Problem 64 - Prealgebra

Question

Explain why each number is a rational number. $$0.75$$

Step-by-Step Solution

Answer

0.75 is a rational number because it can be expressed as the fraction \( \frac{3}{4} \).

Step 1: Identify Characteristics of Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, a rational number can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).

Step 2: Convert Decimal to Fraction

To show that 0.75 is a rational number, we need to express it as a fraction. The decimal 0.75 can be converted to a fraction by recognizing that 0.75 is equivalent to \( \frac{75}{100} \) because the digits 75 are in the hundredths place.

Step 3: Simplify the Fraction

Next, simplify the fraction \( \frac{75}{100} \). Find the greatest common divisor (GCD) of 75 and 100, which is 25. Divide both the numerator and the denominator by their GCD: \( \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \).

Step 4: Confirmation

Now that we have expressed 0.75 as the fraction \( \frac{3}{4} \), and since both 3 and 4 are integers and the denominator is not zero, we can confirm that 0.75 is a rational number.

Key Concepts

Understanding Fractions

Fractions are a way to represent numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the top number, showing how many parts we have. The denominator is the bottom number, indicating how many parts make a whole. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.

• Fractions are essential for understanding parts of a whole.
• They are used for calculations in many areas, including science, cooking, and construction.
• Fractions can be converted to decimals for convenience in certain calculations.

Fractions bring insight into dividing objects into parts and dealing with numbers like \( \frac{3}{4} \) allows us to work with money, time, and resources more precisely. With fractions, you can easily compare sizes by looking at the numerators and denominators.

Decimals and their Conversion

Decimals are another way to represent fractions, particularly when dealing with parts of ten, hundred, thousand, etc. A decimal shows a fraction with a denominator of a power of ten. For example, 0.75 is a decimal.

• The digits after the decimal denote tenths, hundredths, thousandths, etc.
• Decimals make certain arithmetic operations simpler, like addition and subtraction.
• You can convert decimals back into fractions by identifying the place value of the last digit.

To convert a decimal like 0.75 into a fraction, recognize that 75 is in the hundredths place. This translates the decimal to \( \frac{75}{100} \). By simplifying this fraction, you see that it equals \( \frac{3}{4} \). Understanding decimals helps manage everyday math tasks, such as shopping or measuring, where decimal figures are frequently encountered.

The Importance of Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a crucial concept for simplifying fractions. It is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD helps in reducing a fraction to its simplest form. For the fraction \( \frac{75}{100} \), identifying the GCD of 75 and 100 simplifies our understanding.

• The GCD aids in finding equivalent fractions and simplifying mathematical expressions.
• Common methods to find the GCD include prime factorization or using the Euclidean algorithm.
• GCD helps reduce fractions, making calculations more manageable.

In our example, the GCD of 75 and 100 is 25, which simplifies the fraction \( \frac{75}{100} \) to \( \frac{3}{4} \). Mastery of using GCD ensures you can simplify fractions effectively, facilitating clearer insights and simplified math equations.