Problem 21 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-5 \frac{1}{8}$$

Step-by-Step Solution

Answer

The decimal is \(-5.125\). There is no repeating part.

Step 1: Convert Mixed Number to Improper Fraction

The mixed number is given as \(-5 \frac{1}{8}\). We first convert it to an improper fraction:Multiply the whole number by the denominator: \(-5 \times 8 = -40\).Add the numerator: \(-40 + 1 = -41\).So, the improper fraction is \(-\frac{41}{8}\).

Step 2: Divide the Numerator by the Denominator

Now, divide \(-41\) by \(8\) to convert the fraction into a decimal:\(-41 \div 8 = -5.125\).

Step 3: Analyze the Decimal

Since \(-5.125\) is a terminating decimal, there are no repeating numbers.The answer is \(-5.125\) without a repeating bar.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine a whole number with a fraction. For example, when you see something like \(-5 \frac{1}{8}\), you're looking at a mixed number. In such numbers, the integer part represents full units, while the fractional part represents additional parts of a unit. To work with these numbers in calculations or to convert them into decimals, it's often helpful to convert them into improper fractions first.

Here’s how:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to the result from the multiplication.
• The newly found number becomes the numerator of the improper fraction, while the denominator remains the same.

In our exercise, this process transforms \(-5 \frac{1}{8}\) into the improper fraction \(-\frac{41}{8}\). This conversion simplifies further calculations and conversions to other formats, such as decimals.

Improper Fractions

Improper fractions are those fractions where the numerator is greater than or equal to the denominator. For instance, \(-\frac{41}{8}\) is an improper fraction.
Why use improper fractions?

• They are particularly useful for conversion between fraction formats—such as from mixed numbers to decimals.
• Calculations such as addition and subtraction or finding common denominators become more straightforward when fractions are in improper form.

In our exercise, converting the mixed number \(-5 \frac{1}{8}\) into the improper fraction \(-\frac{41}{8}\) allows for seamless mathematical operations. To find the decimal value, you divide the numerator by the denominator, yielding the decimal value which can then be analyzed for patterns or properties.

Terminating Decimals

Terminating decimals are decimal numbers that have a finite number of digits after the decimal point. This means they "terminate" or end without repeating. In our example, converting the fraction \(-\frac{41}{8}\) yields \(-5.125\), a number that comes to a complete stop after a few digits.

How do you know it's terminating?

• If the decimal does not repeat and has a finite number of digits, it's terminating.
• Generally, when the prime factors of the denominator only include 2s and 5s, the result is a terminating decimal.

Terminating decimals are easier to work with because they are precise and definite, unlike repeating decimals which may require estimation or special notation. They give us exact values that are simple to apply in computations or practical measurements.