Problem 28 - Prealgebra

Question

Write each decimal as a fraction in lowest terms. $$0.475$$

Step-by-Step Solution

Answer

\(0.475 = \frac{19}{40}\)

Step 1: Write the Decimal as a Fraction

Express the decimal 0.475 as a fraction. Since there are three digits after the decimal point, place 475 over 1000: \[ 0.475 = \frac{475}{1000} \] This forms our initial fraction.

Step 2: Reduce the Fraction

Find the greatest common divisor (GCD) of the numerator 475 and the denominator 1000. Start by listing the factors: \( 475 = 5 \times 95 = 5 \times 5 \times 19 \) and \( 1000 = 10^3 = 2^3 \times 5^3 \). The common factor is 25. Divide both the numerator and the denominator by 25: \[ \frac{475}{1000} = \frac{475 \div 25}{1000 \div 25} = \frac{19}{40} \] This fraction is already in its simplest form as 19 and 40 have no common factors besides 1.

Key Concepts

Simplifying Fractions

Turning decimals into fractions and then simplifying them helps create cleaner and easier-to-understand expressions. Simplifying means reducing the fraction to its smallest form. To simplify, divide the numerator and the denominator by their greatest common divisor (GCD). This results in a fraction that uses the smallest possible whole numbers, which is often referred to as being in 'lowest terms'. Simplified fractions make math work and comparisons between fractions more straightforward. It streamlines calculations, making them clearer and less prone to errors.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial when reducing fractions to their simplest form. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For the fraction \(\frac{475}{1000}\), you first need to break down each number into its prime factors:

• 475 can be expressed as \(5 \times 5 \times 19\).
• 1000 breaks down into \(2^3 \times 5^3\).

Identifying common factors, we find both numbers share the factor 25. By dividing both the numerator and denominator by their GCD (25), we simplify the fraction to \(\frac{19}{40}\). This technique ensures the fraction is as reduced as possible, making it easier to interpret and use in further calculations.

Place Value in Decimals

Understanding place value is essential when converting decimals to fractions. Each position in a decimal represents a specific power of ten:

• The first digit to the right of the decimal point is the tenths place.
• The second is the hundredths place.
• The third is the thousandths place, and so on.

For example, in the decimal 0.475, the digit 4 is in the tenths place, 7 in the hundredths, and 5 in the thousandths. We convert 0.475 to the fraction \(\frac{475}{1000}\), as the thousandths place guides the denominator to be 1000. Grasping place value helps ensure you correctly set the initial fraction before simplifying, thus maintaining accuracy through the steps of conversion.