Problem 21 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$0.3 \text { to } 3$$

Step-by-Step Solution

Answer

The ratio 0.3 to 3 as a fraction in lowest terms is \(\frac{1}{10}\).

Step 1: Understand the Given Ratio

The given ratio is \(0.3\) to \(3\). Ratios can be expressed as fractions by writing the first number as the numerator and the second number as the denominator.

Step 2: Convert the Ratio to a Fraction

Write the ratio \(0.3\) to \(3\) as a fraction: \(\frac{0.3}{3}\).

Step 3: Eliminate the Decimal

To remove the decimal, multiply both the numerator and the denominator by 10. This gives \(\frac{3}{30}\).

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of \(3\) and \(30\). The GCD is \(3\). Now, divide both the numerator and the denominator by \(3\): \(\frac{3 \div 3}{30 \div 3} = \frac{1}{10}\).

Step 5: Verify Your Solution

Check that the fraction \(\frac{1}{10}\) is in its simplest form by ensuring the numerator and denominator have no common factors other than 1. This confirms the solution is correct.

Key Concepts

Fraction Conversion

When you encounter a ratio, one of the first steps is to convert it into a fraction. This helps in simplifying or mathematically manipulating the ratio when necessary. In a ratio, the first number is placed as the numerator and the second number becomes the denominator. For instance, in the problem given, the ratio "0.3 to 3" gets rewritten as a fraction: \( \frac{0.3}{3} \). This conversion is a straightforward process as it sets the stage for additional operations like simplifying the fraction.

• Identify the ratio components.
• Write the ratio as \( \frac{\text{first number}}{\text{second number}} \).
• This fraction represents the original ratio in another form.

Decimal Elimination

Having decimals in fractions can make them cumbersome to work with, especially when you want to simplify them further. Therefore, it's a good idea to eliminate the decimal by converting it into a whole number. In most cases, you can achieve this by multiplying both the numerator and the denominator by the same power of 10, which shifts the decimal point to the right. In our example, multiplying both 0.3 and 3 by 10 transforms \( \frac{0.3}{3} \) into \( \frac{3}{30} \). This keeps the value of the fraction the same but with whole numbers, making the next steps easier.

• Identify how many places to move the decimal to the right.
• Multiply numerator and denominator by 10, 100, or a higher power of 10 as needed.
• Check your new fraction to ensure no decimals remain.

GCD (Greatest Common Divisor)

Finding the Greatest Common Divisor (GCD) is crucial for reducing fractions to their simplest form. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In our exercise, we found that the GCD of 3 and 30 is 3. Knowing this allows you to reduce the fraction \( \frac{3}{30} \) by dividing both terms by 3, resulting in a simpler form: \( \frac{1}{10} \).

• List the factors of both the numerator and the denominator.
• Determine the largest factor common to both.
• Divide both numerator and denominator by this GCD to simplify.

Simplifying Fractions

Simplifying fractions means expressing the fraction in the smallest possible whole numbers. This makes it easier to understand and work with in further calculations. Once you have the GCD, you can divide both the numerator and denominator to reduce the fraction. For example, \( \frac{3}{30} \) is simplified to \( \frac{1}{10} \) by dividing both 3 and 30 by 3. The goal is to ensure that no number other than 1 can evenly divide both the numerator and denominator. This process verifies that the fraction is in its simplest form.

• Use the GCD to simplify fractions.
• Apply the division to both the numerator and denominator.
• Ensure the fraction cannot be reduced further.