Problem 75 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ \frac{99}{100} \quad \text{_____}\quad0.99 $$

Step-by-Step Solution

Answer

The correct symbol is \(=\).

Step 1: Convert fraction to decimal

Convert the fraction \( \frac{99}{100} \) to a decimal number. You do this by dividing the numerator by the denominator: \(99 \div 100 = 0.99\).

Step 2: Compare the two decimal numbers

Now, you have two numbers to compare: \(0.99\) and \(0.99\). Since both numbers are equal, the correct symbol to use is \(=\).

Step 3: Choose the correct symbol

The statement after inserting the correct symbol is: \( \frac{99}{100} = 0.99 \).

Key Concepts

Understanding Fractions

Fractions are a way of representing numbers by showing the relationship between a part and a whole. In this method, a fraction consists of two numbers: the numerator and the denominator.
The numerator is the top number in the fraction, which represents the number of parts you have. The denominator is the bottom number, indicating the total number of parts that make up a whole.
For example, in the fraction \( \frac{99}{100} \), 99 is the numerator and 100 is the denominator. This means you have 99 parts out of 100, or 99% of a whole.
Converting fractions to other forms, like decimals, can help make comparisons easier as it gives a uniform method of comparison.

Decimals Demystified

Decimals are another way to express parts of a whole, similar to fractions. They are written on a base-ten system, using the decimal point to separate the whole number from the fractional part.
For instance, 0.99 can be read as 99 hundredths, which makes it equivalent to the fraction \( \frac{99}{100} \). Using decimals can simplify comparison because decimals are aligned by their place values.
Decimals provide precise representation of fractions and are widely used in both everyday contexts, like in prices and measurements, and in scientific and technical settings.
Understanding decimals is crucial for accurately comparing and converting from fractions, ensuring you can tackle problems like these with confidence.

The Power of Mathematical Symbols

Mathematical symbols are key in expressing operations and relationships between numbers. They allow us to succinctly convey whether numbers are equal, greater than, or less than each other.
In this context, the symbols \( > \), \( < \), and \( = \) play an important role in comparisons:

• Use \( > \) when the first number is greater than the second number.
• Use \( < \) when the first number is less than the second number.
• Use \( = \) when both numbers are exactly equal.

For example, in comparing \( \frac{99}{100} \) with its decimal counterpart 0.99, the symbol \( = \) is used, since both represent the same value. Understanding how to correctly use these symbols is essential for solving mathematical problems efficiently and accurately.