Problem 29 - Prealgebra

Question

The problems below are a review of the four basic operations with fractions and decimals. Add $$2.03+11.958+0.002$$

Step-by-Step Solution

Answer

The sum of the given numbers is 13.990.

Step 1: Align Decimals

Before adding, align the numbers by their decimal points to ensure that each place value lines up correctly. If needed, append zeros to ensure uniformity in the number of decimal places. For example, write the numbers as follows: \[\begin{align*}2.030 \11.958 \0.002 \\end{align*}\]

Step 2: Add Column by Column

Start adding from the rightmost end (the thousandths place). Add each column, being careful to carry over any excess to the next left column. Proceed through each column:Thousandths: \(0 + 8 + 2 = 10\), place 0 and carry over 1.Hundredths: \(3 + 5 + 0 + 1 = 9\).Tenths: \(0 + 9 + 0 = 9\).Units: \(2 + 1 = 3\), note there is no carryover as there are no higher place values.

Step 3: Write the Final Sum

Combine your results from each column to write down the complete sum:\[2.030 + 11.958 + 0.002 = 13.990.\]

Key Concepts

Fractions and Decimals

Understanding both fractions and decimals forms the foundation of basic arithmetic. They are two different ways of expressing non-whole numbers.
Fractions show a number as a part of a whole, with a numerator and a denominator, such as \( \frac{1}{2} \). On the other hand, decimals use a decimal point to express parts less than one, like 0.5.
Decimals are frequently used in everyday situations and calculations because they are easier to read and use in calculations.

• Fractions are perfect for showing exact portions, particularly in recipes or dividing things evenly.
• Decimals help in operations like adding, subtracting, and comparing because they align with the base-10 number system.

Becoming fluent in converting and operating with fractions and decimals is crucial for mastering more advanced math topics.

Aligning Decimals

Aligning decimals correctly is a key step when performing arithmetic operations like addition or subtraction with decimal numbers. Proper alignment ensures that digits are lined up correctly according to their place value.
To align decimals:

• Place the numbers one under the other so that the decimal points form a straight, vertical line.
• Add zeros to the ends if necessary to balance the number of digits after the decimal point.

For example, when lining up the numbers 2.03, 11.958, and 0.002, add necessary zeros to represent them as 2.030, 11.958, and 0.002. This alignment helps you correctly add each decimal by focusing on the position, such as hundredths and thousandths, which is crucial for precise calculations.

Basic Operations in Arithmetic

Basic operations in arithmetic include addition, subtraction, multiplication, and division. Each has its role in solving numerical problems efficiently.
For decimal addition, after aligning the numbers correctly by their decimal places, start adding from the smallest place value, which is usually to the rightmost decimal digit. This method not only simplifies addition but also reduces errors.
When performing addition:

• Always start from the rightmost position (smallest place value).
• If numbers in a column add up to ten or more, write down the last digit and carry over the additional value to the next column.

This technique helps to systematically add each place value, similar to traditional whole number addition, ensuring accuracy and clarity in your calculations.