Problem 45 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-9.829-(-17.33)\)

Step-by-Step Solution

Answer

7.501

Step 1: Identify the Operation

Identify whether you need to add or subtract the numbers in the equation. The problem given is \(-9.829 - (-17.33)\). Here, you are subtracting a negative number, which is equivalent to addition.

Step 2: Rewrite the Expression

Rewrite the expression as an addition problem: \(-9.829 + 17.33\). Converting \(-(-17.33)\) to addition simplifies subtraction of a negative to sum up a positive number.

Step 3: Align the Decimals for Addition

Align the decimal points of the numbers vertically to ensure correct addition: \[\begin{array}{r} -9.829 \+17.330 \\hlined \end{array} \]

Step 4: Perform the Addition

Add the two numbers by starting from the rightmost digits and moving to the left:\[7.501\]

Step 5: Sign of the Result

Since 17.33 is greater than 9.829 in absolute value, and it is positive, the result will also be positive.

Key Concepts

Addition of Decimals

Adding decimals is quite similar to adding whole numbers; however, you need to be careful with the placement of the decimal point. Here's how to add them step-by-step:

• **Align the Decimal Points**: Place the numbers one on top of the other, so their decimal points line up. This ensures you're adding digits from the same place value.
• **Add from Right to Left**: Just like with whole numbers, start adding from the rightmost digits first, moving to the left. If a column sums to 10 or more, carry over to the next column.
• **Place the Decimal Point in the Result**: After adding, place the decimal point in the sum directly below the other decimal points.

It's crucial to practice aligning the decimal points properly. This step maintains the integrity of the place values — tens, ones, tenths, hundredths, etc. — and ensures accurate addition.

Subtraction of Decimals

Subtraction of decimals works in much the same way as their addition, with careful attention to the decimal point. To subtract decimals, follow these steps:

• **Align Decimal Points**: Write the numbers vertically, ensuring their decimal points are aligned. This sets up the numbers in a way that reflects their actual value.
• **Subtract from Right to Left**: Begin with the rightmost digits and subtract, moving left. If necessary, borrow from the next column just like with whole numbers.
• **Place the Decimal Point**: In your result, the decimal point should go directly below the aligned points from the numbers above.

Aligning correctly is particularly important in subtraction because it helps prevent mistakes due to misplaced digits, ensuring accuracy in your calculations.

Negative Numbers

Negative numbers can sometimes seem tricky, but they're just as manageable as positive numbers once you understand them. They represent values less than zero and are denoted by a minus sign (−).

• **Adding a Negative Number**: This is equivalent to regular subtraction. For instance, adding \(-9\) to \(5\) is just subtracting \(9\) from \(5\), resulting in \(-4\).
• **Subtracting a Negative**: This action changes the operation to addition. The problem \(-9.829 - (-17.33)\) effectively becomes \(-9.829 + 17.33\), simplifying what seemed like a subtraction into addition.
• **Hierarchy in Arithmetic Operations**: When you see negative signs, remember that subtracting a negative results in a positive. This simple rule clarifies many confusing problems.

Understanding these rules allows you to approach problems involving negative numbers with confidence and precision.