Problem 8 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{7}{175}\)

Step-by-Step Solution

Answer

The terminating decimal of \(\frac{7}{175}\) is 0.04.

Step 1: Simplify the Fraction

Before converting the fraction to a decimal, let's simplify it. We do this by finding the greatest common divisor (GCD) of the numerator 7 and the denominator 175. The GCD is 7. So, we divide both the numerator and the denominator by their GCD:\[ \frac{7}{175} = \frac{7 \div 7}{175 \div 7} = \frac{1}{25} \]

Step 2: Convert to a Decimal

Now that the fraction is simplified to \(\frac{1}{25}\), convert it to a decimal. To do this, divide the numerator by the denominator:\( 1 \div 25 = 0.04 \) Hence, the decimal representation is 0.04.

Key Concepts

Simplifying Fractions

Fractions are often not in their simplest form. Simplifying a fraction means reducing it so that the numerator and the denominator have no common factors other than 1.

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 results in a fraction that is equivalent to the original, but in a simpler state.

For example, in the fraction \(\frac{7}{175}\), we find that the GCD is 7. Simplifying it involves dividing both 7 (numerator) and 175 (denominator) by 7, resulting in \(\frac{1}{25}\). This process makes calculations, like converting to decimals, much easier.

Greatest Common Divisor

The greatest common divisor (GCD) is the highest number that divides exactly into two or more numbers.

To find the GCD:

• List the factors of each number.
• Identify the largest factor that both numbers share.

Using the GCD helps in simplifying fractions, as it shows the largest fraction by which both the numerator and denominator can be divided evenly.

For instance, with the numbers 7 and 175, their GCD is 7 because 7 is the largest number that exactly divides both numbers. Thus, \(\frac{7}{175}\) simplifies to \(\frac{1}{25}\). Calculating the GCD can save time and ensures the fraction is simplified correctly.

Terminating Decimals

Decimals can be terminating or non-terminating. A terminating decimal ends after a finite number of digits.

When a fraction has a denominator that is a power of ten, converting it to a decimal will result in a terminating decimal.
To ensure a fraction converts to a terminating decimal:

• Simplify the fraction first, as this may reduce the denominator to a form compatible with powers of ten.
• Perform division of the numerator by the simplified denominator.

For example, the simplified form of \(\frac{7}{175}\) is \(\frac{1}{25}\). Dividing 1 by 25 gives us the terminating decimal 0.04.

Understanding that simplification and division lead to a neat ending decimal makes the conversion process straightforward.