Problem 23 - Algebra A Combined Function

Question

Subtract. \(9.7-16.1\)

Step-by-Step Solution

Answer

The result is -8.4.

Step 1: Understand the Problem

We need to subtract two decimal numbers: 9.7 and 16.1. Since 16.1 is larger than 9.7, the result will be a negative number.

Step 2: Set Up the Subtraction

Write the numbers vertically for easy subtraction. Make sure to align the decimal points:\[\begin{array}{r} 9.7 \-16.1 \\hline\end{array}\]

Step 3: Subtract the Decimal Portions

Subtract the decimal parts of the numbers. Since 0.1 is greater than 0.7, we will borrow 1 from the whole number part (making the 9 into 8 and the 7 into 17):\[17 - 1 = 6\]

Step 4: Subtract the Whole Number Portions

Now subtract the whole numbers after borrowing:\[8 - 16 = -8\]

Step 5: Combine the Results

Combine the results from Steps 3 and 4:\[-8 + 0.6 = -8.4\]

Step 6: Verification

Double-check the calculation to ensure accuracy: Subtracting the adjusted numbers gives us a final result of -8.4, confirming our calculation.

Key Concepts

Understanding Negative Numbers

Negative numbers can be a bit tricky at first, but with practice, they make perfect sense. A negative number is a number that is less than zero. On a number line, these numbers are to the left of zero. They are often used to represent things like a loss or a deficit. In the problem of subtracting 16.1 from 9.7, we expect a negative result because 16.1 is larger than 9.7. When you subtract a larger number from a smaller one, the answer will be negative. This is similar to having a debt, where the amount owed is greater than what one currently possesses.

In mathematical terms, when you have an equation like 9.7 - 16.1, this is equivalent to saying 9.7 + (-16.1). Here, subtracting 16.1 is the same as adding its negative counterpart. Remember, understanding the concept of negative numbers helps in solving arithmetic problems confidently and accurately.

Borrowing in Subtraction

Subtraction can sometimes require us to use a method called 'borrowing'. This is crucial when the digits in the minuend (a number from which another number is subtracted) are smaller than those in the subtrahend (the number to be subtracted). Take, for instance, the decimal subtraction of 9.7 - 16.1. When we set up this subtraction vertically, we write:

• 9.7
• 16.1

Borrowing occurs when we want to subtract the tenths of 1 from 7 in 9.7, but can't because 0.1 in 16.1 is larger. To resolve this, we 'borrow' 1 from the whole number 9, turning it into 8 and adding 10 to the 7 in the decimal part, making it 17. Now it’s possible to perform the subtraction: 17 - 1 = 6.

Once the decimal subtraction is complete, we can proceed to subtract the whole numbers. But remember, since borrowing occurred, only use the adjusted whole number. Practicing borrowing helps to ensure subtraction problems are handled smoothly, even when dealing with decimals.

Decimal Alignment

Aligning decimals is a vital step in performing accurate arithmetic operations with decimal numbers. For subtraction (as well as addition), proper alignment ensures each digit is placed in its correct place value - tenths are above tenths, ones above ones, etc. The problem of 9.7 - 16.1 requires careful alignment.

To align decimals correctly, make certain the decimal points from each number are stacked one directly above the other. This may require adding placeholder zeros to ensure each column is complete, especially when dealing with numbers with varying numbers of decimal places. In our example:

• 9.7 should be noted as 9.7 or 9.70 for more clarity.
• 16.1 remains as 16.1.

This clarity ensures that when subtraction is performed, particularly with borrowing involved, all digits correspond correctly to their respective place values, since even a slight misalignment can lead to errors in calculation. Practicing decimal alignment will enhance one's accuracy and efficiency in solving mathematical problems with decimals.