Problem 21 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 30 days to 24 days

Step-by-Step Solution

Answer

The ratio 30 days to 24 days simplifies to the fraction \( \frac{5}{4} \).

Step 1: Identifying the Ratio

The given ratio is '30 days to 24 days'. This means we are comparing 30 days with 24 days.

Step 2: Convert Ratio to Fraction

Express the ratio as a fraction: \( \frac{30}{24} \). This fraction compares the two quantities directly.

Step 3: Finding the Greatest Common Divisor (GCD)

To simplify the fraction, we need to find the greatest common divisor of 30 and 24. Both numbers are divisible by 6, which is their greatest common divisor.

Step 4: Simplifying the Fraction

Divide both the numerator and denominator by their GCD (6): \( \frac{30 \div 6}{24 \div 6} = \frac{5}{4} \).

Step 5: Result in Simplest Form

Now the ratio is expressed as the fraction \( \frac{5}{4} \), which is in its simplest form.

Key Concepts

Understanding Ratios

Ratios are a way of comparing two quantities to each other. They tell us how much of one thing we have relative to another. For example, in the ratio '30 days to 24 days,' we are comparing two periods of time. A ratio can be presented in several ways:

• 30 to 24
• 30:24
• 30/24 as a fraction

When dealing with ratios, it's important to maintain the order, as swapping the numbers will change their meaning entirely. Translating the ratio into a fraction is a useful method to work with these numbers more easily. A fraction explicitly shows the relationship between the two numbers, allowing for further simplification.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two or more integers without leaving a remainder. Understanding how to find the GCD is essential to simplifying fractions.

In our example, to simplify the fraction \(\frac{30}{24}\), we first need to determine the GCD of 30 and 24. Here are the steps to find it:

• First, list the factors of each number.
• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
• The largest number that appears in both lists is 6. So, the GCD is 6.

Having found the GCD, you can use it to simplify fractions effectively by reducing both the numerator and the denominator by this number.

Simplifying Fractions

Once we have the greatest common divisor (GCD), we can simplify the fraction to its simplest form. Simplifying a fraction involves dividing the numerator and the denominator by their GCD. This process will reduce the fraction to its smallest equivalent values without changing its inherent ratio.

Let's use the fraction \(\frac{30}{24}\) from our example. We found that the GCD is 6, so we simplify as follows:

• Divide the numerator (30) by 6, which equals 5.
• Divide the denominator (24) by 6, which equals 4.
• The simplified fraction is \(\frac{5}{4}\).

This leads us to the simplest form of the original ratio, \(\frac{5}{4}\). Simplifying fractions is a valuable skill used in many areas, such as mathematics, science, and everyday situations, to make numbers more manageable and easier to work with.