Problem 24 - Prealgebra

Question

Write each decimal as a fraction in lowest terms. $$0.45$$

Step-by-Step Solution

Answer

The decimal 0.45 is equivalent to the fraction \( \frac{9}{20} \).

Step 1: Write the Decimal as a Fraction

Start by expressing the decimal number as a fraction. The decimal 0.45 can be written as \( \frac{45}{100} \), since the decimal 0.45 means 45 hundredths.

Step 2: Simplify the Fraction

Next, simplify the fraction \( \frac{45}{100} \). Find the greatest common divisor (GCD) of 45 and 100. The GCD of 45 and 100 is 5.

Step 3: Divide by the GCD

Divide both the numerator and the denominator of the fraction \( \frac{45}{100} \) by the GCD, which is 5. This gives \( \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \). Thus, the fraction in lowest terms is \( \frac{9}{20} \).

Key Concepts

Simplifying Fractions

Simplifying fractions involves breaking them down to their simplest or most reduced form. To achieve this, we must find a number that divides both the numerator (the top number) and the denominator (the bottom number). This common number should be as large as possible to ensure full simplification.

For instance, in our example with the fraction \( \frac{45}{100} \), we simplified it to \( \frac{9}{20} \). The process typically involves dividing both the numerator and the denominator by their greatest common factor. When a fraction is in its simplest form, the numerator and the denominator no longer share any common divisors besides 1. This makes it easier to understand and use in further calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is essential because it helps reduce a fraction to its lowest terms, making it simpler to work with.

In the case of the fraction \( \frac{45}{100} \), we determined that the GCD of 45 and 100 is 5. This means 5 is the largest number that can divide both 45 and 100 without leaving a remainder. By dividing both the numerator and the denominator by 5, we successfully simplify the fraction. The simplified fraction \( \frac{9}{20} \) is much easier to interpret and work with in mathematical operations.

Decimal Representation

Decimal representation is a way of expressing fractions using a decimal point. It provides an alternative to writing numbers in fractional form and is commonly used in day-to-day scenarios. Decimal notation is especially handy because it's often easier to read and compare at a glance.

Decimal numbers can be converted to fractions by understanding the place value of each number after the decimal point. Our example, 0.45, means 45 hundredths, which can be written as the fraction \( \frac{45}{100} \). From there, we further simplify the fraction using its GCD, becoming \( \frac{9}{20} \).

• Remembering that each position to the right of the decimal point represents a power of ten (tenths, hundredths, thousandths, etc.) is key in this conversion.
• Converting decimals to fractions and then simplifying helps strengthen understanding of numbers and their relationships.

Understanding decimal representation allows a seamless transition between different forms of a number, increasing comfort with various mathematical processes.