Problem 24 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 20 inches to 2 feet

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{5}{6} \).

Step 1: Understand the Units

The problem gives a ratio between two measurements: 20 inches and 2 feet. To properly compare them, both measurements need to be in the same unit. We will convert the feet measurement to inches.

Step 2: Convert Feet to Inches

Since there are 12 inches in 1 foot, we need to convert 2 feet to inches. \[ 2 \text{ feet} \times 12 \text{ inches/foot} = 24 \text{ inches} \] Now we have the ratio 20 inches to 24 inches.

Step 3: Write the Ratio as a Fraction

With both measurements in inches, express the ratio as a fraction: \[ \frac{20}{24} \]

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 20 and 24, which is 4. Divide both the numerator and the denominator by 4 to simplify the fraction:\[ \frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6} \] So, the fraction in simplest form is \( \frac{5}{6} \).

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing the relative size of one quantity to another. It's like saying how many times one number fits into another. Think of it as a pair of linked numbers that show how two things are related. Ratios can be written in different ways:

• Using the colon format, for example, 3:4 means 3 units of the first is compared to 4 units of the second.
• In fraction form, where there is a numerator and a denominator, like \( \frac{3}{4} \).

In our exercise, we are given a ratio of 20 inches to 2 feet. The goal is to express this relationship in its simplest form. To do that, the units need to be the same, which is why unit conversion is crucial. We'll delve deeper into this in the next section.

The Importance of Unit Conversion

Unit conversion is crucial when you need to compare or combine different measurements. If you try to compare inches to feet without converting, it's like comparing apples to oranges. It simply doesn't work.

To convert units effectively, know the conversion factors. For length, remember that 1 foot is equal to 12 inches. In our case, we convert 2 feet into inches:

• Multiply the number of feet by the conversion rate: \( 2 \text{ feet} \times 12 \text{ inches per foot} = 24 \text{ inches} \).

Now, the original ratio of 20 inches to 2 feet can be expressed with both measurements in inches: 20 inches to 24 inches. This lays the groundwork for the next step, simplifying the fraction.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible, or as we call it, reduced to its lowest terms. The simplest form of a fraction is when the numerator and the denominator have no common factors other than 1.

Here’s how you simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{20}{24} \), the GCD is 4.
• Divide both the numerator and the denominator by this GCD to simplify the fraction: \( \frac{20}{24} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6} \).

Now, \( \frac{5}{6} \) is the fraction in its simplest form, representing the ratio of 20 inches to 24 inches more clearly and efficiently. Simplifying fractions is not just a step in math exercises, but a useful skill for various real-world applications, making calculations easier and comparisons clearer.