Problem 33 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 80 \\ -61 \end{array} $$

Step-by-Step Solution

Answer

19

Step 1: Arrange the Numbers

Write the numbers in a column format with the larger number on top, aligning the digits by their place value. Here, place 80 above 61.

Step 2: Subtract the Ones Place

Subtract the digit in the ones place of the second number from the digit in the ones place of the first number. Here, subtract 1 from 0. Since 0 is smaller than 1, we need to borrow from the tens place.

Step 3: Borrow from Tens Place

Reduce the tens place of 80 by 1, making it 7, and add 10 to the ones place of 80, making it 10. Now, subtract 1 from 10, resulting in 9.

Step 4: Subtract the Tens Place

After borrowing, subtract the tens place of the second number from the digits in the tens place of the first number. Subtract 6 from 7 to get 1.

Step 5: Write the Result

Combining the differences from the ones and tens place, write the final result of the subtraction. The result is 19.

Key Concepts

Borrowing in Subtraction

When performing subtraction, especially with larger numbers, you may encounter a situation where the digit in the top number is smaller than the corresponding digit in the number below it. This is when borrowing becomes essential.
Consider the example of subtracting 61 from 80:

• In the ones place, 0 is smaller than 1. In such cases, borrowing is necessary.
• We 'borrow' 1 from the tens place of the number 80, reducing it from 8 to 7.
• Then, add that borrowed 10 to the ones place, making it 10, allowing us to subtract 1 from 10 easily.

Borrowing can ensure each place value can be subtracted without the top number being less than the bottom number. It’s a critical skill in subtraction that helps keep our calculations accurate.

Place Value Alignment

Place value is key when performing subtraction. This means lining up each digit according to its place value, such as ones, tens, hundreds, etc. Proper alignment can prevent mistakes and make subtraction easier.
Here's a step-by-step on how to align them correctly:

• Start by writing the larger number above and the smaller number below.
• Ensure digits of the same place value appear in the same column. For example, ones align with ones, tens with tens.
• In our example, 80 and 61 are arranged such that 8 and 6 line up in the tens column, and 0 and 1 line up in the ones column.

Achieving correct alignment means you can effectively utilize borrowing or carrying during calculations, streamlining the process and preventing errors.

Checking Subtraction with a Calculator

After performing a manual subtraction, it’s useful to verify your result using a calculator. This quick method can enhance confidence in your solution and reinforce number sense. Here's how to do it:

• Once you've calculated the subtraction manually, like finding that 80 minus 61 equals 19, input the numbers into the calculator.
• Type 80, press the subtraction key, enter 61, and then press equals. The calculator should display 19 if your manual computation is correct.
• If the results differ, revisit each step of your manual calculation. Double-check for any misalignment or borrowing issues.

Using a calculator as a double-checking tool can bring peace of mind, knowing your manual arithmetic is spot-on. It’s especially useful while learning or when precise results are essential.