Question

Perform the indicated operations. Give exact answers. \(3.005+4.308+2.47=\) ______

Short answer

The exact answer is 9.783.

Step-by-step solution

Understand the Problem

We need to find the sum of three decimal numbers: 3.005, 4.308, and 2.47. The goal is to perform the addition of these numbers and provide an exact numeric result.

Align the Decimals

To add decimal numbers, align them by the decimal point to ensure that each digit is in the correct place value position. This will be helpful when adding they are neatly organized:\[\begin{array}{c} \ \, \, 3.005 \\ \, \, + 4.308 \\ \, \, + 2.470\end{array}\]

Add the Numbers Column by Column

Start adding from the rightmost side (the thousandths place) and move toward the leftmost side (the units place). When necessary, carry over any extra value to the next column on the left.- Thousandths: 5 + 0 + 0 = 5.- Hundredths: 0 + 8 + 7 = 15 (write 5 and carry over 1).- Tenths: 0 + 3 + 4 + 1 (carry over) = 8.- Units: 3 + 4 + 2 = 9.Thus, the total sum is:\[9.783\]

Verify Your Answer

Ensure all digits have been considered correctly and the carryovers were accurately executed. Check addition again if necessary. The calculated result is 9.783.

Key concepts

Place Value Alignment

When it comes to adding decimals, aligning the decimal points ensures each digit occupies the correct position by value. Just like when we add whole numbers, each number must be in the correct place value – thousands, hundreds, tens, units, etc. With decimals, the principle is the same, but now you include tenths, hundredths, and thousandths.
Always line up decimal points directly under each other. This makes sure that each digit aligns with its corresponding value category, making addition straightforward and reducing errors. Consider the example:

• 3.005
• 4.308
• 2.470 (note the added zero)

The zero added to 2.47 does not change its value but ensures all numbers reach the same decimal length, making addition clear and easy.

Carryover in Addition

The carryover step in addition is crucial when the sum of digits in one column exceeds 9, and you need to add that extra value to the next column to the left. This process helps in maintaining the proper numeric value throughout the operation. In our example:

• Thousandths column: 5 + 0 + 0 = 5, no carryover needed.
• Hundredths column: 0 + 8 + 7 = 15. Here, you write down 5 and carry over 1 to the tenths column.
• Tenths column: 0 + 3 + 4 = 7, plus the carryover of 1 makes it 8.
• Units column: 3 + 4 + 2 = 9. No carryover is necessary since the sum is a single-digit number.

By accurately applying carryover, you ensure that the final result reflects the numbers' true combined value.

Step-by-Step Problem Solving

A step-by-step approach in problem-solving is a methodical way to tackle any task, ensuring every aspect is addressed in sequence. For decimal addition, this involves:

• Identifying the numbers you need to sum.
• Aligning decimals for organized addition.
• Adding digits starting from the smallest place value.
• Checking your work for any possible errors.

Taking each step carefully helps ensure no detail is missed, leading to successful completion and an accurate result. Even when a mistake occurs, it is easier to pinpoint exactly where it happened and correct it when each step is clearly laid out.

Exact Numeric Result

Achieving an exact numeric result is vital in decimal addition. Unlike estimations, exact results yield precise values, especially when using decimals, since fractions and small units can accumulate and cause significant changes based on minor additions. In our problem: - The exact answer is computed as 9.783. After completing addition, reviewing each step is crucial to verify no digits have been miscalculated and every carryover was handled appropriately. It provides confidence that the final result truly represents the precise sum of the involved numbers. This accuracy is essential in various real-world contexts, such as financial calculations, scientific measurements, and any other scenario where precision is a priority.