easier to work with. The LCD is essentially the smallest number that each of the original denominators can divide into without leaving a remainder.

• Identify each denominator in the fractions you're working with.
• Determine the Least Common Multiple (LCM) of these denominators.
• This LCM is your LCD, which we use to rewrite the fractions.

For example, with the fractions \( \frac{11}{12} \), \( \frac{7}{15} \), and \( \frac{9}{20} \), the denominators are 12, 15, and 20. We find the LCM of 12, 15, and 20 to be 60, making it our LCD. By converting each fraction to an equivalent one with a denominator of 60, subtraction becomes much more straightforward.

Grasping Equivalent Fractions

Equivalent fractions are fractions that represent the exact same value, even though they may look different. They are created by multiplying or dividing both the numerator and the denominator by the same number. This does not alter the fraction's value, it merely changes its form.

• Ensure consistency by multiplying or dividing both the top and bottom by the same number.
• Equivalent fractions are crucial for comparing, adding, or subtracting fractions with different denominators.

In our example, to make \( \frac{11}{12} \), \( \frac{7}{15} \), and \( \frac{9}{20} \) have the same denominator of 60, we multiply: For \( \frac{11}{12} \): Multiply both the numerator and the denominator by 5 to get \( \frac{55}{60} \). For \( \frac{7}{15} \): Multiply both by 4 to get \( \frac{28}{60} \). For \( \frac{9}{20} \): Multiply both by 3 to get \( \frac{27}{60} \). Once these fractions have the same denominator, they become easy to subtract or add.

Subtracting Fractions Effectively

Subtracting fractions involves a few steps. However, once you have a common denominator, it’s straightforward. This process allows you to handle fractions by turning them into forms with matching bottoms (denominators).

• Convert the fractions to have the Least Common Denominator (as outlined above).
• With a common denominator established, subtract the numerators directly.
• Simplify the result if possible, though sometimes it may already be as simple as can be.

For the example given with fractions \( \frac{55}{60} \), \( \frac{28}{60} \), and \( \frac{27}{60} \), we subtract by: Step 1: Subtract \( 55 - 28 \) to get \( 27 \). Step 2: Subtract \( 27 - 27 \) to finally get \( 0 \). Step 3: This gives us \( \frac{0}{60} \), which simplifies neatly to 0. The key is ensuring all fractions share the same denominator before subtracting their numerators, and always check if the result can be simplified further.