Problem 108 - Elementary Algebra

Question

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{28}{5} \quad \frac{47}{9} $$

Step-by-Step Solution

Answer

\(\frac{28}{5} > \frac{47}{9}\)

Step 1: Convert to Decimal Form

First, convert each fraction to its decimal form. This allows for easier comparison between the two.

Step 2: Convert \(\frac{28}{5}\)

Divide 28 by 5 to convert \(\frac{28}{5}\) to its decimal form. \(\frac{28}{5} = 5.6\)

Step 3: Convert \(\frac{47}{9}\)

Divide 47 by 9 to convert \(\frac{47}{9}\) to its decimal form. \(\frac{47}{9} \approx 5.222\)

Step 4: Compare the Values

Now, compare the decimal equivalent of the two fractions. \(\frac{28}{5} = 5.6\) and \(\frac{47}{9} \approx 5.222\). Since 5.6 is greater than 5.222, \(\frac{28}{5} > \frac{47}{9}\).

Step 5: Write the Comparison Statement

Finally, write the comparison statement using '>' or '<'. In this case: \(\frac{28}{5} > \frac{47}{9}\).

Key Concepts

Decimal Conversion

Decimal conversion is key in comparing fractions. By converting fractions to decimal form, it’s easier to see which is larger or smaller. Let’s look at how to convert fractions into decimals. First, you divide the numerator (the top number) by the denominator (the bottom number). For example, with the fraction \(\frac{28}{5}\):

• Divide 28 by 5.
• You get 5.6.

Similarly, for the fraction \(\frac{47}{9}\):

• Divide 47 by 9.
• You get approximately 5.222.

So, \(\frac{28}{5} \) is 5.6 and \(\frac{47}{9}\) is approximately 5.222. Converting fractions to decimals helps in better comparing their values.

Inequalities

Inequalities compare the relative size of two numbers. The symbols '>' means 'greater than' and '<' means 'less than'. When you use decimal conversion, it becomes straightforward to apply these symbols. For instance, after converting, we have 5.6 and 5.222:

• Since 5.6 is more than 5.222, we use the '>' symbol.
• Thus, the inequality statement is \(\frac{28}{5} > \frac{47}{9}\).

Understanding inequalities is essential for comparing and ordering fractions. Once you know your decimals, applying the appropriate inequality is simple.

Mathematical Operations

Mathematical operations are fundamental in solving fraction comparison problems. Operations like division are crucial for converting fractions to decimals. Here's what you need to do:

• For \(\frac{28}{5}\), you divide 28 by 5, resulting in 5.6.
• For \(\frac{47}{9}\), you divide 47 by 9, resulting in roughly 5.222.

Once you have converted fractions to decimals, comparing is easy. It’s like lining up numbers on a ruler! Remember, accurate division is key to correct comparisons.