Problem 64 - Prealgebra

Question

Convert the given decimal to a fraction. Reduce your answer to lowest terms. 0.84

Step-by-Step Solution

Answer

0.84 as a fraction is \( \frac{21}{25} \).

Step 1: Express the Decimal as a Fraction

To convert the decimal 0.84 to a fraction, first recognize that 0.84 equals 84 hundredths. Write this as a fraction: \( \frac{84}{100} \).

Step 2: Simplify the Fraction

Now that we have \( \frac{84}{100} \), we need to simplify it to its lowest terms. To do this, find the greatest common divisor (GCD) of 84 and 100. The GCD of 84 and 100 is 4. Divide both the numerator and the denominator by 4: \( \frac{84 \div 4}{100 \div 4} = \frac{21}{25} \).

Step 3: Verify the Simplified Fraction

Verify whether \( \frac{21}{25} \) is in its simplest form. Check if there are any common divisors of 21 and 25 besides 1. Since 21 is 3 x 7, and 25 is 5 x 5, there are no common factors. Therefore, \( \frac{21}{25} \) is in its simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form. This involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). Simplifying makes the fraction easier to understand and use. For instance, in the fraction \( \frac{84}{100} \), both 84 and 100 can be divided by their GCD, which is 4. This process converts \( \frac{84}{100} \) to \( \frac{21}{25} \). This is the simplified version of the fraction. It represents the same value as the original but is less complex to work with.

• Simplifying fractions makes calculations simpler.
• It helps in comparing fractions efficiently.

The simplest form of a fraction is always smaller and easier to handle. Always simplify fractions whenever possible to make your math work tidier.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more integers without leaving a remainder. It is essential for simplifying fractions effectively. Finding the GCD helps in reducing a fraction to its simplest form, ensuring that both the numerator and the denominator are as small as possible.
Let's take the example of \( \frac{84}{100} \). You must find the GCD of 84 and 100 to simplify this fraction. Here, the GCD is 4, which means:

• 4 is the largest number that divides both 84 and 100 evenly.
• Using 4, we can evenly divide 84 to get 21 and 100 to get 25.

By dividing by their GCD, \( \frac{84}{100} \) becomes \( \frac{21}{25} \). Knowing how to find the GCD is a key skill in simplifying fractions and solving other mathematical problems.

Fraction in Lowest Terms

A fraction in lowest terms is when the numerator and the denominator have no common factors other than 1. Achieving this form means the fraction is as simple as possible, while still being equivalent to the original fraction. It's also known as reducing a fraction to its simplest form. When a fraction is in lowest terms, it cannot be simplified any further.
Take the fraction \( \frac{21}{25} \) as an example. To check whether it is in lowest terms, ensure that the numerator 21 and the denominator 25 do not have any common factors other than 1, which they don't:

• 21 factors are 1, 3, 7, and 21.
• 25 factors are 1, 5, and 25.

Since there are no common factors apart from 1, the fraction \( \frac{21}{25} \) is confirmed to be in its simplest form. Ensuring fractions are in lowest terms is vital for clarity and efficient mathematical communication.