Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to order the numbers from least to greatest.\(\frac{7}{2}, 2 \sqrt{3}, 3.45, \frac{204}{60}, \frac{31}{9}\)

Step-by-Step Solution

Answer

The numbers in increasing order are \( \frac{204}{60} \), 3.45, \( \frac{31}{9} \), \( \frac{7}{2} \), and \( 2\sqrt{3} \).

Step 1: Convert the fractions to decimals

To convert a fraction like \( \frac{7}{2} \) or \( \frac{204}{60} \) or \( \frac{31}{9} \) into a decimal, perform the division operation. Using a calculator, \( \frac{7}{2} \) equals 3.5, \( \frac{204}{60} \) equals 3.4, and \( \frac{31}{9} \) equals approximately 3.44444.

Step 2: Evaluate the square root

Next, evaluate the square root which is \( 2\sqrt{3} \). To calculate this value, first find \( \sqrt{3} \) and then multiply the result by 2. This equals approximately 3.464.

Step 3: Compare the decimal numbers

Finally, compare all these decimal approximations including the original decimal given, which is 3.45, to order them from least to greatest. The decimal numbers to order are 3.5, 3.4, 3.44444, 3.464, and 3.45.

Step 4: Order the numbers

The numbers in increasing order are 3.4, 3.45, 3.44444, 3.5, and 3.464. Don't forget to convert these decimal numbers back to their original forms, if required.

Key Concepts

Fractions to Decimals

When working with numbers, it's often useful to convert fractions into decimals, especially when comparing them. A fraction like \( \frac{7}{2} \) represents a division operation.

• Simply divide the numerator (the top number) by the denominator (the bottom number) to get the decimal.
• For example, \( \frac{7}{2} \) divided gives you 3.5.
• This method is direct, but a calculator can ensure accuracy, especially for complex fractions.

Converting fractions can make it easier to compare numbers since decimals provide a clear way to see which numbers are larger or smaller relative to each other.

Square Root Calculation

Understanding square roots is an essential mathematical skill, particularly when faced with expressions like \( 2 \sqrt{3} \). This involves two main steps:

• First, find the square root of the number. In this case, \( \sqrt{3} \) is approximately 1.732.
• Next, multiply this result by 2, since the expression is \( 2 \times \sqrt{3} \).
• This results in a product of approximately 3.464.

Using a calculator is helpful to ensure precision, as square roots often lead to irrational numbers which cannot be precisely expressed in decimal form.

Decimal Comparison

After converting fractions and calculating square roots, the next step is comparing the resulting decimal numbers. This process is crucial for ordering numbers correctly from least to greatest.

• Write down each decimal number obtained. For this exercise, these were 3.5, 3.4, 3.44444, 3.464, and 3.45.
• Look at each decimal from left to right, comparing the digits in each decimal place sequentially.
• This systematic approach helps to identify which decimals are larger and which are smaller.

Understanding the place values in decimals will aid you significantly in making accurate comparisons.

Mathematical Problem Solving

Putting together all these calculations demonstrates a crucial part of mathematical problem-solving: the ability to process and manipulate numbers in different forms.

• Start by breaking down complex instructions into manageable steps, such as converting each number into a decimal.
• Make sure to use tools like a calculator to handle complicated calculations.
• Then, apply logical steps to arrange numbers based on their values.

This approach not only improves your number handling skills but also enhances your overall mathematical thinking, helping you tackle more challenging problems efficiently.