Problem 73 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=\). \(2.7 ? \frac{14}{5}\)

Step-by-Step Solution

Answer

\(<\)

Step 1: Convert the fraction to a decimal number

Convert \(\frac{14}{5}\) into a decimal by dividing 14 (numerator) by 5 (denominator). This gives you 2.8.

Step 2: Compare the decimal numbers

Compare the two decimal numbers 2.7 and 2.8. Since 2.7 is less than 2.8, the correct comparison operator to be used is \(<\).

Key Concepts

Decimal Conversion

Understanding decimal conversion is crucial when working with numerical values in different forms such as whole numbers, fractions, and percentages. A decimal is a number that uses a decimal point to separate the whole number part from the fractional part. In our daily lives, we encounter decimals in prices, measurements, and statistics.
To convert a fraction or other number type into a decimal, you need to express it in terms of 10, 100, 1000, etc. This conversion often involves simple division. For example, the fraction \(\frac{1}{2}\) written as a decimal is 0.5 because 1 divided by 2 equals 0.5. Similarly, the number 7 as a decimal is 7.0, which clarifies that there are no fractional parts.

Fraction to Decimal

Converting fractions to decimals is a key math skill that allows for easier comparison and mathematical operations. A fraction consists of a numerator (the top number) and a denominator (the bottom number). To convert a fraction to a decimal, divide the numerator by the denominator.
For instance, let's look at our example \(\frac{14}{5}\). Divide 14 by 5. The result is a decimal 2.8.
Some fractions, like \(\frac{1}{4}\), convert to exact decimals, in this case, 0.25. However, some fractions may produce repeating decimals, such as \(\frac{1}{3}\) which equals approximately 0.333.... Always confirm if a decimal terminates or repeats to ensure accuracy in solving problems.

• Perform the division of the numerator by the denominator.
• Check if the decimal terminates or repeats.
• If repeating, you can round the decimal for practical uses.

Comparing Decimals

Once you have numbers in decimal form, comparing them becomes straightforward. Decimals are compared by looking at each digit from left to right, beginning with the whole number part. If the numbers have the same whole part, you then compare the decimal parts digit by digit.
Consider the decimals 2.7 and 2.8 from our earlier example. Since the whole number is the same (both are 2), start comparing the tenths place. Here, 7 is less than 8. Therefore, we know that 2.7 is less than 2.8, so we use the symbol \(<\).
When comparing decimals:

• Align the numbers vertically to compare digit by digit.
• Compare whole number parts first.
• Then, compare each subsequent decimal place value.
• Use inequality symbols \(<, >, =\) to express the relationship.

This process helps in ordering numbers, assessing sizes, or determining greater values in practical situations.