Problem 51 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-4.560-2.335\)

Step-by-Step Solution

Answer

The result is \(-6.895\).

Step 1: Set Up The Problem

First, write down the two numbers you are subtracting, ensuring their decimal points are aligned vertically:\[-4.560\]\[-2.335\]

Step 2: Subtract Each Digit Starting from the Right

Begin subtracting the numbers from the rightmost decimal place (thousandths place) and move leftwards towards the whole number. Subtract each digit of the bottom number from the corresponding digit above. Note: If a digit in the top number is smaller than the corresponding digit in the bottom number, you will need to borrow from the next left column.

Step 3: Borrow if Necessary

After borrowing, perform the subtraction:- Thousandths: \(0 - 5\) is not possible, so borrow 1 from the hundredths column making it \(10 - 5 = 5\).- Hundredths: Now with borrowing hundredths becomes \(5 - 3 = 2\).- Tenths: \(5 - 3 = 2\) remains.- Units: Borrow from the tens place, making the subtraction \(14 - 2 = 12\).Now, you have:\[-6.895\]

Step 4: Finalize the Borrowed Column

After borrowing for the units digit, adjust the tens digit by decreasing it by 1 (due to the previous borrowing) which results in \(-6 - 2 = -8\).

Step 5: Reconstruct and Present the Final Answer

Put all the digits together, keeping the negative sign from the original number, to show the final result of the subtraction:\[-6.895\]

Key Concepts

Subtracting Decimals

Subtracting decimals might seem complicated at first, but it follows similar principles to subtracting whole numbers, with an additional focus on decimal places. Just like in whole number subtraction, you need to find out how much one number decreases when the other number is taken away. What makes subtracting decimals unique is the presence of places like tenths, hundredths, and thousandths, which require careful handling.

When subtracting decimals like \(-4.560 - 2.335\), your goal is to find the difference between these two numbers. The result will help you understand how much smaller \(-4.560\) is compared to the sum of \(2.335\) taken from it. Always remember to keep track of the negative sign when dealing with negative numbers, as it will affect your final outcome.

Align the Decimal Points

Aligning the decimal points is a crucial step in ensuring your subtraction of decimals is accurate. This means that when you write the numbers being subtracted, their decimal points should line up vertically. This alignment helps in making sure that each digit is in its correct column representing its place value—units with units, tenths with tenths, and so forth.

For example:

• Write the numbers so that each value after the decimal point is directly under its counterpart in the other number.
• In our exercise, you set up \[-4.560\] directly above \[-2.335\].

This allows you to perform subtraction column by column, without mixing the different decimal place values.

Borrowing in Subtraction

Borrowing in subtraction is a technique used when a digit in the top number is smaller than the corresponding digit in the bottom number. This occurs often during decimal subtraction and requires you to take 1 from the nearest left column. This helps to make the necessary subtraction possible.

In our exercise:

• Start from the rightmost column: \(0 - 5\) is impossible, so you "borrow" 1 from the next column (hundredths), making it \(10 - 5 = 5.\)
• For hundredths: After borrowing, the subtraction becomes \(5 - 3 = 2.\)
Each borrowing action requires adjusting the other digits by reducing their value by 1.
This ensures each subtraction step follows through without error.

Place Value Understanding

Understanding place value is essential in decimal arithmetic, especially while subtracting decimals. Place value refers to the value of where a digit is in the number; like units, tenths, and hundredths. This understanding helps you know what each digit in a number represents and is crucial when aligning decimal points.

Each place to the right of the decimal point is a fraction of 10 of the previous one, meaning:

• Units are whole numbers.
• We move to tenths (\(0.1\)), hundredths (\(0.01\)), thousandths (\(0.001\)), etc.

Grasping place value helps you follow the subtraction process smoothly, as you can see exactly what value you are subtracting from at every column. It also helps in understanding where you need to borrow correctly, preserving the balance of the equation.