Problem 28 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$1 \frac{1}{2} \odot \sqrt{2.25}$$

Step-by-Step Solution

Answer

The symbol is \(=\).

Step 1: Convert Mixed Number to Improper Fraction

Start by converting the mixed number \(1 \frac{1}{2}\) into an improper fraction. \(1 \frac{1}{2}\) is equal to \(\frac{3}{2}\).

Step 2: Evaluate the Square Root

Now let's determine \(\sqrt{2.25}\). Recognize that \(2.25 = \left(\frac{3}{2}\right)^2\). Thus, \(\sqrt{2.25} = \frac{3}{2}\).

Step 3: Compare Fractions

Compare \(\frac{3}{2}\) with \(\frac{3}{2}\). Since both fractions are equal, we can conclude that \(\frac{3}{2} = \frac{3}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine whole numbers and fractions, such as \(1 \frac{1}{2}\). They are often used for expressing quantities that are more than one unit but not quite whole numbers either.
To convert a mixed number to an improper fraction, follow these simple steps:

• Multiply the whole number by the denominator of the fraction part.
• Add the numerator of the fraction part to this product.
• Place this sum over the original denominator.

For example, \(1 \frac{1}{2}\) becomes \(\frac{3}{2}\), since \(1 \times 2 + 1 = 3\), which is divided by the original denominator \(2\). This conversion is essential for comparing mixed numbers with other fractions or numbers.

Square Roots

The square root is a mathematical function that finds a number which, when multiplied by itself, gives the original number.
In the given exercise, the square root of \(2.25\) is calculated. To simplify this, realize that \(2.25\) can be expressed as \(\left(\frac{3}{2}\right)^2\).

Square roots can be tricky but understanding perfect squares helps simplify calculations. Recognizing that \(\sqrt{2.25} = \frac{3}{2}\) means we're looking for a number multiplied by itself to return the value \(2.25\). Knowing this can help in solving math problems that involve estimation or simple algebra.

Improper Fractions

Improper fractions are fractions where the numerator (top number) is larger than the denominator (bottom number). For instance, \(\frac{3}{2}\) is an improper fraction because \(3\) is larger than \(2\).
They are very useful in calculations because they avoid dealing with separate whole and fractional parts.

Converting between mixed numbers and improper fractions is often necessary to simplify expressions or solve equations. An improper fraction helps to maintain precision and ease during operations.
Ensure the fraction is in its simplest form to accurately compare it with other numerical values.