Problem 33 - Elementary Algebra

Question

For exercises 27-34, evaluate. $$ \frac{15}{19}-\frac{8}{11} $$

Step-by-Step Solution

Answer

\( \frac{13}{209} \)

Step 1: Find the Least Common Denominator (LCD)

Identify the least common denominator for the fractions \( \frac{15}{19} \) and \( \frac{8}{11} \). To find the LCD, multiply the denominators: 19 and 11. LCD = 19 * 11 = 209.

Step 2: Convert the Fractions to Have the LCD

Convert each fraction to an equivalent fraction with the denominator of 209. Convert \( \frac{15}{19} \): \( \frac{15}{19} = \frac{15 \times 11}{19 \times 11} = \frac{165}{209} \). Convert \( \frac{8}{11} \): \( \frac{8}{11} = \frac{8 \times 19}{11 \times 19} = \frac{152}{209} \).

Step 3: Subtract the Numerators

Subtract the numerators of the fractions while keeping the common denominator. \( \frac{165}{209} - \frac{152}{209} = \frac{165 - 152}{209} = \frac{13}{209} \).

Key Concepts

least common denominator

When subtracting fractions, the first step is to find a common denominator. This is known as the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder. In our example, the fractions are \(\frac{15}{19}\) and \(\frac{8}{11}\). To find the LCD, we multiply the denominators together: \(19 \times 11 = 209\). By using the LCD, we can bring the fractions to a common base, making it easier to perform the subtraction.

equivalent fractions

After finding the least common denominator, the next step is to convert each fraction to an equivalent fraction with the denominator of 209. An equivalent fraction is a fraction that has the same value but a different numerator and denominator. To convert \(\frac{15}{19}\): We multiply both the numerator and the denominator by 11 (since 11 is the denominator of the second fraction). \( \frac{15}{19} = \frac{15 \times 11}{19 \times 11} = \frac{165}{209} \)

To convert \(\frac{8}{11}\): We multiply both the numerator and the denominator by 19 (since 19 is the denominator of the first fraction). \( \frac{8}{11} = \frac{8 \times 19}{11 \times 19} = \frac{152}{209} \)

Now, both fractions \(\frac{165}{209}\) and \(\frac{152}{209}\) have the same denominator, making subtraction straightforward.

numerator subtraction

Once the fractions have been converted to equivalent fractions with the same denominator, we proceed to subtract the numerators. The denominator remains the same throughout this process. For our example, we have: \( \frac{165}{209} - \frac{152}{209} \ \)

To subtract, simply subtract the numerators: 165 - 152 = 13.

Then, place the result over the common denominator: \( \frac{165}{209} - \frac{152}{209} = \frac{13}{209} \)

The final fraction \(\frac{13}{209}\) is already in its simplest form. Therefore, our solution is \(\frac{13}{209}\).