Problem 68 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ \frac{21}{50} \quad 0.4 $$

Step-by-Step Solution

Answer

\( \frac{21}{50} > 0.4 \)

Step 1: Convert the Fraction to Decimal

First, convert the fraction \( \frac{21}{50} \) into a decimal by performing the division. Divide 21 by 50 using long division or a calculator. After performing the calculation, you get \( 0.42 \).

Step 2: Compare the Decimals

Now that you have both numbers in decimal form, compare \( 0.42 \) and \( 0.4 \). Since \( 0.42 \) is greater than \( 0.4 \), you will use the greater than symbol \( > \).

Step 3: Insert the Correct Symbol

Based on the comparison, insert the appropriate symbol. The statement should read \( \frac{21}{50} > 0.4 \).

Key Concepts

Fractions to Decimals

Converting fractions to decimals is a fundamental skill in math. It involves transforming a fraction, which represents a part of a whole, into its equivalent decimal form. The process is simple: divide the numerator (top number) by the denominator (bottom number). By converting fractions into decimals, you make it easier to compare them with other numbers. In our example, we converted \( \frac{21}{50} \) to a decimal by dividing 21 by 50, which gives us \( 0.42 \). This way, you can see the value represented by the fraction in a more accessible way, making subsequent comparisons a breeze.

Decimal Comparison

Once you've converted fractions to decimals, comparing them is straightforward. You line them up and see which is larger or smaller. Decimal comparison hinges on analyzing each digit from left to right. Start with the digits before the decimal point, then move to those after it. In our example, we compare \( 0.42 \) and \( 0.4 \). First, look at the tenths place: both have a 4. Move to the hundredths place: \( 0.42 \) has a 2, while \( 0.4 \) has an implicit 0. Clearly, \( 0.42 \) is greater than \( 0.4 \), making the comparison simple and direct.

Mathematical Symbols

Mathematical symbols help us succinctly communicate complex ideas. They serve as shortcuts to express relationships like equality or inequality. In inequalities, symbols like \( < \) (less than) and \( > \) (greater than) show how numbers relate. In the exercise, you were asked to insert either \( > \) or \( < \) between \( \frac{21}{50} \) and \( 0.4 \). Since the decimal \( 0.42 \) is greater than \( 0.4 \), you use the greater than symbol: \( > \). Understanding these symbols is key to clearly expressing mathematical relationships and making your point quickly.