Problem 4 - Elementary Algebra

Question

Express each ratio in reduced form. 240 miles4 hours

Step-by-Step Solution

Answer

The reduced form of the ratio is 60:1.

Step 1: Understand the Ratio

The ratio given is between 240 miles and 4 hours. Ratios compare two quantities by showing how many times one value contains or is contained within the other. In this case, it is the miles to hours.

Step 2: Write the Ratio as a Fraction

Express the ratio as a fraction: \( \frac{240\text{ miles}}{4\text{ hours}} \). This fraction shows the relationship between miles and hours.

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{240}{4} \) by dividing both the numerator and denominator by their greatest common divisor (GCD). Here, the GCD of 240 and 4 is 4.

Step 4: Perform the Division

Divide the numerator and denominator by 4: \( \frac{240 \div 4}{4 \div 4} = \frac{60}{1} \). This gives us the simplified ratio.

Key Concepts

Understanding Fractions

When we talk about fractions, we are discussing how a whole is divided into parts. A fraction consists of two numbers: a numerator and a denominator.

• The numerator is the number of parts we are interested in.
• The denominator is the total number of equal parts the whole is divided into.

A fraction can also represent a division operation, where the numerator is divided by the denominator. For example, the fraction \( \frac{240}{4} \) means 240 divided by 4.

Fractions can be used to express ratios, which compare two quantities, like the miles per hour in the example you've been working with. This comparison helps us understand how many times one quantity fits into another.

Simplifying Fractions

Simplifying fractions is a key concept in mathematics. We simplify fractions to make them easier to understand and work with. Simplifying involves reducing the fraction to its simplest form, where the numerator and denominator do not have any common factors other than 1.

To simplify, you need to find the greatest number that divides both the numerator and the denominator without leaving a remainder. For example:

• Take the fraction \( \frac{240}{4} \). The greatest common factor of 240 and 4 is 4.
• Divide both the numerator and denominator by this number: \( \frac{240 \div 4}{4 \div 4} = \frac{60}{1} \).

This process makes the fraction simpler while keeping the value unchanged. A simplified fraction is easier to interpret, especially in practical scenarios like calculating speed or density.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can divide two or more numbers without leaving a remainder. Finding the GCD is an essential step in simplifying fractions.

Here's how you can find the GCD of two numbers:

• List the factors of each number. Let’s take 240 and 4 as an example:
• Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
• Factors of 4: 1, 2, 4
• Identify the common factors: 1, 2, 4
• The greatest number in this list is 4, so the GCD of 240 and 4 is 4.

Using the GCD to simplify fractions makes the entire process more systematic and ensures that you achieve the most reduced form. This makes it much easier to work with fractions in complex calculations.