Problem 50 - College Algebra

Question

Express each repeating decimal as a fraction in lowest terms. $$0.529$$

Step-by-Step Solution

Answer

The fraction form of 0.529 in lowest terms is \( \frac{529}{999} \)

Step 1: Assign a variable

We'll call the repeating decimal \( x \). So, \( x = 0.529 \)

Step 2: Multiply by 1000

To get rid of the decimal we multiply from both sides by 1000, getting \( 1000x = 529.529 \).

Step 3: Subtract the original equation from this new one

When we subtract the equation \( x = 0.529 \) from \( 1000x = 529.529 \), we get \( 999x = 529 \)

Step 4: Simplify by dividing from both sides

To find the value of \( x \), we divide from both sides by 999, giving us \( x = 529/999 \)

Step 5: Reduce to the lowest term

After checking we see the fraction cannot be reduced further, so \( 529/999 \) is the fraction in its simplest form.

Key Concepts

Fractions

Fractions are numbers that represent a part of a whole. They are composed of two parts: a numerator and a denominator. The numerator is the top number indicating how many parts we have. The denominator is the bottom number showing the total number of equal parts the whole is divided into. In our exercise, we converted a repeating decimal into a fraction. This allows us to express numbers that aren't whole in a different form.

Understanding how to manipulate fractions is crucial. You can add, subtract, multiply, and divide them. Fractions can also be equivalent, meaning they represent the same value even if their numerators and denominators are different. This is an essential concept when working with fractions, especially when converting from decimals.

Decimal Conversion

Decimal conversion is the process of changing a fraction or other numeric representation into a decimal. For repeating decimals, this can be a bit challenging but follows a systematic approach.

To convert a repeating decimal to a fraction, we assign a variable to the repeating decimal, multiply by a power of ten to move the decimal point right after the repeat ends, and form an equation. This technique allowed us to transform the repeating decimal 0.529 into the equation 1000x = 529.529, and further subtracting an initial equation x = 0.529.

By solving the resulting equation, we eliminated the decimal, simplifying our work to regular fraction operations.

Lowest Terms Reduction

Reduction to the lowest terms refers to simplifying a fraction so that the numerator and denominator have no common factor other than one. This ensures the fraction is in its simplest form.

In our exercise, once we obtained the fraction \( \frac{529}{999} \), we checked for any common factors between the numerator and denominator. When there are no common factors to divide both by, then we have successfully reduced the fraction.

Simplifying fractions is important because it makes them easier to work with. It also forms a standard way of reporting fractions, making it simpler for others to understand and utilize them.