Problem 10 - Technical Mathematics with Calculus

Question

Adding and Subtracting Approximate Numbers Combine each set of approximate numbers as indicated. Round your answer. $$39.75+27.4$$

Step-by-Step Solution

Answer

The sum of 39.75 and 27.4 rounded to the nearest whole number is 67.

Step 1: Add the Numbers

Combine the two approximate numbers using addition. Just like regular addition, line up the decimal places and add the numbers column by column. This will give you the exact sum before rounding.

Step 2: Round the Answer

Round the sum to the nearest whole number. Check the digit right after the decimal point. If it is 5 or more, round up. If it is less than 5, round down. Since the number after the decimal is 5 or more (1.15), we round up.

Key Concepts

Rounding Decimals

When working with approximate numbers, rounding decimals is an essential skill that simplifies figures to make them easier to work with and understand. When you round a number, you're adjusting it to the nearest whole number or specified decimal place based on certain rules. Here's how it typically works:

• If the digit to the right of your rounding place is 5 or higher, you round up the digit in your rounding place.
• If the digit is less than 5, you leave the digit in the rounding place as it is (round down).

For example, when rounding 39.75 to the nearest whole number, you check the first decimal place, which is 7 in this case. Since 7 is greater than 5, you'd round up to 40. It's a straightforward process that ensures consistency in your calculations and aids in easier interpretation of results.

Decimal Addition

The process of adding decimal numbers, or 'decimal addition,' is similar to adding whole numbers, but with the added step of aligning the decimal points. It's important to ensure that all the decimal places line up vertically before you begin adding, just like in our example with 39.75 and 27.4:

• First, write the numbers one under the other with the decimal points aligned.
• Second, add each column of digits, starting from the right-most digit and moving left.
• Finally, place the decimal point in the sum directly below the decimal points of the numbers you're adding.

In our exercise, once you add 39.75 and 27.4, you would positionally write the numbers, carry if necessary, and make sure the decimal point in the sum aligns with those of the numbers being added. This ensures accurate results before any rounding takes place.

Significant Figures

Significant figures encompass the digits within a number that are meaningful in terms of accuracy and precision. They include all the non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion. Here's what you need to keep in mind:

• The initial zeros are not significant as they are only placeholders.
• Any trailing zeros to the right of the decimal are considered significant.

In the context of our addition problem (39.75 + 27.4), every digit is significant because they all contribute to the precision of the numbers. When combining required exactness with everyday use, it's essential to recognize which digits in your calculation carry meaning to ensure the reliability of your final rounded answer.