Problem 97 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{11}{12}-\frac{3}{4} $$

Step-by-Step Solution

Answer

\( \frac{11}{12} - \frac{3}{4} = \frac{1}{6} \)

Step 1 - Find a common denominator

To subtract fractions, both fractions must have the same denominator. Identify the least common multiple (LCM) of the denominators 12 and 4. The LCM of 12 and 4 is 12.

Step 2 - Convert the fractions to have the common denominator

The first fraction \(\frac{11}{12}\) already has the denominator 12. The second fraction \(\frac{3}{4}\) needs to be converted. Multiply both the numerator and denominator of \(\frac{3}{4}\) by 3 to get \(\frac{9}{12}\).

Step 3 - Subtract the fractions

Now subtract \(\frac{9}{12}\) from \(\frac{11}{12}\): \(\frac{11}{12} - \frac{9}{12} = \frac{2}{12}\).

Step 4 - Simplify the fraction

Simplify the resulting fraction \(\frac{2}{12}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2: \(\frac{2}{12} = \frac{1}{6}\).

Key Concepts

least common multiple

To subtract fractions effectively, you must first find a common denominator. The common denominator ensures that both fractions are on the same base for easy subtraction. This involves finding the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. In our case, we have denominators 12 and 4. To find the LCM, list the multiples:

• Multiples of 12: 12, 24, 36...
• Multiples of 4: 4, 8, 12...

The smallest common multiple in both lists is 12. Hence, 12 is our LCM.

simplifying fractions

After performing operations with fractions, the final step is often to simplify the result. Simplifying fractions means reducing the fraction to its simplest form. This involves dividing both the numerator and denominator by their Greatest Common Divisor (GCD). For our problem, after subtracting the fractions, we get \( \frac{2}{12} \). To simplify, find the GCD of 2 and 12, which is 2, and then divide both the numerator and denominator by 2: \( \frac{2}{12} = \frac{1}{6} \).

greatest common divisor

The Greatest Common Divisor (GCD) plays a critical role in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCD of 2 and 12:

• List the factors of 2: 1, 2
• List the factors of 12: 1, 2, 3, 4, 6, 12

The greatest number common to both lists is 2. So, the GCD of 2 and 12 is 2. By dividing the numerator and the denominator by their GCD, you reduce the fraction to its simplest form: \( \frac{2}{12} = \frac{1}{6} \).

common denominators

To subtract fractions with different denominators, convert them to have the same denominator, known as a common denominator. This lets you combine the fractions easily. For our fractions, \( \frac{11}{12} \) and \( \frac{3}{4} \), the common denominator is the LCM of 12 and 4, which is 12. \( \frac{11}{12} \) already has this denominator. Convert \( \frac{3}{4} \) into \( \frac{9}{12} \) by multiplying both the numerator and denominator by 3. Now, both fractions have the common denominator, letting you subtract: \( \frac{11}{12} - \frac{9}{12} = \frac{2}{12} \).