Problem 35 - Prealgebra

Question

Subtract. $$\begin{array}{r}34.07 \\\\-6.18 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The result of the subtraction is 27.89.

Step 1: Align the Numbers

To subtract, the first step is to ensure the numbers are properly aligned by their decimal points. Place 34.07 above 6.18 and ensure the decimal points are vertically aligned.\(\begin{array}{r}34.07 \-6.18 \\hline\end{array}\)

Step 2: Subtract the Hundredths Place

Subtract the digits in the hundredths place: 7 minus 8. Since 7 is smaller than 8, we need to borrow 1 from the tenths place, making 7 into 17. Therefore, 17 minus 8 equals 9.

Step 3: Subtract the Tenths Place

After borrowing from the tenths place, subtract the digits in the tenths place: 0 minus 1 (with 1 borrowed, it is adjusted to 9 in the hundredths place). You have to borrow 1 from the ones place, making it 10. So, 10 minus 1 equals 9.

Step 4: Subtract the Ones Place

With borrowing done, subtract the ones place digits: 4 (after borrowing, it's 3) minus 6. Since 3 is less than 6, borrow 1 from the tens place, making it 14. Thus, 13 (after borrowing) minus 6 equals 7.

Step 5: Subtract the Tens Place

Subtract the tens place digits. Originally, you had a 3 on top, but after borrowing, it becomes 2. So, 2 minus 0 equals 2.

Step 6: Write the Final Answer

Combine the results from each place value to write the final answer of the subtraction: 27.89. Double-check to ensure each place value operation is correct and that it aligns properly under subtraction rules.

Key Concepts

Place Value

Understanding place value is crucial when working with decimals. Each digit in a number is part of a different place value, representing powers of ten. In the number 34.07 for instance:

• The "3" is in the tens place, meaning it represents 3 tens or 30.
• The "4" is in the ones place, contributing 4 to the total.
• The "0" is in the tenths place, a part of the decimal portion, indicating zero tenths.
• The "7" is in the hundredths place, representing seven hundredths.

Place value helps us understand the exact value of numbers when aligning them for arithmetic operations like subtraction.

Borrowing in Subtraction

Borrowing in subtraction helps when a digit in the subtrahend is larger than the corresponding digit in the minuend. For instance, when subtracting the hundredths place:

• You encounter 7 (from 34.07) minus 8 (from 6.18).
• Because 7 is smaller, borrow 1 from the tenths place to make it 17.
• This process continues across the digits when necessary.

Borrowing ensures each subtraction problem is realistic and doable, even when numbers in a particular column of the minuend are smaller than those in the subtrahend.

Aligned Numbers

Properly aligning numbers is key when performing decimal subtraction. This means setting up the digits so that similar place values are in a vertical line.

• Align the decimal points to the same vertical line.
• This helps keep each digit in its correct position: tens above tens, ones above ones, and so on.
• In our example, 34.07 over 6.18 aligns tens, ones, tenths, and hundredths properly.

Always check the alignment before you start subtracting to avoid mistakes and make calculations easier.

Step-by-Step Subtraction

Carrying out subtraction step by step is fundamental to getting the correct result. Here's a simplified look at the complete steps:

• First, ensure numbers are aligned by their decimal points.
• Start subtracting from the rightmost side (hundredths place) moving to the left.
• Borrow between places when necessary, adjusting digits accordingly.
• Continue through the tenths, ones, and finally tens.
• Each move involves checking and re-evaluating if additional borrowing is needed.
• Once all positions are subtracted, write the final answer.

This meticulous, step-by-step process guarantees accuracy, particularly in decimal operations.