Problem 27 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$149.37-28.96$$

Step-by-Step Solution

Answer

The difference is 120.41.

Step 1: Align Decimal Points

To subtract the numbers 149.37 and 28.96, we first write them with the decimal points aligned. This helps in ensuring that each digit is in the correct position relative to the decimal point.\[ \begin{array}{r} 149.37 \ - 28.96 \ \hline \end{array}\]

Step 2: Subtract Rightmost Column

Subtract the numbers in the hundredths place (rightmost column). Subtract 6 from 7, which gives us 1. Write 1 down in the result.\[ \begin{array}{r} 149.37 \ - 28.96 \ \hline . \cdot 1 \end{array}\]

Step 3: Subtract Tenths Column

Subtract the numbers in the tenths place. Subtract 9 from 3, which requires borrowing since 3 is smaller than 9. Borrow 1 from the 4 in the tenths place, turning it from 4 to 3, and increasing 3 to 13. Subtract 9 from 13 to get 4.\[ \begin{array}{r} 149.3_{3}^{13}7 \ - 28.96 \ \hline \cdot .41 \end{array}\]

Step 4: Borrow for Units Place

Subtract the units place digit by digit. We need to handle the next borrowing. Subtract 8 from 8 (after borrowing), resulting in 0.\[ \begin{array}{r} 149.3_{13}7 \ - 28.96 \ \hline 0 .41 \end{array}\]

Step 5: Subtract Tens Place

Subtract the tens place. Subtract 2 from 4 and get 1. Write down the result.\[ \begin{array}{r} 149.37 \ - 28.96 \ \hline 1{0}.41 \end{array}\]

Step 6: Subtract Hundreds Place

Finally, subtract the hundreds place. 1 from 1 leaves 0, which completes the subtraction.\[ \begin{array}{r} 149.37 \ - 28.96 \ \hline 120.41 \end{array}\]

Step 7: Write the Final Answer

After completing the subtraction in each column, we find that the difference between 149.37 and 28.96 is 120.41. This value is our final answer.Result:\[149.37 - 28.96 = 120.41\]

Key Concepts

Aligning Decimal Points

Before performing decimal subtraction, it's crucial to align the decimal points of the numbers being subtracted. Aligning the decimals is similar to ensuring that columns in a column-based math operation are straight and organized. It allows for an accurate subtraction.

• Write down both numbers vertically, ensuring that the decimal points are directly under each other.
• This helps keep each digit in the correct column, like the tenths with tenths, hundredths with hundredths, etc.
• For example, placing the numbers 149.37 and 28.96 directly under one another ensures that calculations are made accurately without any misplaced digits.

Aligning decimal points might seem trivial, but it prevents mistakes that could arise from misplacement of digits. This step sets the stage for an error-free subtraction task.

Borrowing in Subtraction

Borrowing is a technique used in subtraction when a digit in a minuend (the number from which another number is subtracted) is smaller than the corresponding digit in the subtrahend. Consider the example of subtracting 28.96 from 149.37.

• When we reach the tenths column, we notice that 3 is smaller than 9. To solve this, we borrow 1 from the next column to the left (the ones column).
• This borrowing increases the digit in the tenths place from 3 to 13. By doing so, the calculation becomes feasible: subtracting 9 from 13 yields 4.
• This borrowing adjusts the digit in the next column, reducing it usually by 1, so be sure to account for that change in subsequent steps.

This process is essential because it maintains the numerical integrity of each place value, ensuring our subtraction remains consistent and accurate.

Subtracting Decimals

Subtracting decimals follows the same principles as regular subtraction but requires special attention to decimal places. Proper execution involves:

• Begin with aligning the decimal points as discussed, which ensures clarity in digit arrangement.
• Start subtraction from the rightmost column, which is the hundredths place in this example, and proceed leftwards. Subtract each column consecutively.
• Apply borrowing where necessary to handle cases where the top digit is smaller than the bottom digit.
• After subtracting all the columns, ensure that the decimal point in the answer is placed directly below the originating decimal points.

Using the problem example, subtracting 28.96 from 149.37 results in 120.41, indicating careful subtraction leads to accurate results. This methodical process ensures no decimal place is left unaccounted for, maintaining the precision which decimals require.