Question

Find each sum or difference without using a calculator. \(73.7654-5\)

Short answer

68.7654

Step-by-step solution

Understand the Problem

The exercise requires finding the difference between two numbers: 73.7654 and 5. This can be solved through basic subtraction.

Align the Decimal Points

Write the numbers in a column, aligning their decimal points. This helps to ensure accurate subtraction. 73.7654- 5.0000

Subtract from Right to Left

Start subtracting from the rightmost digit (the thousandths place) and move left. If necessary, borrow from the next left digit. 73.7654- 5.0000--------- 68.7654

Combine the Numbers

After performing the column subtraction, combine the digits to get the final answer. The result is the difference between 73.7654 and 5, which is 68.7654.

Key concepts

Aligning Decimal Points

When performing subtraction involving decimal numbers, it's crucial to align the decimal points. This ensures that each place value corresponds correctly, avoiding errors. For example, to subtract 5 from 73.7654, first, write the numbers in a column with the decimal points lined up:
73.7654
- 5.0000
By aligning the decimals, you correctly place the whole number 5 as 5.0000, making it easier to see that you need to subtract from each place value correctly. Proper alignment helps maintain the integrity of place values like ones, tenths, hundredths, and more.

Column Subtraction

Column subtraction is a method of subtracting numbers by writing them in a column format. This approach helps to manage and visualize the numbers effectively. After aligning the decimals, write each digit under the corresponding place value:
73.7654
- 5.0000
Then, start subtracting from the rightmost digit (in this case, the thousandths place) and go left. Ensure that you subtract each digit separately:

• 4 - 0 = 4 (in the thousandths place)
• 5 - 0 = 5 (in the hundredths place)
• 6 - 0 = 6 (in the tenths place)
• 7 - 0 = 7 (in the ones place)
• 3 - 5 (in the tens place requires borrowing, which we will cover next)

Finally, write out the result for each column, the answer will be visible as you go down the column, making it easier to combine the digits for the final answer.

Borrowing in Subtraction

Sometimes, when subtracting, you encounter a situation where a digit in the minuend (the number from which you subtract) is smaller than the corresponding digit in the subtrahend (the number you are subtracting). In these cases, borrowing is necessary.
Looking at the tens place in our example, we have the calculation 3 - 5, which is not possible without borrowing. To handle this, we 'borrow' 1 from the next left place value (the tens place, in this case, the 7), turning it into 6, and converting the 3 into 13 to facilitate subtraction:
Borrow 1: 13 (tens place)
Decrement by 1: 6 (ones place)
Now, we can perform the subtraction: 13 - 5 = 8 (tens place)
After borrowing, our final operation looks like this:

• 13 - 5 = 8
• 6 - 0 = 6 (from the borrowing adjustment)
• Final result: 68.7654

Borrowing helps in managing these tricky spots by redistributing values, making each digit subtraction feasible.