Problem 12 - Calculus: One and Several Variables

Question

Replace the symbol \(*\) by \(<,>,\) or \(=\) to make the statement true. \(0.33 * \frac{1}{3}\).

Step-by-Step Solution

Answer

The statement that makes the comparison true is: \(0.33 < \frac{1}{3}\).

Step 1: Understand the Exercise

The exercise wants you to compare two quantities. The decimal 0.33 and the fraction 1/3. The task is to determine whether 0.33 is less than, equal to or greater than 1/3.

Step 2: Convert Fraction to Decimal

Convert the fraction 1/3 to a decimal. After making the division, you find out that the result is approximately 0.3333.

Step 3: Compare Decimal Values

With the decimal representation of the fraction, compare 0.33 with the fraction’s decimal equivalent, 0.3333. It is clear that 0.33 is less than 0.3333.

Key Concepts

Converting Fractions to Decimals

To solve problems involving the comparison of decimals and fractions, it's often helpful to convert fractions to decimals. This conversion makes it easier to directly compare values.- A fraction represents a part of a whole, while a decimal is another representation of that fraction.- To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number).In our exercise, the fraction \(\frac{1}{3}\) needs to be converted to a decimal to make the comparison with 0.33. When you perform the division \(1 \div 3\), you get an approximate result of 0.3333. Why approximate? Because the division of 1 by 3 is a repeating decimal that goes infinitely. Knowing how to convert fractions to decimals is a crucial math skill that simplifies the process of comparing numbers and understanding their relative values.

Understanding Inequalities

Inequalities are mathematical expressions used to compare the sizes of two numbers or expressions. When we examine inequalities, we aren't just looking to determine if two numbers are equal, we're also interested to see how they relate in terms of size.- The symbols for inequalities include \(<\) for less than, \(>\) for greater than, and \(=\) for equal to.- When comparing two decimals, we use these symbols to describe their relationship.For instance in the exercise, we have decimal \(0.33\) and fraction \(\frac{1}{3}\), which we have converted to 0.3333. By comparing these two decimals, we determine the lesser or greater one. This helps us assign the correct inequality symbol between them. The understanding of inequalities helps in many areas, like solving equations and analyzing data.

Comparing Numbers

Comparing numbers is a fundamental skill in mathematics that allows us to analyze and understand their relationships.- To compare numbers, especially decimals, we start from the leftmost digit and work our way right until we find a difference.- Even a slight difference past the decimal point can change the result of a comparison.In the example from the exercise, when comparing \(0.33\) with \(0.3333\), the stored decimal places make a significant difference. The smaller decimal, 0.33, is less than 0.3333 due to fewer repeating threes. Identifying which number is larger or smaller is important for making informed decisions in statistical analyses, everyday calculations, and many aspects of scientific studies.