Problem 9 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.95 $$

Step-by-Step Solution

Answer

\( \frac{19}{20} \)

Step 1: Write the decimal as a fraction

The decimal 0.95 can be written as \ \[ \frac{95}{100} \ \].

Step 2: Simplify the fraction

To simplify \( \frac{95}{100} \), find the greatest common divisor (GCD) of 95 and 100. The GCD of 95 and 100 is 5. Now, divide both the numerator and the denominator by 5: \ \[ \frac{95 \div 5}{100 \div 5} = \frac{19}{20} \ \].

Key Concepts

simplifying fractions

Simplifying fractions makes them easier to work with and understand. It involves reducing the numerator (top number) and the denominator (bottom number) to their smallest possible values while still keeping the same overall value. For example, the fraction \(\frac{95}{100}\) was simplified to \(\frac{19}{20}\). This process involves finding common factors of the numerator and denominator and dividing them out. By simplifying, you ensure the fraction represents its simplest and most fundamental form, which can be very useful in math.

greatest common divisor

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, you need to look for the largest number that both numbers are divisible by. For example, for the fraction \(\frac{95}{100}\), the GCD of 95 and 100 is 5. This is because 5 is the largest number that can evenly divide both 95 and 100. Using the GCD, you can simplify fractions by dividing both the numerator and the denominator by this number, effectively shrinking the fraction to its simplest form.

fraction reduction

Fraction reduction is another term for simplifying fractions. It involves reducing the size of the numerator and the denominator. Take the fraction \(\frac{95}{100}\) for example. By dividing both the top and bottom by their GCD of 5, we reduce the fraction to \(\frac{19}{20}\). This process ensures that the fraction is in its simplest form. The resulting fraction retains the same exact value but is expressed in simpler terms. This makes it easier to handle in calculations and comparisons. Remember, the goal is to make the fraction as simple as possible by identifying and removing common factors.