Problem 71 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{10}{6}$$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{5}{3}\).

Step 1: Identify the Greatest Common Divisor (GCD)

Find the greatest common divisor (GCD) of the numerator and the denominator in the fraction \(\frac{10}{6}\). The numbers are 10 and 6. We list the divisors of each: the divisors of 10 are 1, 2, 5, 10, and the divisors of 6 are 1, 2, 3, 6. The largest common divisor is 2.

Step 2: Divide by the GCD

Divide both the numerator and the denominator of the fraction \(\frac{10}{6}\) by the greatest common divisor, which is 2. This gives \(\frac{10 \div 2}{6 \div 2} = \frac{5}{3}\).

Step 3: Simplified Fraction

Ensure that no further simplification is possible. The fraction is now \(\frac{5}{3}\), and the only divisors of 5 and 3 are 1, therefore \(\frac{5}{3}\) is in its lowest terms.

Key Concepts

Greatest Common Divisor

When reducing fractions, one important step is finding the Greatest Common Divisor (GCD). This is the largest number that can evenly divide both the numerator and the denominator of a fraction. It helps us simplify fractions to make them easier to understand and work with.
To find the GCD, list out all the divisors for each number. For example, let's consider the fraction \(\frac{10}{6}\). The divisors of 10 are 1, 2, 5, and 10. The divisors of 6 are 1, 2, 3, and 6. The common divisors are 1 and 2, and the greatest one is 2.

• Listing divisors helps clearly identify common factors.
• Picking the greatest one ensures the fraction is reduced as much as possible.
• Always compare all divisors to find the GCD accurately.

Simplifying Fractions

Once you have determined the GCD, the next step is simplifying the fraction by dividing both the numerator and the denominator by this GCD. This step is essential because it transforms the fraction into a simpler, but equal expression. For example, in our fraction \(\frac{10}{6}\), the GCD is 2. We divide both 10 and 6 by 2, resulting in the fraction \(\frac{5}{3}\).
Simplifying fractions makes them easier to compare and use in calculations.

• Divide both top and bottom by the GCD.
• Check your result by multiplying back to ensure equality.
• Recall that the value of the fraction remains unchanged.

Lowest Terms

A fraction is in its lowest terms when the numerator and the denominator have no common divisors other than 1. This means it cannot be simplified further. Ensuring a fraction is in its lowest terms is crucial for clarity, especially in math problems or when comparing fractions. In the example of \(\frac{5}{3}\), since 5 and 3 have no common factors except for 1, this fraction is in its lowest terms.
Checking lowest terms confirms the simplification process is complete.

• A simple fraction means simpler math.
• Always check if more simplification is possible.
• Using lowest terms aids in comparison and clarity.