Problem 85 - Prealgebra

Question

Place the correct inequality symbol, \(<\) or \(>\) between each pair of numbers. $$\frac{3}{8} \quad \frac{5}{6}$$

Step-by-Step Solution

Answer

\( \frac{3}{8} < \frac{5}{6} \)

Step 1: Understand the Problem

We need to determine which of the two fractions, \( \frac{3}{8} \) and \( \frac{5}{6} \), is larger. To do this, we'll compare their values by expressing them with a common denominator.

Step 2: Find a Common Denominator

Determine the least common denominator (LCD) for the fractions \( \frac{3}{8} \) and \( \frac{5}{6} \). The denominators are 8 and 6, and their least common multiple is 24. So, we'll use 24 as the common denominator.

Step 3: Convert Each Fraction

Convert \( \frac{3}{8} \) to a fraction with a denominator of 24 by multiplying both the numerator and denominator by 3: \( \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \). Convert \( \frac{5}{6} \) to a fraction with a denominator of 24 by multiplying both the numerator and denominator by 4: \( \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \).

Step 4: Compare the Converted Fractions

Now, compare \( \frac{9}{24} \) and \( \frac{20}{24} \). Since they have the same denominator, simply compare the numerators: 9 and 20. Since 9 is less than 20, \( \frac{9}{24} < \frac{20}{24} \).

Step 5: Write the Conclusion

The original fractions \( \frac{3}{8} \) and \( \frac{5}{6} \) must follow the same inequality as their converted versions, so we conclude that \( \frac{3}{8} < \frac{5}{6} \).

Key Concepts

Least Common Denominator

When you're comparing fractions, it can be tricky because they often have different denominators. The easiest way to compare them is to convert them into equivalent fractions that have the same denominator. This will give you a clear picture of which fraction is larger or smaller.

To do this, we use the **least common denominator (LCD)**. The LCD is the smallest number that both denominators can divide into evenly. Think of it as the lowest common multiple of the denominators.

Let's use the fractions from our example: \( \frac{3}{8} \) and \( \frac{5}{6} \). Here, the numbers we're concerned with are 8 and 6. To find the LCD, list the multiples of each number:
* 8, 16, 24, 32, 40...* 6, 12, 18, 24, 30...
As you see, **24** is the smallest number in both lists. That makes it your least common denominator. Once you have this, you can convert both fractions into new ones with 24 as their denominator. This simplifies the comparison.

Fraction Conversion

Once we've identified the least common denominator, we need to convert our fractions so they share this new denominator. This process is essential because it allows us to make accurate comparisons.

Here's how you convert a fraction: Take a fraction like \( \frac{3}{8} \) and multiply both the numerator and the denominator by the factor that will turn the denominator into the LCD. Since 8 times 3 equals 24, we multiply like so: \( \frac{3 imes 3}{8 imes 3} = \frac{9}{24} \).

Next, take \( \frac{5}{6} \). We know 6 times 4 gives us 24, so multiply both parts: \( \frac{5 imes 4}{6 imes 4} = \frac{20}{24} \). Now both fractions have 24 as their denominator.

• \( \frac{3}{8} = \frac{9}{24} \)
• \( \frac{5}{6} = \frac{20}{24} \)

Now, you’re ready to compare them directly.

Inequality Symbols

After converting fractions to have the same denominator, you can easily compare them using inequality symbols. These symbols help us express if one number is larger or smaller than another. The main symbols we use are:

• < "less than"
• > "greater than"

Now, let's apply this to our converted fractions \( \frac{9}{24} \) and \( \frac{20}{24} \). You only need to compare the numerators when the denominators are the same.

In this case, 9 is less than 20. So, you place a "less than" symbol in between: \( \frac{9}{24} < \frac{20}{24} \). Hence, the original fractions \( \frac{3}{8} \) and \( \frac{5}{6} \) fit the same pattern: \( \frac{3}{8} < \frac{5}{6} \).

Using inequality symbols correctly shows the relationship between numbers in a clear, concise way.