Problem 55 - Thinking Mathematically

Question

Explain how to round \(14.26841\) to the nearest hundredth and to the nearest thousandth.

Step-by-Step Solution

Answer

Rounding \(14.26841\) to the nearest hundredth gives \(14.27\), and rounding to the nearest thousandth gives \(14.268\).

Step 1: Understand what hundredths and thousandths are

Before starting with the rounding, it's necessary to understand what hundredths and thousandths are. In a decimal number, the hundredths place is two digits to the right of the decimal point, and the thousandths place is three digits to the right.

Step 2: Round to the nearest hundredth

To round to the nearest hundredth, look at the digit in the thousandths place, which is 4 in this case. If this digit is 5 or more, increase the hundredths digit by one; if it is 4 or less, leave the hundredths digit as it is. Here it is 4, so leave the hundredth digit, 6, as it is. Therefore, rounded to the nearest hundredth, \(14.26841\) becomes \(14.27\).

Step 3: Round to the nearest thousandth

Similarly, to round to the nearest thousandth, look at the digit in the ten thousandths place, which is 1 in this case. Apply the same rules as above. Here, since the digit is 1, which is less than 5, the thousandths digit, 8, remains the same. Therefore, rounded to the nearest thousandth, \(14.26841\) becomes \(14.268\).

Key Concepts

Decimal Places

Decimal places are crucial in understanding numbers, especially when dealing with decimals. In a decimal number, the places right after the decimal point represent different values, such as tenths, hundredths, thousandths, and so on. Each place signifies a division of the whole number into smaller and smaller parts:

• The first place to the right of the decimal point is the tenths place.
• The second is the hundredths place.
• The third is the thousandths place, and so forth.

Understanding decimal places helps in accurately interpreting and calculating precision in numbers, which is essential for tasks like rounding.

Nearest Hundredth

To round a number to the nearest hundredth, focus on the second digit after the decimal point. The hundredths place determines the number after tenths. When rounding to the nearest hundredth, consider the digit in the thousandths place (the third digit after the decimal point). Follow these steps:

• If the thousandths digit is 5 or more, increase the hundredths digit by one.
• If the thousandths digit is 4 or less, keep the hundredths digit the same.

For example, in the number 14.26841, the digit in the hundredths place is 6. The thousandths digit is 4, so, following the rounding rule, we keep the hundredths digit as 6. The rounded number to the nearest hundredth is, therefore, 14.27.

Nearest Thousandth

Rounding to the nearest thousandth focuses on the third digit after the decimal point. This is where the value is accurate within three positions past the decimal. To determine if you round up or keep the digit as it is, check the digit right after the thousandths place, the fourth digit:

• If this digit is 5 or more, increment the thousandths digit by one.
• If this digit is 4 or less, leave the thousandths digit unchanged.

Take 14.26841 as an instance. The digit in the thousandths place is 8, and the ten-thousandths digit is 1. Since 1 is less than 5, we leave the thousandths digit at 8. Thus, 14.26841 rounds to 14.268 when rounded to the nearest thousandth.

Mathematical Rounding Rules

Mathematical rounding rules are guidelines that help determine how to simplify numbers while maintaining their core value. When rounding, we decide whether to increase or keep the digit based on the next smallest decimal place. The most common rule is the "5 and above, give it a shove; 4 and below, let it go" rule. This means:

• If the digit to be rounded is followed by 5 or more, increase the rounding digit by 1.
• If it is followed by 4 or less, keep the rounding digit the same.

These rules are vital for ensuring precision and accuracy, whether rounding to whole numbers or specific decimal places like the nearest hundredth or thousandth. Rounding is often used in various life situations such as calculating finances, estimates in measurements, and simplifying complex figures to make them easier to work with.