Problem 45 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation. $$ 3.71+1.054 $$

Step-by-Step Solution

Answer

The sum of 3.71 and 1.054 is 4.764.

Step 1: Line up the numbers

First, write the two numbers with their decimal points lined up. This is done so that like terms (ones with ones, tenths with tenths, etc.) can be added together. \[ \begin{array}{{r}} 3.71 \ + 1.054 \end{array} \]

Step 2: Add from right to left

Start adding up from the furthest column to the right. In other words, move from the smallest place-value to the largest, carrying any excess to the next column as needed. \[ \begin{array}{{r}} 3.71 \ + 1.054 \ \hline 4.764 \end{array} \]

Key Concepts

Place Value

In decimal numbers, place value is crucial because it helps us understand what each digit represents. Each digit is part of a position that signifies powers of ten. For instance, in the number 3.71, the '3' is in the ones place, the '7' is in the tenths place, and the '1' is in the hundredths place. Similarly, for 1.054, the '1' is in the ones place, the '0' is in the tenths place, the '5' is in the hundredths place, and the '4' is in the thousandths place.
Recognizing place values allows you to organize numbers properly when doing operations such as addition. This ensures that every digit aligns with the digit of the same type and contributes correctly to the final sum.

• Ones: The whole number part.
• Tenths: One tenth of a whole (0.1).
• Hundredths: One hundredth of a whole (0.01).
• Thousandths: One thousandth of a whole (0.001).

Understanding this concept is pivotal for lining up decimals effectively, which is the next step in adding decimals.

Lining Up Decimals

To line up decimals means to make sure the decimal points are vertically aligned. This helps to organize the numbers by place value, which is vital when adding. You want all ones to line up with ones, tenths with tenths, and so on. In our exercise, you should write 3.71 above 1.054, making sure their decimal points are in the same column. This creates a clear structure for addition.
Lining up the decimals is not just about aesthetics. It serves a practical purpose:

• Ensures each digit adds with a digit of the same value. This prevents mistakes.
• Keeps the decimal point consistent in the answer, crucial for accurate results.
• Makes it easier to spot where you might need to "carry" a number over to the next place value.

A well-organized setup paves the way for a seamless addition process.

Adding Decimals

Once your decimals are lined up properly, you start adding from the rightmost column, moving to the left. This mirrors how you would in integer addition but includes handling of decimal points.
Starting with the Rightmost Column:

• Add the digits in the thousandths column first, if present.
• Continue to the hundredths, tenths, and finally the ones column, carrying over any extra if the sum exceeds 9.

This method ensures that all place values contribute correctly to the final result. In our example:

• First, add the thousandths: 0 (from 3.71) + 4 = 4.
• Next, the hundredths: 1 + 5 = 6.
• Then, the tenths: 7 + 0 = 7.
• Finally, the ones: 3 + 1 = 4.

Hence, the resulting sum is 4.764, showing you how precise alignment and careful addition lead to an accurate answer.