Problem 105 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{7}{21}$$

Step-by-Step Solution

Answer

The fraction \(\frac{7}{21}\) reduces to \(\frac{1}{3}\).

Step 1: Identify the greatest common divisor (GCD)

To reduce the fraction \(\frac{7}{21}\) to its lowest terms, we must first identify the greatest common divisor (GCD) of the numerator (7) and the denominator (21).

Step 2: Find the GCD

The factors of 7 are 1 and 7. The factors of 21 are 1, 3, 7, and 21. The greatest common divisor of 7 and 21 is 7, as it is the largest number common to both sets of factors.

Step 3: Divide by the GCD

Divide both the numerator and the denominator by the GCD, 7. This gives us: \(\frac{7 \div 7}{21 \div 7} = \frac{1}{3}\).

Step 4: Confirm the reduced form

Check that the fraction \(\frac{1}{3}\) is in its simplest form. Since 1 and 3 have no common factors other than 1, \(\frac{1}{3}\) is the fraction in its lowest terms.

Key Concepts

Greatest Common Divisor (GCD)

To tackle reducing fractions, the first step is often to identify the greatest common divisor, or GCD. The GCD is the largest number that can evenly divide two or more numbers. Finding the GCD is crucial because it helps us simplify fractions by reducing both the numerator and the denominator by this common factor.
For example, in the fraction \(\frac{7}{21}\), the GCD of 7 and 21 is 7. This number is the key to simplifying the fraction.
To find the GCD, list the factors of each number:

• Factors of 7: 1, 7.
• Factors of 21: 1, 3, 7, 21.

Both numbers share the factor 7, which is the largest factor common to both. Thus, GCD is helpful not only in fraction simplifications but also in solving many algebraic problems.

Reducing Fractions

Once we determine the greatest common divisor (GCD), the next step involves reducing fractions. Reducing a fraction means expressing it in its simplest form, where the numerator and denominator are as small as possible while still keeping the same value.
Let's take \(\frac{7}{21}\) as an example. We identified that the GCD is 7. To reduce the fraction, divide both the numerator and the denominator by this GCD.

• Divide the numerator: 7 ÷ 7 = 1.
• Divide the denominator: 21 ÷ 7 = 3.

After this division, the fraction becomes \(\frac{1}{3}\). This fraction is equivalent to \(\frac{7}{21}\) but is in a simpler form. Reducing fractions makes calculations easier and helps in comparing and understanding the fractions better.
Remember, always check that you've used the largest common divisor, which ensures you have reduced the fraction fully.

Simplest Form

A fraction is said to be in its simplest form when the numerator and denominator have no common factors other than 1. This representation is the most basic form of the fraction, without any possibility of further reduction.
When you simplify a fraction to its simplest form, you involve dividing both parts by their greatest common divisor (GCD). For the fraction \(\frac{1}{3}\) (derived from \(\frac{7}{21}\)), it is already in its simplest form because:

• The numerator 1 has no other factors than itself.
• The denominator 3 is a prime number, meaning its only factors are 1 and 3.

Therefore, there is no way to further reduce \(\frac{1}{3}\). Simplifying fractions to the simplest form is important for clarity in mathematics, as it provides the clearest understanding, making it easier to perform operations like addition, subtraction, and comparison with other fractions.