Problem 26 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{36}{60}$$

Step-by-Step Solution

Answer

The reduced form of \( \frac{36}{60} \) is \( \frac{3}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify \( \frac{36}{60} \), first, identify the greatest common divisor of the numerator and the denominator. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, and the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The greatest common factor is 12.

Step 2: Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their greatest common divisor, 12. Doing this, \( \frac{36}{60} \) becomes \( \frac{36 \div 12}{60 \div 12} \), which simplifies to \( \frac{3}{5} \).

Step 3: Confirm the Fractions are Equivalent

To ensure that the calculation is correct, check if the fractions \( \frac{36}{60} \) and \( \frac{3}{5} \) are equivalent. Multiply both the numerator and denominator of \( \frac{3}{5} \) by 12 to see if you arrive back at \( \frac{36}{60} \), thereby confirming the simplification is correct.

Key Concepts

Greatest Common Divisor

When simplifying fractions, finding the greatest common divisor (GCD) is an essential step. The GCD is the largest whole number that can evenly divide both the numerator and the denominator without leaving a remainder. In the case of the fraction \( \frac{36}{60} \), finding the GCD involves listing out the factors of both the numerator and denominator. For 36, the factors are:

• 1
• 2
• 3
• 4
• 6
• 9
• 12
• 18
• 36

Similarly, for 60, the factors are:

• 1
• 2
• 3
• 4
• 5
• 6
• 10
• 12
• 15
• 20
• 30
• 60

The greatest factor shared by both numbers is 12, hence the GCD of 36 and 60 is 12. By using the GCD, you can simplify the fraction without altering its value.

Lowest Terms

Reducing a fraction to its lowest terms means simplifying it so that the numerator and denominator have no common factors other than 1. Using the GCD that we've identified, dividing both the numerator and the denominator by this number simplifies the fraction. For example, the fraction \( \frac{36}{60} \) can be simplified by dividing both the numerator and the denominator by 12:

• 36 divided by 12 gives you 3.
• 60 divided by 12 gives you 5.

Thus, \( \frac{36}{60} \) becomes \( \frac{3}{5} \). This fraction, \( \frac{3}{5} \), is now in its simplest form or its lowest terms. No further simplification is possible, as the only common factor of 3 and 5 is 1.

Equivalent Fractions

Understanding equivalent fractions is essential to verifying that your simplified fraction is correct. Equivalent fractions represent the same value or part of a whole, even if their numerators and denominators differ.To verify equivalence of \( \frac{36}{60} \) and \( \frac{3}{5} \), do a quick backward calculation. This means multiplying \( \frac{3}{5} \) by 12 (the original GCD) to check if you return to \( \frac{36}{60} \):

• 3 times 12 equals 36.
• 5 times 12 equals 60.

Hence, \( \frac{36}{60} \) and \( \frac{3}{5} \) are indeed equivalent fractions. This step takes the guesswork out of simplifying fractions and ensures accuracy by confirming that both have the same value when expressed differently.