Problem 17 - Fundamentals of Mathematics

Question

Reduce each fraction to lowest terms. \(\frac{150}{240}\)

Step-by-Step Solution

Answer

The fraction \(\frac{150}{240}\) simplifies to \(\frac{5}{8}\).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of 150 and 240. To do this, let's break both numbers down into their prime factors:- 150: The prime factorization is \(150 = 2 \times 3 \times 5^2 \).- 240: The prime factorization is \(240 = 2^4 \times 3 \times 5 \).Identify the common factors: 2, 3, and 5. The GCD is the product of the lowest powers of these common factors: \(2^1 \times 3^1 \times 5^1 = 30\).

Step 2: Divide the Numerator and Denominator by the GCD

Now divide both the numerator and the denominator by their GCD to simplify the fraction:- Numerator: \(\frac{150}{30} = 5\)- Denominator: \(\frac{240}{30} = 8\) Thus, the fraction \(\frac{150}{240}\) simplifies to \(\frac{5}{8}\).

Step 3: Verify the Fraction is in Lowest Terms

Check the simplified fraction \(\frac{5}{8}\) to ensure it is in its lowest terms by seeing if there are any common factors between 5 and 8.Since 5 and 8 have no common factors other than 1, \(\frac{5}{8}\) is indeed in its lowest terms.

Key Concepts

Greatest Common Divisor

The term "Greatest Common Divisor" (GCD) refers to the largest number that divides two or more integers without leaving a remainder. It plays an essential role in simplifying fractions. To find the GCD, a common method involves identifying all common factors of the numbers involved.

For example, with numbers like 150 and 240, we look for all numbers that can evenly divide both. The GCD is determined by comparing the product of these common factors. This is especially handy when reducing fractions to their simplest form. Finding the GCD simplifies the arithmetic involved in making fractions easier to work with.

• The GCD helps reduce computational effort, allowing fractions to express values in the most uncomplicated terms.
• It ensures no further simplification is possible, bringing clarity to fraction operations.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number components. Primes are numbers greater than 1, only divisible by 1 and themselves. This breakdown can be extremely useful in many mathematical operations, such as finding the GCD.

In the context of our example:- 150 is expressed as the product of its prime factors: \(150 = 2 \times 3 \times 5^2\)- 240 breaks down to \(240 = 2^4 \times 3 \times 5\)
Prime factorization helps by providing a clear view of all divisors of a number, making it simpler to identify common divisors.

• It aids in simplifying fractions by exposing common factors between the numerator and denominator.
• Knowing the prime factors of a number can reveal insights about its divisibility and other properties.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest possible size, where the numerator and denominator have no common factors other than 1. This process is often referred to as reducing a fraction.

The first step in simplification is to find the greatest common divisor (GCD) of the numerator and denominator. Once the GCD is determined, divide both parts of the fraction by this number:- For \(\frac{150}{240}\), divide both by their GCD of 30, giving \(\frac{5}{8}\).
After simplification, the fraction becomes easier to work with in arithmetic calculations, comparisons, and problem-solving.

• It minimizes the numbers being used, making calculations less cumbersome.
• Simplified fractions are easier to understand and interpret.

Lowest Terms

A fraction is considered to be in its lowest terms when the numerator and the denominator are as small as possible, having no common factors aside from 1. This means the fraction cannot be reduced any further.

To check if a fraction is in its lowest terms, ensure that the greatest common divisor (GCD) of the numerator and denominator is 1. If so, the fraction is fully simplified. Taking our example of \(\frac{5}{8}\), 5 and 8 share no common divisors other than 1, confirming that it is in its lowest terms.

• Reaching the lowest terms means achieving the ultimate simplicity possible for a fraction.
• This simplifies arithmetic operations and comparisons with other fractions, providing clarity.

Rendering fractions down to their lowest terms also helps with recognizing and manipulating them in broader mathematical contexts, such as equation solving or when combining with other numeric values.