Problem 14 - Prealgebra

Question

In Exercises 13-24, subtract the decimals. \(98.14-7.27\)

Step-by-Step Solution

Answer

The difference is 90.87.

Step 1: Align the Numbers

Write the numbers vertically and ensure that the decimal points are aligned. This means you write 98.14 on top and 7.27 below it, aligning their corresponding digits.

Step 2: Subtract the Hundredths Place

Start from the rightmost digit (hundredths place) and subtract 7 from 4. Since 4 is smaller than 7, we need to borrow 1 from the tenths place.

Step 3: Subtract the Tenths Place

After borrowing, the digit on the tenths place in 98.14 becomes 0 (since we borrowed 1 to make the hundredths digit 14), now subtract 2 from 0. Again, we need to borrow 1 from the unit place to turn the tenths into 10.

Step 4: Subtract the Units Place

Subtract 7 from 7, which gives 0. Remember that the borrow affected this column resulting in 7 turning to 6.

Step 5: Subtract the Tens Place

Finally, subtract 0 from 9 (since 9 remains unaffected). This results in a 9.

Step 6: Write the Final Answer

The result is 90.87 after all borrowing and subtraction operations are completed.

Key Concepts

Borrow Method in Decimal Subtraction

When you subtract decimals, sometimes you need to borrow. To "borrow" means to take one unit from a more significant place value to make subtraction possible in a smaller place value where the top number is smaller than the bottom number.
Essentially, the borrow method ensures you have enough value to perform the subtraction at each place.

• **Identify where borrowing is needed:** In the decimal subtraction of 98.14 minus 7.27, you can't directly subtract 7 (hundredths) from 4. So, borrowing from the tenths place makes the subtraction 14 - 7.
• **Borrowing chain reaction:** Once you borrow from one place value, it affects the rest. Here, borrowing from the tenths changes that column too, which in turn requires borrowing from the units, etc.
• **End with clean subtraction:** After borrowing, perform the subtraction easily down the line starting from right to left.

Each borrowing step gets you closer to your final answer. It might look complex, but it's like borrowing sugar from your neighbor – just ensuring you have what you need to get through the process.

Understanding Place Value in Decimal Subtraction

Place value is the foundation of decimal subtraction. Each digit in a number represents a value depending on its position.
Understanding place value is crucial when it comes to alignment in subtraction.

• **Right order subtracting:** Start from the smallest place value (like hundredths) and move left. This maintains the correct order and flow.
• **Aligning decimals is key:** Write the numbers vertically with decimal points aligned, ensuring each respective digit is in the right column. For example, in 98.14 and 7.27, ensure units align with units, tenths with tenths, etc.
• **Significance of each place:** Recognize how each place can influence neighboring digits when borrowing.

The importance of place value cannot be overstated. It holds the key to performing precise arithmetic operations and prevents errors in subtraction.

Basic Arithmetical Operations in Decimal Subtraction

Subtraction is one of the four core arithmetical operations. In decimals, it requires careful attention to detail when aligning digits and handling the borrowing process.
Notably, dealing with decimals isn't too different from "regular" subtraction, but more attention is needed due to decimal places.

• **Align correctly:** Like stated before, make sure decimal points are aligned. This sets the foundation for a correct operation.
• **Sequential subtraction:** Start subtracting from right to left, from the smallest place value to the largest. This is typically from the hundredths place through to the tens place.
• **Borrowing revisited:** Borrow wherever necessary to facilitate proper subtraction, as some decimal hands out might require more borrowing than integer operations.

Practicing these arithmetic operations step-by-step promotes accuracy and confidence in tackling more complex calculations later.