Problem 8 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.16 $$

Step-by-Step Solution

Answer

0.16 = \(\frac{4}{25}\).

Step 1 - Identify the Place Value

The decimal 0.16 has the digit 1 in the tenths place and the digit 6 in the hundredths place.

Step 2 - Write as a Fraction

Write the decimal 0.16 as a fraction: \[0.16 = \frac{16}{100}\].

Step 3 - Simplify the Fraction

Simplify the fraction \(\frac{16}{100}\) by finding the greatest common divisor (GCD) of 16 and 100. The GCD of 16 and 100 is 4. Divide both the numerator and the denominator by 4: \[ \frac{16 \/ 4}{100 \/ 4} = \frac{4}{25} \].

Key Concepts

Place Value

Place value is fundamental in understanding and converting decimals to fractions. It tells you where each digit stands in a number. In the decimal number 0.16:

• The digit 1 is in the tenths place.
• The digit 6 is in the hundredths place.

This means 0.16 can be read as 16 hundredths, and it can be written using the fraction \(\frac{16}{100}\).
Recognizing place value makes converting decimals to fractions straightforward. Just place the digits at the appropriate denominator: 10 for tenths, 100 for hundredths, 1000 for thousandths, and so on.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form. This is done by dividing the numerator and denominator by their greatest common divisor.
For the fraction \(\frac{16}{100}\), we find the GCD of 16 and 100 is 4. To simplify:

• Divide the numerator 16 by 4 to get 4.
• Divide the denominator 100 by 4 to get 25.

So, \(\frac{16}{100} = \frac{4}{25}\).
The simplified fraction is \(\frac{4}{25}\). Simplifying helps in making fractions easier to work with and compare.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the highest number that divides exactly into two or more numbers. Finding it helps in simplifying fractions effectively.
To find the GCD of 16 and 100:

• List the factors of 16: 1, 2, 4, 8, 16.
• List the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

The common factors are 1, 2, and 4. The greatest of these is 4, so the GCD is 4.
By dividing both the numerator and the denominator by their GCD, fractions are simplified: \(\frac{16}{100}\) becomes \(\frac{4}{25}\). Understanding the GCD is crucial for fraction operations.