Problem 1 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$\frac{7}{8}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to convert the fraction \( \frac{7}{8} \) into a decimal format. This means we have to find how many times 8 fits into 7, and then keep dividing further to get the decimal form.

Step 2: Performing the Division

Divide 7 by 8. Since 7 is smaller than 8, we consider it as 7.0 and divide: \( 7.0 \div 8 \). The quotient starts as 0. Divide 70 by 8 gives us 8, because 8 times 8 is 64. After subtracting 64 from 70, we have a remainder of 6.

Step 3: Continuing the Division

Bring down another 0 making it 60. Divide 60 by 8, which gives 7, because 8 times 7 is 56. Subtract 56 from 60, leaving a remainder of 4.

Step 4: Completing the Division

Bring down another 0 making it 40. Divide 40 by 8, which gives 5 exactly since 8 times 5 equals 40. Subtract 40 from 40 leaving a remainder of zero. Thus, the division terminates here.

Step 5: Writing the Decimal Format

Now, we have obtained the decimal form 0.875 from the division. Since it terminates, there is no need to use a bar for repeating decimals.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal is a common task in mathematics. Essentially, this process involves division. A fraction signifies division, where the numerator is divided by the denominator. For example, in the fraction \( \frac{7}{8} \), you take 7 (the numerator) and divide it by 8 (the denominator) to change it into a decimal.
To break this down:

• First, perform the division as you would with any two numbers.
• If the numerator is smaller than the denominator, think of the numerator as a decimal by adding a zero. For instance, treat 7 as 70, 700, and so on.
• Continue the division until the remainder is zero or you identify a repeating pattern.

This method helps in transforming fractions into decimals, making them easier to understand and compare. It's useful for dealing with fractions in everyday calculations.

Terminating Decimals

When converting a fraction to a decimal, a terminating decimal is one that comes to an end. It doesn't go on indefinitely, making it simpler and more convenient to work with.
A decimal terminates when, after performing the division, you reach a zero remainder after a finite number of steps. Taking \( \frac{7}{8} \) as an example, after dividing, we reach a decimal representation of 0.875 with no remaining number to divide further.

• Terminating decimals are crucial because they show the exact value of fractions as decimals without any approximation.
• These decimals can often be expressed as fractions where the denominator has only the prime factors 2 and/or 5.

Understanding terminating decimals helps us identify precisely when the division of two numbers will complete without recurring digits.

Long Division Method

The long division method is a reliable process to convert fractions into decimals, especially when the relationship between numbers is not apparent. It is methodical and involves a series of straightforward steps:

• Set up the division with the numerator inside the division bracket and the denominator outside.
• Perform the division as you would with whole numbers – see how many times the denominator fits into the numerator and subtract.
• Bring down any remaining figures after each subtraction to continue the division.
• Continue this process until you reach a zero remainder or identify a repeating pattern.

Unlike simple fractions, complex calculations benefit greatly from this method. It provides a step-by-step breakdown, which illuminates how fractions extend into their decimal forms. With our example of \( \frac{7}{8} \), the long division process leads smoothly to a terminating decimal of 0.875. It is essential for students to grasp this technique for both academic and practical applications.