Problem 71 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ \frac{27}{22} \quad 1.2 \overline{28} $$

Step-by-Step Solution

Answer

Insert \(<\) for a true statement: \( \frac{27}{22} < 1.2 \overline{28} \).

Step 1: Understanding the Problem

We have to determine which symbol, either \(<\) or \(>\), makes the statement true for the numbers \(\frac{27}{22}\) and \(1.2\overline{28}\).

Step 2: Convert the Fraction to Decimal

Convert \(\frac{27}{22}\) to a decimal by performing the division: \(27 \div 22 = 1.22727\ldots\), which approximates to \(1.2273\).

Step 3: Understand Recurring Decimal

The number \(1.2\overline{28}\) means \(1.2282828\ldots\) with the "28" repeating indefinitely.

Step 4: Compare the Decimals

Now, compare the decimal forms: \(1.2273\) and \(1.2282828\ldots\). Notice that \(1.2273\) is less than \(1.2282828\ldots\) since \(1.2273\) is less than \(1.2283\).

Step 5: Establish the Relationship

Since \(1.2273 < 1.2282828\ldots\), we find that \(\frac{27}{22} < 1.2\overline{28}\).

Key Concepts

Decimal Conversion

Decimal conversion is the process of transforming a fraction or any other form of number into a decimal form. This conversion can help make calculations easier. For instance, when comparing fractions or evaluating expressions, decimals can often give a clearer picture.

To convert a fraction into a decimal, you simply divide the numerator by the denominator. For example, for the fraction \(\frac{27}{22}\), you divide 27 by 22, which results in approximately 1.2273. This method offers a straightforward way to see the decimal equivalent of a fraction.

Converting fractions to decimals is especially useful when dealing with non-standard numbers or when trying to compare to other decimals.

Comparing Fractions and Decimals

Comparing fractions and decimals is a crucial skill in mathematics, especially when solving problems involving inequalities. When comparing these two forms of numbers, it's often helpful to convert fractions into decimals first. This helps because decimals can be easier to compare directly, especially when it comes to identifying greater or lesser values.

Once both numbers are represented in decimal form, such as turning \(\frac{27}{22}\) into \(1.2273\), it simply becomes a matter of comparing decimal values. Just like comparing numbers, align them with their decimal points and compare digits one by one from the left to right.

• If the decimal places are equal, keep checking until a difference is found.
• In our exercise, after conversion, \(1.2273\) was less than \(1.2\overline{28}\), thus proving \(1.2273 < 1.2282828\ldots\).

Comparisons become significantly easier when decimals are used, helping ensure quick and accurate decisions.

Repeating Decimals

Repeating decimals, also known as recurring decimals, are decimals that have one or more repeating sequences of numbers after the decimal point. For example, in the number "\(1.2\overline{28}\)", the digits "28" repeat indefinitely.

When you encounter repeating decimals, it means you will see the same digits continue in the same order. It's noted by placing a line over the repeating digits, like the bar in "\(\overline{28}\)". Identifying repeating patterns is crucial to understanding and accurately representing the value of the number.

To effectively compare repeating decimals to other numbers, treat them as infinite series. In calculations, you usually take a few extra repeating digits to ensure a thorough comparison. A repeating decimal like "1.2\overline{28}" indicates the decimal continues as "1.2282828\ldots". This concept ensures you understand the actual size of the number when making comparisons, such as deciding whether it is greater than or less than another decimal, fraction, or whole number.