Problem 6 - Prealgebra

Question

Find each of the following sums. (Add.) $$7.65+3.8$$

Step-by-Step Solution

Answer

The sum is 11.45.

Step 1: Align the Decimal Points

To add the numbers, first align the decimal points vertically. Write 7.65 and 3.8, filling in any missing digit places with zero for clarity:\[\begin{array}{c} 7.65 \+3.80 \\hline \end{array}\]

Step 2: Add the Hundredths Place

Add the numbers in the hundredths place from the rightmost column. In this case, it's \(5 + 0 = 5\).

Step 3: Add the Tenths Place

Move to the next column to the left, the tenths place. Add these numbers: \(6 + 8 = 14\). Write 4 below the line in the tenths place and carry over the 1 to the next column on the left.

Step 4: Add the Units and Carryover

Lastly, add the units place numbers and any carryover from the previous column. Add \(1\) (carryover) to \(7 + 3 = 10\) to get 11. Write 1 in the units place and carry over the next 1.

Step 5: Finalize the Sum

Since there was an additional carry from the last step, add the 1 carry to the zero in the tens place giving a final sum:\[\begin{array}{c} 11.45 \\end{array}\]Thus, the complete sum is 11.45.

Key Concepts

Aligning Decimal Points

When adding decimal numbers, it's crucial to align the decimal points vertically. This means each digit falls under the correct place value column, making the addition process clearer and more accurate.

For example, in the problem of adding 7.65 + 3.8, write 7.65 first, then line up 3.8 directly below it, making sure the decimals are in a straight line. Since 3.8 has one less digit after the decimal, append a zero to facilitate the addition:

• 7.65
• +3.80

This approach ensures that each digit is added within its correct place value column, with the tenths matching the tenths and the hundredths matching the hundredths. This precise alignment is the foundation of performing accurate arithmetic with decimals.

Understanding Carry Over

Carrying over is a technique used when the sum of digits in a column exceeds nine. This process helps us handle extra values and functions similarly in decimal addition as it does in whole numbers.

In the exercise, when adding the tenths place (6 + 8 = 14), the number 14 exceeds 10. We write 4 as the result for that place value and carry over the 1. This 1 is then added to the next column on the left (units column), modifying its sum. Thus, carry over allows us to appropriately handle and transfer values across column boundaries during the addition. This way, each sum respects its respective place value.

Importance of Place Value

Place value is the idea that the position of a digit in a number determines its value. In decimal numbers, each digit to the left of the decimal point represents whole units, while each digit to the right represents fractions of a unit.

In this exercise:

• The number 7.65 is made up of 7 whole units, 6 tenths (0.6), and 5 hundredths (0.05).
• 3.8 includes 3 whole units and 8 tenths (0.8).

Understanding place value is essential for adding decimals correctly. It helps us realize why digits in the same decimal place should be aligned and added together. This understanding assures that digits stand in their assigned place, securing the integrity of the number's worth during addition.