Problem 46 - Prealgebra

Question

Change each decimal to a fraction, and then reduce to lowest terms. $$0.375$$

Step-by-Step Solution

Answer

The decimal 0.375 as a fraction is \( \frac{3}{8} \) in lowest terms.

Step 1: Express the Decimal as a Fraction

To convert the decimal 0.375 into a fraction, first note that 0.375 is the same as 375 in the thousandths place. Thus, express it as a fraction with a denominator of 1000: \[ 0.375 = \frac{375}{1000} \]

Step 2: Simplify the Fraction

To simplify the fraction \( \frac{375}{1000} \), find the greatest common divisor (GCD) of 375 and 1000. The GCD of 375 and 1000 is 125. Divide both the numerator and the denominator by 125: \[\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}\] The fraction \( \frac{3}{8} \) is in its simplest form.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we're looking to make a fraction as simple as possible. This means having the smallest possible numerator and denominator while still being equal to the original fraction. To simplify a fraction, you need to divide the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that can evenly divide both numbers.

• For example, with the fraction \( \frac{375}{1000} \), simplifying it requires dividing both 375 and 1000 by their GCD, which is 125.
• After dividing, you get \( \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \).
• Thus, \( \frac{3}{8} \) is the simplest form of the original fraction.

Simplifying fractions helps us see the most essential part of the fraction, making it easier to understand and use in calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept in simplifying fractions. It's the largest number that can divide two numbers without leaving a remainder. Finding the GCD can be done using several methods:

• **Prime Factorization:** Break down each number into its prime factors and multiply the common factors.
• **Listing Factors:** List all factors of both numbers and find the largest common one.
  Example: To find the GCD of 375 and 1000, list the factors or use another method like the Euclidean algorithm to arrive at the GCD of 125.

Knowing the GCD helps you simplify fractions because you divide both the numerator and denominator by this number. This process reduces the fraction to its simplest form. This is what we did when we turned \( \frac{375}{1000} \) into \( \frac{3}{8} \). Working with the GCD ensures your fraction simplification is accurate and saves time when dealing with larger numbers.

Decimal to Fraction Conversion

Converting decimals to fractions is all about finding a way to express the decimal as a fraction of whole numbers. This might seem tricky at first, but it's quite straightforward when you break it down.Let's take the example of 0.375:

• First, recognize that 0.375 is equivalent to 375 thousandths because it's in the thousandths place. This means you can write it as \( \frac{375}{1000} \).
• The next step is simplifying this fraction. You do this by finding the GCD of 375 and 1000, which is 125, and then divide both the numerator and the denominator by this common divisor.
• Once simplified, you will have \( \frac{3}{8} \).

By using this method, you not only convert the decimal into a fraction but also ensure it is expressed in its simplest form. Understanding this process can really help make decimal to fraction conversions a breeze! It's like converting from a long scientific notation to a neat, usable number.