Problem 9 - Fundamentals of Mathematics

Question

For the following problems, complete the table by rounding each number to the indicated positions. 106,007 $$ \begin{array}{|l|l|l|l|} \hline \text { hundred } & \text { thousand } & \text { ten thousand } & \text { million } \\ \hline & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Hundred: 106,000; Thousand: 106,000; Ten thousand: 110,000; Million: 0.

Step 1: Rounding to the Nearest Hundred

Identify the hundreds, tens, and units places in the number 106,007. The number in the hundreds place is 0, the tens place is 0, and the units place is 7. Since the tens digit is 0, which is less than 5, we round down to the nearest hundred. Thus, 106,007 rounded to the nearest hundred is 106,000.

Step 2: Rounding to the Nearest Thousand

Identify the thousands, hundreds, and tens places in the number 106,007. The number in the thousands place is 6, the hundreds place is 0, and the tens place is 0. Since the hundreds digit is 0, which is less than 5, we round down to the nearest thousand. Thus, 106,007 rounded to the nearest thousand is 106,000.

Step 3: Rounding to the Nearest Ten Thousand

Identify the ten thousands, thousands, and hundreds places in the number 106,007. The number in the ten thousands place is 0, the thousands place is 6, and the hundreds place is 0. Since the thousands digit is 6, which is 5 or greater, we round up to the nearest ten thousand. Thus, 106,007 rounded to the nearest ten thousand is 110,000.

Step 4: Rounding to the Nearest Million

Identify the millions and hundred thousands places in the number 106,007. Since our number is less than 1 million overall (the digit in the millions place is implied to be 0), we round down. Therefore, 106,007 rounded to the nearest million is 0.

Key Concepts

Nearest Hundred

Rounding a number to the nearest hundred involves examining the tens digit. The rule is simple: if the tens digit is 5 or more, you round up. If it is less than 5, you round down.

For our example number, 106,007, let's look at the hundreds, tens, and units place:

• The digit in the hundreds place is 0.
• The digit in the tens place is 0.
• The digit in the units place is 7.

Since the tens digit is 0, which is less than 5, you do not increase the hundreds digit when rounding. Simply replace the tens and units digits with zeros. Hence, 106,007 becomes 106,000 when rounded to the nearest hundred.

This method ensures simplicity by focusing only on the nearby numbers, making rounding a "smoothing" process to facilitate easier calculations and approximations.

Nearest Thousand

When rounding a number to the nearest thousand, it requires attention to the hundreds place. Just like before, if the hundreds place digit is 5 or greater, round the current thousand digit up by one; if it’s less than 5, keep your thousand digit as is and replace the following digits with zeros.

For instance, with 106,007:

• The digit in the thousands place is 6.
• The digit in the hundreds place is 0.
• The digit in the tens place is 0.

The hundreds digit, 0, is less than 5, so you do not round up. Instead, the number remains 106,000 upon rounding to the nearest thousand.

This approach makes practical estimations easier, offering a balance between accuracy and simplicity in daily calculations.

Nearest Ten Thousand

To round to the nearest ten thousand, observing the digit in the thousands place is key. The rounding rule is consistent: increase the ten thousand digit by one if the thousands digit is 5 or more. Otherwise, let it be.

Consider 106,007:

• The digit in the ten thousands place is 0.
• The digit in the thousands place is 6.
• The digit in the hundreds place is 0.

Here, the thousands digit is 6, which is greater than 5, prompting a round up. Consequently, 106,007 becomes 110,000 when rounded to the nearest ten thousand.

This rounding technique aids in simplifying data and makes figures manageable while preserving accuracy relative to its scale.

Nearest Million

Rounding to the nearest million involves digits in the millions and hundred thousands places. The idea is to check the hundred thousands digit: if 5 or more, go up; if less, go down.

Analyzing 106,007:

• The digit in the millions place is implicitly 0 (since 106,007 is less than a million).
• The digit in the hundred thousands place is 1.

Since the hundred thousands digit (1) is less than 5, the entire number rounds down to 0 for the nearest million.

Rounding to the nearest million is especially useful for simplifying large numbers in extensive data analysis, keeping estimates straightforward without losing sight of macro trends.