Problem 51 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Write the answer in lowers ferms. See Example 7. $$ \frac{5}{22}-\frac{5}{33} $$

Step-by-Step Solution

Answer

\(\frac{5}{66}\)

Step 1: Identify the Least Common Denominator (LCD)

The denominators of the fractions are 22 and 33. To add or subtract fractions, they need a common denominator. Identify the least common multiple of 22 and 33. The prime factors of 22 are 2 and 11, and for 33, they are 3 and 11. The least common multiple is 66.

Step 2: Convert Each Fraction to the LCD

Convert each fraction to have 66 as the denominator. For \(\frac{5}{22}\), multiply both the numerator and denominator by 3: \(\frac{5 \times 3}{22 \times 3} = \frac{15}{66}\). For \( \frac{5}{33} \), multiply both by 2: \( \frac{5 \times 2}{33 \times 2} = \frac{10}{66}\).

Step 3: Subtract the Fractions

Now that both fractions have the common denominator of 66, subtract the numerators: \(\frac{15}{66} - \frac{10}{66} = \frac{5}{66}\).

Step 4: Simplify the Result

Check if \(\frac{5}{66}\) can be simplified. Since there are no common factors between 5 and 66 other than 1, \(\frac{5}{66}\) is already in its simplest form.

Key Concepts

Least Common Denominator

When working with fraction addition or subtraction, finding the Least Common Denominator (LCD) is crucial. The LCD is the smallest number that both denominators can divide into without leaving a remainder. Having a common denominator allows us to perform operations directly on the numerators.
To identify the LCD, find the Least Common Multiple (LCM) of the denominators. This involves considering the prime factors of each denominator. For instance:

• For 22, the prime factors are 2 and 11.
• For 33, the prime factors are 3 and 11.

The LCM is found by taking the highest power of each prime occurring in the factorizations. In our example, 2, 3, and 11 are the primes. Thus, the LCM is \[2 \times 3 \times 11 = 66\] which becomes our LCD.

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks, which are prime numbers. This step is essential for finding the LCD when adding or subtracting fractions with different denominators.
To perform prime factorization, start by dividing the number by the smallest prime (usually 2), and continue dividing by the next smallest prime until the number is a prime number itself.
For example:

• 22 can be divided by 2, leading to 11, which is a prime. So, 22 = \(2 \times 11\).
• 33 is divisible by 3, resulting in 11, so 33 = \(3 \times 11\).

This step simplifies finding the LCM, which is used as the LCD, ensuring the fractions can be directly compared and calculated upon.

Simplifying Fractions

Once you have performed the operation on fractions, it's essential to simplify them. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator no longer have any common factors other than 1.
To simplify:

• Check for the greatest common factor (GCF) of both numerator and denominator.
• Divide both by the GCF.

In the exercise, \[ \frac{5}{66} \] was checked for simplification. Since 5 and 66 share no common factors other than 1, it was already in its simplest form, demonstrating how simplification verifies the final answer's accuracy.

Numerator Subtraction

After aligning the fractions under a common denominator, focus on subtracting or adding the numerators. This is the key step where you perform the actual operation.
In subtraction:

• Subtract the second numerator from the first while keeping the common denominator unchanged.
• In our problem, subtracting the numerators, 15 and 10, with the common denominator of 66 gives: \(15 - 10 = 5\).

This operation shows the importance of aligning denominators first, as it preserves the fractions' value while allowing us to compute the difference or sum directly in the numerators, keeping calculations straightforward and clear.