Problem 9 - Contemporary Precalculus

Question

Use a calculator and list the given numbers in order from smallest to largest. $$\frac{189}{37}, \frac{4587}{691}, \quad \sqrt{47}, \quad 6.735, \quad \sqrt{27}, \quad \frac{2040}{523}$$

Step-by-Step Solution

Answer

In the given list of numbers, which includes fractions, square roots, and a decimal, arrange them in order from smallest to largest. Answer: $$\frac{2040}{523}, \quad \frac{189}{37}, \quad \sqrt{27}, \quad \frac{4587}{691}, \quad 6.735, \quad \sqrt{47}$$

Step 1: Convert Fractions to Decimals

Using a calculator, convert the three fractions into decimals: $$\frac{189}{37} \approx 5.108, \quad \frac{4587}{691} \approx 6.636, \quad \frac{2040}{523} \approx 3.899$$

Step 2: Compute the Square Roots

Using a calculator, find the decimal approximations of the square roots: $$\sqrt{47}\approx 6.856, \quad \sqrt{27} \approx 5.196$$

Step 3: List all the Numbers as Decimals

Now that all numbers are in decimal form, list them together: $$5.108, \quad 6.636, \quad 3.899, \quad 6.735, \quad 5.196, \quad 6.856$$

Step 4: Order the Decimals from Smallest to Largest

Comparing each decimal value, rearrange the list in order from smallest to largest: $$3.899, \quad 5.108, \quad 5.196, \quad 6.636, \quad 6.735, \quad 6.856$$

Step 5: Write the Original Numbers in the Same Order

Match the ordered decimals back to their original numbers and write them in the same order: $$\frac{2040}{523}, \quad \frac{189}{37}, \quad \sqrt{27}, \quad \frac{4587}{691}, \quad 6.735, \quad \sqrt{47}$$ That's the given numbers listed in order from smallest to largest.

Key Concepts

Fractions to Decimals

Converting fractions into decimals is a fundamental concept in precalculus that can simplify many calculations. Essentially, a fraction is a division problem: the numerator (top number) is divided by the denominator (bottom number). Using a calculator, you can quickly convert these fractions into decimals. For instance:

• To convert \( \frac{189}{37} \), divide 189 by 37 to get approximately 5.108.
• Similarly, \( \frac{4587}{691} \) becomes approximately 6.636.
• And \( \frac{2040}{523} \) converts to around 3.899.

Converting fractions into decimals allows for easier comparison with other numerical formats. This helps when ordering numbers or performing additional calculations, as decimals are often more intuitive to work with, especially when using a calculator.

Ordering Numbers

Ordering numbers requires converting them into a common format to accurately compare their sizes. In this exercise, we've converted all numbers, whether they are fractions, pure decimals, or square roots, to decimal form. Once in decimal form, it's straightforward to arrange them from smallest to largest:

• Convert each number: fractions to decimals and square roots to decimal approximations.
• List out all decimals.
• Compare the decimal values and reorder them from the lowest value to the highest.
• Match these ordered decimals back to their original forms.

This process ensures that even numbers originally presented in different forms can be orderly understood. Ordering is especially practical in contexts requiring decision-making based on numerical values, such as budgeting or statistics.

Square Roots

Square roots are another essential component of precalculus, providing a way to find a number that, when multiplied by itself, equals the original number under the root symbol. Using a calculator, we can find decimal approximations. For instance:

• \( \sqrt{47} \) is approximately 6.856.
• \( \sqrt{27} \) is around 5.196.

Understanding square roots in decimal form makes it easier to integrate them into operations with other numerical formats, like ordering numbers or comparing values. Calculators simplify this process, offering precise decimal approximations to help with valid comparisons and calculations. In mathematics, familiarity with square roots is fundamental, especially when solving quadratic equations or analyzing geometric figures.