Problem 65 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$\frac{13}{28}-\frac{1}{21}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{5}{12}\).

Step 1: Find the Least Common Denominator

To add or subtract fractions, they must have the same denominator. The denominators here are 28 and 21. We need to find the least common multiple (LCM) of these numbers. The prime factorization of 28 is \(2^2 \times 7\) and for 21 it is \(3 \times 7\). The LCM is \(2^2 \times 3 \times 7 = 84\).

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with a denominator of 84. \(\frac{13}{28}\) becomes \(\frac{13 \times 3}{28 \times 3} = \frac{39}{84}\).\(\frac{1}{21}\) becomes \(\frac{1 \times 4}{21 \times 4} = \frac{4}{84}\).

Step 3: Subtract the Fractions

Now, with the same denominators, the fractions can be subtracted:\[\frac{39}{84} - \frac{4}{84} = \frac{39 - 4}{84} = \frac{35}{84}.\]

Step 4: Simplify the Fraction

To simplify \(\frac{35}{84}\), find the greatest common factor (GCF) of 35 and 84. The prime factorization of 35 is \(5 \times 7\) and for 84 it is \(2^2 \times 3 \times 7\). The GCF is 7.Divide both the numerator and the denominator by 7:\[\frac{35}{84} = \frac{35 \div 7}{84 \div 7} = \frac{5}{12}.\]

Step 5: Result

The simplified result of the subtraction is \(\frac{5}{12}\).

Key Concepts

Least Common Denominator

Fractions need a common denominator to be subtracted or added easily. The least common denominator (LCD) is the smallest number that both denominators divide into without a remainder. To find the LCD, you need the least common multiple (LCM) of the denominators—here, 28 and 21.

To find the LCM:

• Use prime factorization.
• For 28, it breaks down to \(2^2 \times 7\).
• For 21, it's \(3 \times 7\).
• Combine these factors using each prime the greatest number of times it appears in either factorization.
• This gives you \(2^2 \times 3 \times 7 = 84\).

By converting each fraction to have this common denominator, subtraction becomes straightforward.

Simplifying Fractions

Once you've performed operations on fractions, simplifying the result makes it more comprehensible. Simplifying fractions is all about reducing both the numerator and denominator to their simplest form, where they only meet at 1 using division.

Here's how to simplify:

• Identify a number that divides both the numerator and the denominator without leaving a remainder. This number is called a common factor.
• If this process can continue until no number except 1 fits as a factor, you've found the simplest form.

In this problem, \(\frac{35}{84}\) simplifies to \(\frac{5}{12}\) as dividing both by 7 simplifies the fraction fully.

Prime Factorization

Prime factorization involves breaking down a number into the set of prime numbers which multiply together to make the original number. It’s a crucial step for finding both the LCM and the GCF; it's at the heart of dealing with fractions.

How to do prime factorization:

• Start by dividing the number by the smallest prime number (usually 2) and continue dividing by primes.
• Record each factor until you’re left with 1.
• E.g., 28 becomes \(2^2 \times 7\) and 21 becomes \(3 \times 7\).

With the prime factors in hand, it's much simpler to calculate the LCM and GCF required for solving fraction problems.

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides two numbers without any remainder. It’s essential for simplifying fractions effectively.

To find the GCF:

• List out the prime factors of each number.
• Identify common factors in both lists.
• The product of these common factors is the GCF.

In this context, the GCF of 35 and 84 is 7, as it's the highest number that will divide both. This allows you to reduce \(\frac{35}{84}\) to \(\frac{5}{12}\), simplifying the fraction into its simplest form.