Problem 15 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. \(\frac{48}{64}\)

Step-by-Step Solution

Answer

The fraction \( \frac{48}{64} \) reduced to lowest terms is \( \frac{3}{4} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{48}{64} \) to its lowest terms, first identify the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both 48 and 64 without leaving a remainder.

Step 2: Find the GCD of 48 and 64

To find the GCD, list the factors of 48 and 64. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64. The largest common factor is 16, so the GCD is 16.

Step 3: Divide Both the Numerator and Denominator by the GCD

Divide both the numerator (48) and the denominator (64) by their GCD (16). Perform the division: \( 48 \div 16 = 3 \) and \( 64 \div 16 = 4 \).

Step 4: Write the Reduced Fraction

After performing the division from the previous step, write the resulting fraction: \( \frac{3}{4} \). This is the fraction in its lowest terms.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when it comes to reducing fractions. It helps identify the largest number that can divide two integers without leaving a remainder.
This number is essential for simplifying fractions to their simplest form. Here's how to pick the GCD:

• List the factors of both numbers. For example, for 48, the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
• For 64, the factors are 1, 2, 4, 8, 16, 32, and 64.

The greatest common divisor is the largest factor shared by both numbers, which in this case, is 16.
Knowing the GCD allows you to simplify a fraction efficiently by dividing both the numerator and the denominator by this common factor.

Numerator and Denominator

When dealing with fractions, it's important to understand what numerators and denominators are. These are the two essential parts of a fraction:

• The numerator is the top number in a fraction. It indicates how many parts of a whole are being considered. In the fraction \( \frac{48}{64} \), the numerator is 48.
• The denominator is the bottom number. It tells us how many equal parts make up the whole. In \( \frac{48}{64} \), the denominator is 64.

Reducing a fraction involves finding a new numerator and denominator that are smaller but still represent the same value. This is where the GCD comes into play, simplifying both numbers while maintaining the ratio.

Lowest Terms

A fraction is in its lowest terms when the numerator and the denominator have no common divisors other than 1. This means the fraction cannot be simplified any further.Here's how to reduce a fraction like \( \frac{48}{64} \) to its lowest terms:

• First, find the GCD of the numerator and the denominator, which is 16 as discussed earlier.
• Next, divide both the numerator (48) and the denominator (64) by the GCD. For 48, it's \( 48 \div 16 = 3 \), and for 64, it's \( 64 \div 16 = 4 \).
• The reduced fraction is \( \frac{3}{4} \).

This fraction now represents the most simplified or lowest terms version of the original fraction \( \frac{48}{64} \), preserving the same value with smaller numbers.