Problem 63 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{6}{14}$$

Step-by-Step Solution

Answer

The fraction \( \frac{6}{14} \) reduces to \( \frac{3}{7} \).

Step 1: Identify Common Factors

To reduce the fraction \( \frac{6}{14} \), we start by identifying the greatest common factor (GCF) of the numerator (6) and the denominator (14). The factors of 6 are 1, 2, 3, 6, and the factors of 14 are 1, 2, 7, 14. The largest factor they have in common is 2.

Step 2: Divide by the GCF

Now that we've found the greatest common factor to be 2, we divide both the numerator and the denominator by this common factor. \( \frac{6}{14} = \frac{6 \div 2}{14 \div 2} \).

Step 3: Simplify the Fraction

After performing the division, we get \( \frac{3}{7} \). Check if the fraction is in its lowest terms. 3 and 7 have no common factors other than 1.

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a crucial concept when it comes to simplifying fractions. It helps us find the largest number that divides two or more numbers without leaving a remainder.

To determine the GCF of two numbers, such as 6 and 14, we need to list all the factors of these numbers. Factors are the numbers that multiply together to get another number. For example:

• The factors of 6 are 1, 2, 3, and 6.
• The factors of 14 are 1, 2, 7, and 14.

Next, we identify the largest factor that both numbers share.

In this case, both 6 and 14 share the factor 2 as the largest.

Understanding the GCF is essential because it tells us how we can break down fractions into simpler forms by using what the numerator and denominator have in common.

Reducing Fractions

Reducing fractions, sometimes called simplifying fractions, involves dividing the numerator and the denominator by their greatest common factor. This process ensures the fraction is expressed in the simplest form possible.

In the given problem, we have the fraction \( \frac{6}{14} \). We've already identified that the GCF is 2. This means we can simplify the fraction by dividing both the numerator and denominator by this number.

That leads to:

• \( 6 \div 2 = 3 \)
• \( 14 \div 2 = 7 \)

Thus, the fraction \( \frac{6}{14} \) reduces to \( \frac{3}{7} \).

This process not only makes the fraction simpler but also makes it easier to work with, whether you are adding, subtracting, or comparing fractions.

Lowest Terms

When a fraction is in its lowest terms, it means there are no common factors other than 1 between the numerator and denominator.

• This implies the fraction cannot be simplified any further.

In our example, after reducing \( \frac{6}{14} \) to \( \frac{3}{7} \), we should verify it's in the lowest terms.

To do this, check the factors of 3 and 7:

• Factors of 3: 1 and 3.
• Factors of 7: 1 and 7.

The only common factor between 3 and 7 is 1.

Therefore, \( \frac{3}{7} \) is in its lowest terms. Expressing a fraction in lowest terms not only makes it more elegant but also makes computational operations on it straightforward. Most importantly, it helps in better understanding and interpreting mathematical problems.