Problem 19 - Fundamentals of Mathematics

Question

For the following problems, complete the table by rounding each number to the indicated positions. 23,047,991,521 $$ \begin{array}{|l|l|l|l|} \hline \text { hundred } & \text { thousand } & \text { ten thousand } & \text { million } \\ \hline & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

23,047,991,500; 23,047,992,000; 23,048,000,000; 23,000,000,000

Step 1: Round to the nearest hundred

To round 23,047,991,521 to the nearest hundred, look at the tens digit, which is 2. Since 2 is less than 5, you round down. Thus, the number rounded to the nearest hundred is 23,047,991,500.

Step 2: Round to the nearest thousand

To round 23,047,991,521 to the nearest thousand, examine the hundreds digit, which is 5. As 5 is equal to or greater than 5, you round up. Therefore, the number rounded to the nearest thousand is 23,047,992,000.

Step 3: Round to the nearest ten thousand

For rounding to the nearest ten thousand, check the thousands digit, which is 9. Since 9 is greater than 5, round up the ten thousands digit. The rounded number is 23,048,000,000.

Step 4: Round to the nearest million

To round to the nearest million, the hundred thousands digit is considered, which is 7. Since this is greater than 5, round up the millions digit. Therefore, 23,047,991,521 rounded to the nearest million is 23,000,000,000.

Key Concepts

Understanding Place Value

Place value is a fundamental concept in understanding how numbers work. It refers to the position of a digit within a number and determines its value. Each place in a number has a different value:

• The first digit from the right is the "ones" place.
• The next digit to the left is the "tens" place.
• The third place is "hundreds," then "thousands," and it continues up in powers of ten.

This sequence is crucial when rounding numbers, as it tells us which digit to examine and potentially increase or decrease depending on rounding rules. For example, in the number 23,047,991,521, each digit has a specific place that influences the overall magnitude of the number.

Rounding to the Nearest Hundred

When rounding a number to the nearest hundred, you focus on the tens digit, which helps you decide whether to round up or down.

• If the tens digit is 5 or greater, you round up.
• If it's less than 5, you round down.

For instance, in the number 23,047,991,521, the tens digit is 2. Since 2 is less than 5, round down, and the result is 23,047,991,500. This process helps simplify numbers, making them easier to work with or estimate in calculations.

Rounding to the Nearest Thousand

Rounding to the nearest thousand involves checking the hundreds digit of a number.

• Round up if this digit is 5 or more.
• Round down if it is less than 5.

In 23,047,991,521, the hundreds digit is 5, so we round up. This gives us 23,047,992,000. Understanding rounding is important for estimating costs, distances, or anything else where approximate values are sufficient.

Rounding to the Nearest Ten Thousand

To round a number to the nearest ten thousand, look at the thousands digit.

• If the thousands digit is 5 or more, round up.
• If it's less than 5, round down.

Using this rule, the thousands digit in 23,047,991,521 is 9. We therefore round up, resulting in 23,048,000,000. This method is common in financial contexts where accuracy to ten thousand is more practical than exact figures.

Rounding to the Nearest Million

When rounding to the nearest million, the hundred thousands digit informs the decision.

• If this digit is 5 or greater, round up the millions digit.
• If less than 5, round down.

In the number 23,047,991,521, the hundred thousands digit is 7, so we round up, resulting in 23,000,000,000. Rounding to the nearest million often simplifies large numbers in reports and data analysis, allowing for a clearer overview of numbers on such a grand scale.