Problem 105 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{7}{8} $$

Step-by-Step Solution

Answer

The fraction \( \frac{7}{8} \) as a decimal is 0.875, which is a terminating decimal.

Step 1: Understand the Problem

The exercise is asking to convert the fraction \( \frac{7}{8} \) into a decimal.

Step 2: Convert the Fraction Into a Decimal Using Division

To convert a fraction into a decimal, one needs to divide the numerator (top number) by the denominator (bottom number). For this problem, divide 7 by 8.

Step 3: Complete the Division Calculation

Performing the division of 7 by 8 produces the decimal 0.875.

Step 4: Determine the Type of Decimal

Upon observing the decimal, no identifiable pattern of repetition appears, implying that it's a terminating decimal.

Key Concepts

Terminating Decimal

A terminating decimal occurs when a division results in a finite number of decimal places. This means, the division eventually "ends" or "terminates" after a certain number of digits. Unlike repeating decimals, terminating decimals don't go on forever. For example, the division of \( \frac{7}{8} \) results in 0.875, which is a terminating decimal.

Terminating decimals are often easier to work with because there are no repeating sequences to track. This makes them straightforward to convert back into fractions, or use directly for further calculations.

Repeating Decimal

Repeating decimals, on the other hand, occur when the division leads to a digit or group of digits repeating indefinitely. During division, if you keep getting the same remainder, the decimal digits will start forming a pattern.

For example, the fraction \( \frac{1}{3} \) converts into a repeating decimal of 0.333... with the 3 repeating infinitely. These decimals are often noted with a bar over the repeating digits, like \( 0.\overline{3} \).

• Repeating decimals can appear more complex, but identifying the pattern allows you to work with them effectively.
• They also have equivalent fraction representations even if their decimal form goes on forever.

Division

Division is a mathematical operation used to determine how many times one number is contained within another. When converting a fraction to a decimal, division is used to separate the numerator by the denominator.

In the original problem, dividing 7 by 8 gives us the decimal 0.875. This simple division transforms the fraction into a more understandable decimal form. Understanding this process highlights the relationship between fractions and decimals, and reinforces our arithmetic skills.

• Establishes a foundation for converting any fraction into a decimal.
• Helps in identifying whether the resulting decimal terminates or repeats.

Numerator and Denominator

Fractions are expressions that contain two parts: the numerator and the denominator. The numerator is the number above the division line, representing how many parts of the whole are being considered. Denominator is the number below the line, indicating the total number of equal parts into which the whole is divided.

In the fraction \( \frac{7}{8} \), the 7 is the numerator and the 8 is the denominator.

• This relationship is crucial for division since the numerator is divided by the denominator to convert the fraction into a decimal.
• Knowing the role of these components aids in understanding and performing arithmetic operations more effectively.