Problem 17 - Basic Engineering Mathematics

Question

Evaluate \(42.7+3.04+8.7+0.06\)

Step-by-Step Solution

Answer

The sum is 54.50.

Step 1: Align the Decimals

To add the numbers with decimals, first align them by their decimal points so that the decimal places are directly under each other. Here are the numbers aligned: \[\begin{align*}42.70 \ 3.04 \ 8.70 \ 0.06 \\end{align*}\]

Step 2: Add the Hundredths Place

Start by adding the digits in the hundredths place. The numbers in this column are 0, 4, 0, and 6. Adding them gives us:\[0 + 4 + 0 + 6 = 10\]Write 0 in the hundredths place in the sum and carry over 1 to the tenths place.

Step 3: Add the Tenths Place

Adding the digits in the tenths place (7, 0, 7, and 0) along with the carry-over 1 gives us:\[7 + 0 + 7 + 0 + 1 = 15\]Write 5 in the tenths place in the sum and carry over 1 to the units place.

Step 4: Add the Units Place

Now, add the digits in the units place (2, 3, and 8) along with the carry-over 1:\[2 + 3 + 8 + 1 = 14\]Write 4 in the units place in the sum and carry over 1 to the tens place.

Step 5: Add the Tens Place

Finally, add the digits in the tens place (4) along with the carry-over 1:\[4 + 1 = 5\]Write 5 in the tens place in the sum.

Step 6: Write the Final Sum

The final sum of all the numbers is:\[\boxed{54.50}\]

Key Concepts

Arithmetic

Arithmetic is a branch of mathematics that deals with numbers and the basic operations on them, such as addition, subtraction, multiplication, and division. In the context of our problem, we're focusing on addition, more specifically, decimal addition.

When adding numbers with decimals, the key is to increase the total by summing each individual part of the number. This process breaks down into manageable units that can be solved step by step.

• Add hundreds, tens, units, tenths, and so on, ensuring they are properly aligned.
• Each column or place value should be summed before moving to the next.
• We add any carry over to the next column.

Proper handling of each digit ensures an accurate result. In this exercise, it's important to also understand the role of each digit and how they contribute to form the complete sum.

Place Value

Understanding place value is crucial for performing arithmetic operations accurately. Place value refers to the value of a digit based on its position within a number. Each position corresponds to a power of ten.

In our decimal addition example, each digit falls into a specific category:

• The tens place (represented by the number 4 in 42)
• The units place (represented by the number 2 in 42)
• The tenths place (represented by the number 7 in 42.7)
• The hundredths place (represented by the number 0 in 42.70)

Understanding these distinctions and organizing them correctly by aligning decimal points make it easier to add, as each digit aligns with the same place value and can be directly summed.

Carry Over

The carry over in addition allows for accurate computation by transferring the excess value from one place to the next higher place value. It is crucial when the sum of digits in one column exceeds the base amount (10 in the decimal system).

In this problem, when adding the hundredths place, the sum of 0 + 4 + 0 + 6 gives 10. Thus, 0 is written in the hundredths place, and 1 is "carried over" to the tenths place.

• Carries occur only when digits surpass a single unit of their place (i.e., 10 tenths becoming 1 whole unit).
• Each carry affects subsequent columns, elevating their value by one.
• This ensures that our sum remains consistent with place value principles.

This carrying over is fundamental to achieving the correct sum in addition, ensuring that no value is lost in translation from one place value to the next.