Problem 80 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) (Skills Review pp. \(763,770,771\) ) $$\frac{17}{5} ? 3 \frac{4}{5}$$

Step-by-Step Solution

Answer

\(\frac{17}{5} < 3 \frac{4}{5}\)

Step 1: Convert the Mixed Number to an Improper Fraction

A mixed number is composed of a whole part and a fraction part. To convert \(3\frac{4}{5}\) to an improper fraction, the denominator of the fraction part (5) should be multiplied by the whole number (3) and then add the numerator of the fraction part (4). For this problem, the result is \(5*3+4=19\), so \(3\frac{4}{5}\) can be written as \(\frac{19}{5}\).

Step 2: Compare the Fractions

Now that both numbers are written as fractions, they can be compared directly. In this case, \(\frac{17}{5}\) and \(\frac{19}{5}\) are being compared. Since the denominators are the same (5), the comparison will be between the numerators 17 and 19. Since 17 is less than 19, \(\frac{17}{5} < \frac{19}{5}\). So the solution for \(\frac{17}{5} ? 3 \frac{4}{5}\) is \(\frac{17}{5} < 3 \frac{4}{5}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

Understanding how to convert mixed numbers to improper fractions is essential for comparing fractions effectively. A mixed number consists of a whole number and a fraction, and converting it requires a simple process. Take the mixed number like 3\(\frac{4}{5})\). To convert this to an improper fraction, you multiply the whole number (in this case, 3) by the denominator of the fractional part (which is 5) and then add the numerator of the fraction (which is 4).

The calculation would look like this:

• Multiply the whole number by the denominator: \(3 \times 5 = 15\).
• Add the numerator: \(15 + 4 = 19\).
• Place the sum over the original denominator to get the improper fraction: \(\frac{19}{5}\).

Converting the mixed number to an improper fraction makes it much simpler to perform operations such as comparison, addition, or subtraction with other fractions.

Comparing Fractions With Like Denominators

When you have two fractions with the same denominator, comparison is straightforward. You only need to look at the numerators to determine which fraction is larger or if they are equal. Consider the fractions \(\frac{17}{5}\) and \(\frac{19}{5}\), both with the denominator 5. Compare their numerators - 17 and 19.

Since the denominators are identical, they can be ignored while comparing, and thus:

• If one numerator is greater than the other, that fraction is the larger one.
• If the numerators are equal, the fractions are equal.

In the case of \(\frac{17}{5}\) versus \(\frac{19}{5}\), 17 is less than 19, making \(\frac{17}{5}\) the smaller fraction.

Understanding Numerical Inequality

Numerical inequality is a way of representing the relationship between numbers, indicating whether one number is less than, greater than, or equal to another number. Symbols such as '<' (less than), '>' (greater than), and '=' (equal to) are used to show these relationships.

These symbols are crucial when comparing numbers or quantities. For instance:

• If a number A is smaller than number B, we write \(A < B\).
• If number A is larger than number B, we write \(A > B\).
• If number A is equal to number B, we write \(A = B\).

Recognizing these symbols and understanding how to apply them, particularly in fractions comparison, is fundamental in math. Going back to our fraction comparison, since \(\frac{17}{5}\) is less than \(\frac{19}{5}\), we use \(\frac{17}{5} < 3 \frac{4}{5}\) to show this inequality.