Problem 50 - Prealgebra

Question

Problems Work each of the following problems on your calculator. If rounding is necessary, round to the nearest hundred thousandth. $$429.87+16.925$$

Step-by-Step Solution

Answer

The summed value is 446.795, which does not need rounding.

Step 1: Align the Numbers

To add these two numbers, start by aligning them by their decimal points. This makes sure each digit is added to the correct place value, such as units, tenths, hundredths, and so on.

Step 2: Add the Numbers

Add the numbers column by column, starting from the rightmost digit (hundredths place) moving to the left (units and the hundreds place). Add: \[\begin{align*}429.870 \+ 16.925 \\end{align*}\]

Step 3: Calculate the Total

Performing the addition, you get: \[ \begin{align*} 9 & : \, 5 + 0\, (= 5) \8 & : \, 2 + 7\, (= 9) \7 & : \, 9 + 9\, (= 18, \, \text{write} \, 8, \, \text{carry} \, 1) \6 & : \, 6 + 8\, (= 14, \, \text{plus carry 1 equals} \, 15, \, \text{write} \, 5, \, \text{carry} \, 1) \4 & : \, 1 + 2\, (= 3, \, \text{plus carry 1 equals} \, 4) \2 & : \, 4 \left(\text{hundreds place}\right)\end{align*} \]

Step 4: Finalize the Result

The final answer, after carrying out addition as described above, is: \[ 446.795 \]. Check if any rounding is needed: since the instruction is to round to the nearest hundred thousandth and our result has three decimal places, no rounding is necessary in this case.

Key Concepts

Understanding Place Value in Decimal Addition

In decimal addition, each digit has a specific place value, which represents the weight of that digit in the entire number. Understanding place value is essential when aligning numbers for addition because it helps ensure that each digit is added to its corresponding digit. Think of place value as a system where each decimal place has a name:

• Units (ones)
• Tenths
• Hundredths
• Thousandths

This sequence continues onward. So when you have two numbers like 429.87 and 16.925, you will align them by their decimal points to make sure every digit in similar place values add up together. This means units align with units, tenths with tenths, and so on. Failing to align by place value could lead to incorrect results. Remember, even in decimals, the right place matters.

Rounding to the Nearest Hundred Thousandth

Rounding is a process of simplifying a number while keeping its value close to what it really is. In the context of this exercise, you may need to round a decimal to the nearest hundred thousandth. Even though the calculated result in this exercise is 446.795, and doesn't require rounding because it is already within three decimal places, knowing the rounding process is helpful.
To round to the nearest hundred thousandth:

• First, identify the digit in the relevant place. In this case, it's the sixth digit after the decimal.
• Next, look at the digit immediately following it.
• If this digit is 5 or greater, increase the hundred thousandth digit by one. Otherwise, leave it as is.

Understanding rounding enables you to represent numbers precisely and handle results neatly in scientific and everyday contexts.

Making Use of a Calculator for Decimals

Calculators are incredibly handy tools when you're dealing with decimal operations like addition. They can not only help speed up the process but also ensure accuracy.
When adding decimals using a calculator, follow these steps:

• Carefully enter each number, making sure to input the decimal point correctly.
• Press the addition button.
• After entering all numbers, press the equals button to get the result.

Using a calculator does not only handle the mathematical operation efficiently but also helps to verify your steps if you're performing manual calculations. It's always good to double-check your work. Remember, even small errors, like a misplaced decimal, can lead to big mistakes, so take your time to input numbers accurately.