Problem 20 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$0.21 \text { to } 0.03$$

Step-by-Step Solution

Answer

7

Step 1: Convert Decimals to Fractions

First, convert the given decimals to fractions. The decimal 0.21 can be written as \( \frac{21}{100} \) and the decimal 0.03 can be written as \( \frac{3}{100} \).

Step 2: Set Up the Ratio

The ratio of 0.21 to 0.03 can now be written in fraction form as \( \frac{21}{100} \div \frac{3}{100} \).

Step 3: Simplify the Division of Fractions

To divide the fractions, multiply \( \frac{21}{100} \) by the reciprocal of \( \frac{3}{100} \). This gives: \[ \frac{21}{100} \times \frac{100}{3} = \frac{21 \times 100}{100 \times 3} = \frac{21}{3}. \]

Step 4: Simplify the Fraction to Lowest Terms

Now, simplify \( \frac{21}{3} \). Divide both the numerator and the denominator by their greatest common divisor, which is 3: \[ \frac{21 \div 3}{3 \div 3} = \frac{7}{1}. \] This results in a simplified fraction of 7.

Key Concepts

Fractions

Fractions are a way of expressing numbers as parts of a whole. They consist of a numerator and a denominator.

• The numerator is the top number and represents how many parts we have.
• The denominator, on the bottom, signifies the total number of equal parts into which the whole is divided.

When we deal with ratios, converting them into fractions helps us see the relationship between two numbers visually. For instance, if we take the decimal values "0.21 to 0.03", converting them to fractions makes the comparison more straightforward and eliminates the decimal, which can sometimes be tricky to manage. In this process, both decimals are converted to have a common base (such as 100), which simplifies the calculations.

Decimal Conversion

Converting decimals into fractions is a fundamental skill, especially useful when you want to work with exact numbers without the messiness of decimal points.
To convert a decimal to a fraction:

• Identify the place value of the last digit. For example, in 0.21, the last digit, 1, is in the hundredths place.
• Use this place value to form the fraction's denominator: 0.21 becomes \( \frac{21}{100} \).
• Repeat this for 0.03, which becomes \( \frac{3}{100} \).

By doing this, both decimals are expressed as fractions with a consistent denominator, making further calculations a breeze. This method keeps your work neat and precise, avoiding approximation errors that can occur when decimals are left in their original form.

Simplifying Fractions

Once a fraction is formed, it’s crucial to simplify it. Simplifying means reducing the fraction to its simplest form, where the numerator and denominator have no common divisors other than 1. Here’s how you do it:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both by this GCD.
• This will give you the fraction in its simplest form.

For example, the fraction \( \frac{21}{3} \) can be simplified by dividing both 21 and 3 by their GCD, which is 3. The simplified fraction becomes \( \frac{7}{1} \), which simplifies further to just 7.
Remember, simplifying fractions makes them easier to understand and work with during any mathematical operations.