Problem 72 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ \frac{25}{990} \quad 0.0 \overline{26} $$

Step-by-Step Solution

Answer

The correct symbol is a less than symbol: \( \frac{25}{990} < 0.0 \overline{26} \).

Step 1: Convert the Fraction to Decimal

First, convert the fraction \( \frac{25}{990} \) into a decimal. To do this, divide 25 by 990. When you perform this division, you get approximately 0.0253 as the decimal representation.

Step 2: Understand the Repeating Decimal

Next, consider the repeating decimal, which is given as \( 0.0 \overline{26} \). This implies that the digits '26' repeat indefinitely. In decimal form, it is equivalent to 0.0262626... and so on.

Step 3: Compare the Converted Values

Now compare the two values obtained: the decimal from the fraction (0.0253) and the repeating decimal (0.0262626...). Since 0.0253 is less than 0.0262626..., the correct inequality is \( \frac{25}{990} < 0.0 \overline{26} \).

Key Concepts

Fractions to Decimals

Converting fractions to decimals is a critical skill in math. It allows you to compare different forms of numbers more easily. To transform a fraction into a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, with the fraction \( \frac{25}{990} \), you divide 25 by 990. This will give you approximately 0.0253.

Here's a simple guide to remember:

• Always set up the division, placing the numerator inside the division bracket.
• Divide until you either find the number repeating or get a precise result.
• If the division does not end, it might become a non-terminating decimal, which we'll explore more in repeating decimals.

Converting fractions accurately helps make comparisons straightforward when dealing with complex math problems.

Repeating Decimals

Repeating decimals happen when a decimal number has a digit or a sequence of digits that keeps repeating infinitely. They are often represented with a line over the repeated digits. In the given problem, \(0.0 \overline{26}\) means the sequence '26' repeats indefinitely: 0.0262626... and so forth.

To identify a repeating decimal, look for:

• Digits or a sequence that recurs without end.
• A line or bar over the repeating numbers in its decimal notation.
• If you're converting from a fraction, keep dividing past the decimal point, and watch for any repeated sequences.

Repeating decimals can sometimes be tricky to compare to other numbers, but understanding how they extend indefinitely will enable a proper comparison.

Inequalities

Inequalities involve comparing two different values to determine their relationship. It's similar to determining which number is bigger or smaller. Inequality symbols like \(<\) for less than, and \(>\) for greater than, help express these relationships.

When facing an inequality problem, like comparing \(\frac{25}{990}\) and \(0.0 \overline{26}\), follow these steps to solve it:

• Ensure both numbers are in the same format, either as decimals or fractions. This makes comparison easier.
• Place the numbers side by side and compare digit by digit from the first decimal place.
• If one decimal consistently outpaces the other, use the correct inequality symbol to express their relationship.

In this exercise, since 0.0253 is less than 0.0262626, the inequality \(\frac{25}{990} < 0.0 \overline{26}\) is true. Knowing how to articulate these comparisons is essential for solving math problems efficiently.