Problem 17 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-\frac{8}{25}$$

Step-by-Step Solution

Answer

The decimal representation of \(-\frac{8}{25}\) is \(-0.32\).

Step 1: Convert Fraction to Decimal

Divide the numerator by the denominator to change the fraction to a decimal. For the fraction \(-\frac{8}{25}\), perform the division \(-8 \div 25\).

Step 2: Perform Long Division

Execute the division \(-8 \div 25\). Since 8 is less than 25, add a decimal point and a zero to make it \(-80 \div 25\). This division gives \(-3.2\) because 25 goes into 80 three times (with a remainder of 5), and 50 goes two times into the remaining 30.

Step 3: Identify the Result

The result of the division operation is a terminating decimal, which means no digits repeat. Thus, \(-\frac{8}{25} = -0.32\).

Key Concepts

Long Division

Long division is a method used to divide numbers in a straightforward and organized manner. It breaks down a division problem into a series of easier steps. Here's how long division works:- **Setup the Division**: Start by writing the dividend (the number being divided) inside the division bracket and the divisor (the number you are dividing by) outside.- **Divide and Multiply**: Determine how many times the divisor fits into the current portion of the dividend, then multiply the divisor by this number and subtract the result from the dividend.- **Bring Down and Repeat**: Bring down the next digit from the dividend and repeat the process.If, like in our example of \(-8 \div 25\), the divisor doesn't fit into the dividend, you can extend the division by adding a decimal point and zeros to the dividend, allowing you to continue the division. This gives us a more exact result, overcoming the limits of dealing only with whole numbers.

Terminating Decimals

Terminating decimals are decimals that come to an end after a certain number of digits. In other words, they do not go on indefinitely like repeating decimals do. In the case of dividing \(-8 \div 25\), the process gave us \(-0.32\). Here, there are no digits that repeat endlessly, hence, it's a terminating decimal.When a decimal terminates, it means the long division has finally arrived at a remainder of zero, signaling the end of the division process. They are often cleaner and easier to work with than repeating decimals. Key points to remember about terminating decimals:- They indicate that a fraction can be perfectly divided, with no repeating remainder.- They can be helpful in ensuring exact calculations.- Sometimes signal a neat, finite decimal representation of a fraction.

Numerator and Denominator

In a fraction, the numerator and denominator are key components:- **Numerator**: This is the top number and shows how many parts of a whole are being considered. In our case, \(-8\) is the numerator.- **Denominator**: This is the bottom number, representing the total number of equal parts that make up a whole. Here, \(25\) is the denominator.Understanding the roles of the numerator and denominator is crucial in converting fractions to decimals. They provide the basis for the division we perform during conversion. The relationship between the numerator and denominator determines the value of the fraction:- A larger numerator compared to the denominator results in a value greater than one.- A smaller numerator means the fraction is less than one.Remember, the process of converting these into a decimal involves dividing the numerator by the denominator, a powerful tool in mathematics for switching between fractional and decimal forms.