Question

Compare using \(<,=,\) or \(>.\) $$21.1 \underline{?} 20 .99$$

Short answer

The comparison sign between \(21.1\) and \(20.99\) is \(>\).

Step-by-step solution

Identify the Numbers

The numbers given for comparison are \(21.1\) and \(20.99.\) The numbers are already in decimal form with two digit accuracy.

Compare the Whole Number Part

Look at the whole number part of both numbers. The whole number part of \(21.1\) is \(21\) and of \(20.99\) is \(20.\) Since \(21\) is greater than \(20,\) therefore \(21.1\) will be greater than \(20.99\) irrespective of the decimal parts.

Conclusion

Conclude that \(21.1\) is greater than \(20.99.\) Hence, the correct comparison sign is \(>\).

Key concepts

Decimal Comparison

Understanding how to compare decimals is crucial in both everyday life and in academic settings. When looking at numbers like 21.1 and 20.99, the process of comparing them requires attention to both the whole number and decimal parts. The first step is to compare the digits to the left of the decimal point. If they are not equal, as with numbers 21 (from 21.1) and 20 (from 20.99), the comparison is straightforward: the number with the greater whole number part is larger overall. This basic comparison informs us that 21.1 is greater than 20.99.

To be thorough, even when whole numbers are the same, we would compare the decimal parts digit by digit from left to right. For example, in comparing 21.15 and 21.09, after noting that the whole numbers 21 are equal, we would then look at the tenths place (after the decimal), finding that '1' is the same in both numbers. Moving to the hundredths place, we see '5' is greater than '0', making 21.15 the greater number. This digit-by-digit comparison is necessary when the whole numbers are equal.

Inequalities

Understanding Inequalities

Inequalities are expressions that show the relative size or order of two values. The symbols used in inequalities are less than (<), equal to (=), and greater than (>). When comparing decimals, as in the provided example, these symbols become our tools for expressing the mathematical relationship. The symbol '>' confirms that 21.1 is greater than 20.99.

Inequalities help to visually and conceptually organize information on a number line, with smaller numbers to the left and larger numbers to the right. This spatial representation can make it easier to understand and remember the relationship between numbers. When learning to use inequalities, remember that the open end of the symbol always faces the larger number and the pointed end to the smaller number, like a crocodile's mouth that always wants to eat the bigger meal!

Basic Number Sense

Basic number sense is an intuitive understanding of numbers and their relationships. It enables us to estimate, compare, and understand numerical concepts without relying on cumbersome calculations. For instance, when comparing 21.1 and 20.99, our number sense tells us that 21.1 must be greater than 20.99 because it has a larger whole number part—even before we examine the decimal parts.

Developing a strong number sense involves being comfortable with numbers and recognizing patterns. For example, knowing that any whole number followed by a decimal is greater than any smaller whole number with a decimal (thus, 21.x is always greater than 20.y, regardless of the x and y values). It is a fundamental skill in math that allows for quick and accurate estimations and comparisons, like knowing off-hand that 21.1 is not just greater than 20.99, but also understanding it is close to 21 and far from 22.