Problem 10 - Intermediate Algebra

Question

Show that each of the following numbers is a rational number by expressing it as a fraction with an integer numerator and a nonzero integer denominator: \(7,-7 \frac{3}{5}, \quad 0.007, \quad 700.1\)

Step-by-Step Solution

Answer

The numbers are rational: \(\frac{7}{1}, \frac{-32}{5}, \frac{7}{1000}, \frac{7001}{10}\).

Step 1: Express 7 as a Fraction

The number 7 is already a whole number, which means it can be easily expressed as a fraction. Any integer can be written as itself divided by 1. Thus, \(7\) can be expressed as \(\frac{7}{1}\).

Step 2: Convert Mixed Number to Improper Fraction

The number \(-7 \frac{3}{5}\) is a mixed number. To convert it to an improper fraction, multiply the whole number part by the denominator of the fractional part and add the numerator of the fractional part. That gives \((-7 \times 5) + 3 = -35 + 3 = -32\). The denominator remains the same, which is 5. Thus, \(-7 \frac{3}{5}\) can be expressed as \(\frac{-32}{5}\).

Step 3: Convert Decimal to Fraction

The decimal number 0.007 needs to be converted to a fraction. Since there are three decimal places, this is equivalent to \(\frac{7}{1000}\), as moving the decimal point three places to the right is the same as multiplying by 1000.

Step 4: Convert Decimal Number 700.1 to Fraction

The decimal number 700.1 can be separated into two parts: 700 and 0.1. First, 700 is a whole number and can be expressed as \(\frac{700}{1}\). The decimal \(0.1\) is equivalent to \(\frac{1}{10}\). Adding these two gives \(\frac{7000}{10} + \frac{1}{10} = \frac{7001}{10}\).

Key Concepts

Fractions

A fraction is a way of representing numbers that are not whole. It consists of two parts: the numerator and the denominator. The fraction looks like this \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Here’s a simple breakdown:

• The numerator (the top number) counts how many parts we are considering.
• The denominator (the bottom number) tells how many equal parts make up a whole.

When working with fractions, keep in mind:

• If the numerator is smaller than the denominator, it’s called a proper fraction and indicates a value between 0 and 1.
• If the numerator is equal to the denominator, it simplifies to 1.
• If the numerator is greater than the denominator, it’s known as an improper fraction, indicating a value greater than 1.

Fractions can represent exact values in situations where decimals might not be precise enough, especially in everyday measurements like cooking or construction. Understanding this helps when transitioning between mixed numbers and improper fractions.

Improper Fractions

Improper fractions have numerators that are greater than or equal to their denominators, such as \(\frac{32}{5}\). These fractions represent numbers greater than or equal to 1, which can be useful in various calculations.
To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number part by the fraction's denominator.
• Add the result to the numerator of the fraction part.
• The sum becomes the new numerator and the denominator remains the same.

For example, let’s convert \(-7 \frac{3}{5}\) into an improper fraction:

• Multiply \(-7\) by 5 to get \(-35\).
• Add 3 to \(-35\) to get \(-32\).
• The improper fraction is \(\frac{-32}{5}\).

Transforming mixed numbers into improper fractions allows for simpler calculations, particularly when adding, subtracting, or comparing fractions.

Decimal to Fraction Conversion

Converting decimals to fractions is essential for interpreting values more precisely. The process involves rewriting the decimal in a form where a fraction can naturally describe it. Here's how:

• Identify the place value of the last digit in the decimal. For example, in 0.007, the seven is in the thousandths place.
• Use this place value as the denominator (e.g., \(1000\), in the case of three decimal places).
• The number without the decimal becomes the numerator.

So, for 0.007, move the decimal point three places to the right to get 7, and the fraction becomes \(\frac{7}{1000}\).When working with numbers that contain both a whole number and a decimal, like 700.1, break them down:

• Express the whole number separately as a fraction: \(\frac{700}{1}\).
• Convert the decimal part \(0.1\) into a fraction \(\frac{1}{10}\).
• Add these fractions to form a single fraction: \(\frac{7000}{10} + \frac{1}{10} = \frac{7001}{10}\).

Understanding how to convert decimals to fractions helps communicate values accurately, especially when handling precise calculations in finance and science.