Problem 30 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{360}{700} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{18}{35}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). First, identify the factors of both 360 and 700. - Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. - Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700. The greatest common divisor of 360 and 700 is 20.

Step 2: Divide the Numerator and Denominator by the GCD

Divide both the numerator (360) and the denominator (700) by the GCD (20).- Calculate: \(\frac{360}{20} = 18\)\(\frac{700}{20} = 35\)

Step 3: Write the Simplified Fraction

The fraction \(\frac{360}{700}\) simplifies to \(\frac{18}{35}\). Since 18 and 35 do not have any common divisors other than 1, the fraction is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that divides two or more numbers without leaving a remainder. In the process of simplifying fractions, identifying the GCD is crucial because it helps in reducing the fraction to its simplest form. For example, when working with the fraction \(\frac{360}{700}\), we need the GCD of 360 and 700 to simplify effectively.

To find the GCD, first list all the factors of the numbers involved:

• Factors of 360 include numbers like 1, 2, 3, up to 360 itself.
• Factors of 700 include numbers like 1, 2, 4, up to 700 itself.

From these lists, the greatest common factor is 20, since it's the highest number that appears in both sets of factors. The GCD allows us to divide both the numerator and the denominator, reducing the original fraction to its simplest form. Setting the stage with the GCD is an essential first step in simplifying fractions.

Numerators and Denominators

In any fraction, the numerator is the top number, and it represents how many parts of a whole are being considered. The denominator is the bottom number and represents the total number of equal parts the whole is divided into. Understanding numerators and denominators is key to managing and simplifying fractions.

Consider the fraction \(\frac{360}{700}\):

• The numerator, 360, indicates the part of the total amount.
• The denominator, 700, shows the whole amount that 360 is a part of.

To simplify, we divided both parts by the greatest common divisor, 20, resulting in the simplified fraction \(\frac{18}{35}\). This process maintains the same proportionate relationship between the numerator and the denominator, just with smaller, simpler numbers. Thus, a clear grasp of how numerators and denominators operate is essential for any fraction work.

Factors of Numbers

Factors are numbers that can be multiplied together to produce another number. Understanding factors is helpful for finding the greatest common divisor and simplifying fractions.

To comprehend how factors work, consider:

• Counting the multiples of numbers, like 1, 2, 3, etc.
• Checking which of these multiples fit perfectly into our numbers of interest.

For instance, with 360 and 700, we sought all factors:

• The factors of 360 include numbers such as 1, 2, 4, which multiply in various combinations to give 360.
• The factors of 700 include numbers like 1, 2, 5, and more.

By recognizing the shared factor of 20, which is the largest shared factor among their lists, we can simplify the fraction by dividing both the numerator and the denominator by 20. This approach not only aids in simplifying but also enhances number sense by understanding how numbers relate through their factors.