Problem 86 - Prealgebra

Question

Place the correct inequality symbol, \(<\) or \(>\) between each pair of numbers. $$\frac{9}{10} \quad \frac{10}{11}$$

Step-by-Step Solution

Answer

\(\frac{9}{10} < \frac{10}{11}\)

Step 1: Analyze the Fractions

We are given two fractions, \( \frac{9}{10} \) and \( \frac{10}{11} \). To determine which fraction is larger, we can compare them using a common denominator or by converting them to decimals.

Step 2: Convert Fractions to Decimals

Convert each fraction to a decimal to make comparison easier:\[ \frac{9}{10} = 0.9 \]\[ \frac{10}{11} \approx 0.909 \]

Step 3: Compare the Decimals

Compare the decimal values from Step 2:\(0.9\) vs. approximately \(0.909\).Since \(0.9 < 0.909\), \( \frac{9}{10} \) is less than \( \frac{10}{11} \).

Step 4: Place the Correct Inequality Symbol

Based on the comparison from Step 3, we place the inequality symbol for less than between the fractions:\[\frac{9}{10} < \frac{10}{11}\]

Key Concepts

Understanding Fractions

Fractions are numbers that represent parts of a whole. They consist of two main components: the numerator and the denominator. The numerator is the top number of the fraction and indicates how many parts of the whole are being considered. The denominator is the bottom number and shows the total number of equal parts the whole is divided into. For example, in the fraction \(\frac{9}{10}\):

• The numerator is 9, which means we have 9 parts of the whole.
• The denominator is 10, indicating that the whole is divided into 10 equal parts.

Fractions can be simple to compare when they have the same denominator, but when they have different denominators like \(\frac{9}{10}\) and \(\frac{10}{11}\), we may need additional steps. One effective method is to convert the fractions into decimal form, which simplifies the comparison process.

Decimal Conversion Made Easy

Converting fractions to decimals is a crucial skill for comparing different fractions that don't share the same denominator. For instance, to convert a fraction to a decimal:

• Divide the numerator by the denominator.
• The result of this division is the decimal equivalent.

Let's apply this to our example:

• For \(\frac{9}{10}\), divide 9 by 10 to get 0.9.
• For \(\frac{10}{11}\), dividing 10 by 11 yields approximately 0.909.

These decimal conversions help make the comparison clear, especially when the decimals have the same number of decimal places. It simplifies the process of determining which number is larger or smaller.

How to Compare Decimals

When you have two decimals, comparing them is straightforward. You begin by comparing the digits starting from the left:

• Look at the whole numbers first. If they differ, the larger whole number represents the larger decimal.
• If the whole numbers are the same, compare each subsequent decimal place until a difference is found.

In our previous example:

• Both decimals, 0.9 and 0.909, have the same whole number, 0.
• Next, compare the tenths place. Here, 9 is equal to 9, so move to the next place value.
• Finally, compare the hundredths place. Since 0 in 0.9 is less than 9 in 0.909, 0.9 is less than 0.909.

Thus, the inequality \(\frac{9}{10} < \frac{10}{11}\) is confirmed.