Problem 22 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$0.5 \text { to } 10$$

Step-by-Step Solution

Answer

The ratio 0.5 to 10 as a fraction in lowest terms is \( \frac{1}{20} \).

Step 1: Write As A Fraction

The given ratio is 0.5 to 10. We can express this as a fraction: \( \frac{0.5}{10} \).

Step 2: Eliminate Decimals

To eliminate the decimal from 0.5, we can multiply both the numerator and the denominator by 10, resulting in \( \frac{5}{100} \).

Step 3: Simplify The Fraction

Now, simplify the fraction \( \frac{5}{100} \). To do this, find the greatest common factor of 5 and 100, which is 5. Divide both the numerator and the denominator by this common factor: \( \frac{5 \div 5}{100 \div 5} = \frac{1}{20} \).

Key Concepts

Ratios as Fractions

Ratios are a way to compare two quantities, often representing relationships between them. To express a ratio as a fraction, we simply write the first number over the second number with a fraction bar between them. This turns the ratio "0.5 to 10" into the fraction \( \frac{0.5}{10} \). This conversion is useful because fractions provide a standardized way to perform arithmetic operations and comparisons. When working with ratios, thinking of them as fractions helps in simplifying and handling the numbers more effectively. It's essential, particularly in mathematical contexts, to express fractions in their simplest form, allowing for easier interpretation and computation.

Eliminating Decimals

Decimals often complicate calculations and comparisons. When simplifying fractions from ratios that contain decimals, the first step is to remove these decimals. In our example with the fraction \( \frac{0.5}{10} \), we can eliminate the decimal by multiplying the numerator and the denominator by a power of 10. Since 0.5 has one decimal place, multiplying by 10 shifts the decimal point one place to the right, resulting in \( \frac{5}{100} \). Doing so converts the decimal into a whole number, making the fraction easier to work with. Eliminating decimals is a crucial step toward simplifying fractions, particularly when they come from ratios that involve decimal numbers.

Greatest Common Factor

The greatest common factor (GCF) plays a key role in simplifying fractions. It is the largest number that divides both the numerator and the denominator of the fraction without leaving a remainder. To find the GCF, list the factors of both the numerator and the denominator and identify the largest number they have in common. For our fraction \( \frac{5}{100} \), the factors of 5 are \{1, 5\} and the factors of 100 include \{1, 2, 4, 5, 10, 20, 25, 50, 100\}. The greatest factor common to both is 5. Dividing both the numerator and denominator by this common factor effectively simplifies the fraction to its lowest terms, resulting in \( \frac{1}{20} \). Finding the GCF ensures that the fraction is fully simplified and is a fundamental skill for reducing fractions.