Problem 24 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$5 \frac{1}{4} \odot \sqrt{26}$$

Step-by-Step Solution

Answer

Replace \(\odot\) with \(>\): \(5 \frac{1}{4} > \sqrt{26}\).

Step 1: Convert the Mixed Number

Convert the mixed number into an improper fraction. The mixed number given is \(5 \frac{1}{4}\), which can be rewritten as an improper fraction. To do this, multiply the whole number 5 by the denominator 4, then add the numerator 1: \(5 \times 4 + 1 = 21\). So, \(5 \frac{1}{4} = \frac{21}{4}\).

Step 2: Approximate the Square Root

Estimate the value of \(\sqrt{26}\). Since \(\sqrt{25} = 5\) and \(\sqrt{36} = 6\), we know that \(\sqrt{26}\) is slightly greater than 5.

Step 3: Convert Improper Fraction to Decimal

Convert the improper fraction \(\frac{21}{4}\) to a decimal to compare it with \(\sqrt{26}\). Perform the division: \(21 \div 4 = 5.25\).

Step 4: Compare the Values

We now compare the decimal value of the improper fraction \(5.25\) and the estimated square root value greater than \(5\). Since \(5.25\) is greater than \(\sqrt{26}\), we conclude that \(5.25 > \sqrt{26}\).

Step 5: Conclude with the Correct Symbol

Finally, replace \(\odot\) with \(>\) because \(5 \frac{1}{4} > \sqrt{26}\), making the statement true.

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They allow us to express quantities that are larger than whole numbers but don't completely reach the next whole number. Knowing how to handle mixed numbers is important in math, as we often need to convert them into improper fractions for easier calculations.

Here's how to convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction part.
• The sum is placed over the original denominator to form the improper fraction.

For example, with the mixed number \(5 \frac{1}{4}\):

• Multiply 5 (whole number) by 4 (denominator): \(5 \times 4 = 20\).
• Add the numerator 1 to get 21.
• Write this as \(\frac{21}{4}\), which is the equivalent improper fraction.

By converting mixed numbers to improper fractions, we make comparing and calculating expressions simpler.

Decoding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Knowing how to approximate square roots can be very useful, especially for calculations that don't require precise results.

For example, the square root of 26, denoted by \(\sqrt{26}\), requires us to estimate. We know:

• \(\sqrt{25} = 5\)
• \(\sqrt{36} = 6\)

Therefore, \(\sqrt{26}\) is between 5 and 6, likely closer to 5 as 26 is closer to 25 than 36. Knowing this helps us to quickly compare \(\sqrt{26}\) with other numbers like the mixed number converted to an improper fraction. When approximating square roots, learning the roots of perfect squares (like 4, 9, 16, 25, etc.) is particularly helpful as reference points.

Converting and Comparing Improper Fractions

Improper fractions have numerators larger than their denominators and represent values greater than 1. Converting these fractions into decimals can simplify comparisons with other numerical forms such as roots or mixed numbers.

To illustrate, let's take the improper fraction \(\frac{21}{4}\):

• We convert it by dividing the numerator (21) by the denominator (4): \(21 \div 4 = 5.25\).

This gives us the decimal representation, making it much easier to compare with approximations of square roots or while cross-examining other numerical expressions.

When comparing the decimal \(5.25\) with the estimated value of \(\sqrt{26}\), which is slightly over 5, it's clear that \(5.25\) is greater. Conversions and comparisons like these can help with solving problems that involve different types of numbers.