Problem 70 - Algebra 1

Question

Compare using \(<,=,\) or \(>.\) $$5.68 \underline{?}=5.678$$

Step-by-Step Solution

Answer

5.68 < 5.678

Step 1: Identifying the number of decimal places

First, identify the number of decimal places in each number. In 5.68, there are two decimal places, while in 5.678, there are three decimal places.

Step 2: Make the decimal places the same for easy comparison

Next, add an extra digit to 5.68 to make it have three decimal places for easy comparison. Hence, 5.68 becomes 5.680.

Step 3: Compare the numbers

Now, compare the numbers. Here, 5.680 is less than 5.678.

Key Concepts

Understanding Decimal Places

Decimal places refer to the number of digits to the right of the decimal point. They play a crucial role in determining the value of a decimal number. For example, in the number 5.68, the decimal places are represented by the numbers 6 and 8. Similarly, in 5.678, the decimal places are 6, 7, and 8.

More decimal places can mean a more precise number. But when comparing decimals, difference in decimal places can be tricky. To compare easily, one method is to equalize the number of decimal places.

For instance, turning 5.68 into 5.680 aligns both numbers to three decimal places, making comparison straightforward.

Techniques for Number Comparison

When comparing numbers like decimals, ensure that you are matching the number of decimal places first. This strategy helps avoid errors due to unequal lengths.

Here’s a quick step-by-step way to compare:

• Convert numbers to the same number of decimal places by adding zeros.
• Align the numbers vertically, focusing on each digit from left to right.
• Compare each corresponding digit until a difference is found.

If all digits are the same, the numbers are equal. But if one number has a larger digit in a particular place, then it's the larger of the two.

Using the same technique, you can directly see that once standardized, 5.680 is indeed smaller than 5.678.

The Importance of Place Value

Place value is the value given to a digit based on its position in a number. It's essential for understanding and comparing numbers effectively. In decimal numbers, each position to the right of the decimal point represents a decreasing order of magnitude:

• The first digit after the decimal is the tenths place.
• The second digit is the hundredths place.
• The third is the thousandths place, and so on.

Place value is what allows us to directly compare decimals once their decimal places are aligned. For example, in 5.680, the place values are 5 (units), 6 (tenths), 8 (hundredths), and 0 (thousandths). In contrast, 5.678 has the same initial place value digits but ends with a 7 in the thousandths place, indicating it is larger than 5.680. This systematic understanding of place value is crucial when performing any number comparisons.