Problem 19 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$0.05 \text { to } 0.15$$

Step-by-Step Solution

Answer

The ratio 0.05 to 0.15 as a fraction in lowest terms is \( \frac{1}{3} \).

Step 1: Identify the Ratio

The given ratio is 0.05 to 0.15. This can be expressed as a fraction \( \frac{0.05}{0.15} \).

Step 2: Eliminate Decimals

To eliminate decimals, multiply both the numerator and the denominator by 100 to convert 0.05 and 0.15 into whole numbers. This gives \( \frac{5}{15} \).

Step 3: Find the Greatest Common Divisor (GCD)

Determine the GCD of 5 and 15. The GCD of these numbers is 5.

Step 4: Simplify the Fraction

Divide both the numerator and the denominator by their GCD, which is 5. Thus, \( \frac{5}{15} = \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. A fraction consists of a numerator and a denominator. The numerator is the number above the line, which represents how many parts are being considered. The denominator is the number below the line, which indicates how many equal parts the whole is divided into.
For example, in a pie divided into 4 equal slices, if you take 1 slice, you have eaten 1/4 of the pie, where 1 is the numerator, and 4 is the denominator. Fractions are versatile and can represent values less than 1, exactly 1, or greater than 1.
Fractions are fundamental in expressing quantities in mathematics. In situations like recipes, measurements, and ratios, you might encounter fractions regularly. Understanding how to manipulate and interpret fractions ensures accurate calculations and comparisons.

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes referred to as the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. To simplify a fraction effectively, finding the GCD of the numerator and the denominator is crucial.
For instance, if you have the numbers 12 and 15, factors of 12 are 1, 2, 3, 4, 6, and 12, and factors of 15 are 1, 3, 5, and 15. The largest common factor is 3, making it the GCD.
Knowing how to find the GCD allows you to simplify fractions and make calculations easier. It's particularly useful when you need to add, subtract, or compare fractions, as they must often be expressed in their simplest form. Techniques such as listing out each number's factors or using prime factorization can aid in finding the GCD efficiently.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. This process involves dividing the numerator and denominator by their GCD so that the fraction becomes as simple as possible whilst representing the same value.
Take the fraction \( \frac{8}{12} \) as an example. The GCD of 8 and 12 is 4, and dividing both the numerator and denominator by 4 gives \( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \).
This simplification makes it easier to understand and use fractions in further calculations. Simplified fractions are especially handy in real-world scenarios such as adjusting recipes, converting measurements, or dealing with ratios in everyday life.
Always ensure your fractions are in their simplest form for clarity and ease of comparison. This not only aids in arithmetic operations but also in communicating your results more effectively.