Problem 11 - Prealgebra

Question

Convert each of the following fractions to a decimal. $$\frac{14}{32}$$

Step-by-Step Solution

Answer

The decimal representation is 0.4375.

Step 1: Simplify the Fraction

First, we simplify the fraction \( \frac{14}{32} \). We find the greatest common divisor (GCD) of 14 and 32 to reduce it to its simplest form. The GCD of 14 and 32 is 2. So, we divide both the numerator and the denominator by 2 to simplify the fraction: \( \frac{14 \div 2}{32 \div 2} = \frac{7}{16} \).

Step 2: Convert the Fraction to a Decimal

Now, we convert the fraction \( \frac{7}{16} \) into a decimal by performing the division. Divide 7 by 16. Performing this division yields approximately 0.4375.

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them is a common and necessary task. The process of simplifying a fraction involves reducing it to its simplest form, where the numerator and the denominator have no common divisors other than 1. This is important because it makes the fraction easier to interpret and work with in further calculations.

To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

This will leave you with a fraction that is both clear and concise, making subsequent operations much simpler. Simplifying fractions is not only useful in mathematical exercises but also helps in real-life applications where fractions are involved.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is a crucial concept in simplifying fractions. It is simply the largest number that divides both the numerator and the denominator without leaving a remainder.

To find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the common factors.
• Select the greatest one from this list.

Finding the GCD allows for the effective simplification of fractions, ensuring that both parts of the fraction are in their smallest possible form. This step is essential before attempting to convert fractions to decimals, as having a simplified fraction makes for clearer calculations and more accurate interpretations.

Division

Division is an operation used to calculate how many times one number, known as the divisor, is contained into another number, the dividend. When converting a fraction to a decimal, division becomes especially important. The process involves dividing the numerator (the top part of a fraction) by the denominator (the bottom part).

To convert a simplified fraction to a decimal:

• Write the numerator as the dividend and the denominator as the divisor.
• Carry out the division operation.
• Continue dividing until you achieve a remainder of zero or a repeating decimal pattern.

Practicing division with fractions enhances numerical proficiency, providing the basis for accurately translating fractions into decimal form. With each successful conversion, understanding division becomes more intuitive and seamless.