Problem 67 - Elementary and Intermediate Algebra

Question

Perform the operations. Subtract \(-137\) from \(12 .\)

Step-by-Step Solution

Answer

The result is 149.

Step 1: Understand the problem

To subtract (-137) from (12), you need to perform the operation (12 - (-137)). This subtraction is equivalent to adding the opposite, which is the addition of 12 and 137.

Step 2: Rewrite the subtraction as addition

When you subtract a negative number, it is equivalent to adding the positive of that number. Therefore, rewrite the expression (12 - (-137)) as (12 + 137).

Step 3: Perform the addition

Now, calculate the sum of 12 and 137. Start by adding the ones place, then the tens, and finally the hundreds: - **Ones place**: 2 + 7 = 9 - **Tens place**: 1 + 3 = 4 - **Hundreds place**: 1 (from 137) as there is no hundreds in 12 So, 12 + 137 = 149.

Key Concepts

Subtracting Negative Numbers

Subtracting negative numbers can be a bit tricky at first, but once you get the hang of it, it's quite simple. When we think about subtracting, we often think of taking something away. However, when you subtract a negative number, you're actually doing the opposite. Let's break it down.

If you have a negative number, such as (-137), and you need to subtract it from another number, like 12, you're performing the operation 12 - (-137). The crucial point here is understanding that subtracting a negative is the same as adding its positive counterpart.

A useful rule to remember is: **'Subtracting a negative is the same as adding a positive.'** So, in your example, it transforms as follows:

• Subtracting (-137) becomes adding 137.
• Thus, 12 - (-137) changes to 12 + 137.

Learning that subtraction of negatives converts into addition makes solving these problems easier and more intuitive.

Addition of Integers

When it comes to integer operations, especially addition, it's essential to get comfortable with both positive and negative numbers. In the case of 12 + 137, these are both positive integers, making the addition straightforward.

To add integers, simply align the numbers as you would with whole numbers:

• Start by adding the digits in the ones place.
• Next, move to the tens place.
• Lastly, add any remaining digits at the hundreds place.

Make sure to add from right to left, carrying over values as needed. It's the same process regardless of whether the result involves carrying over to the next digit.

For example, in 12 + 137:

• The ones place: 2 + 7 = 9
• The tens place: 1 + 3 = 4
• The hundreds place: Ample 1 from 137 + (no hundreds in 12)

After confirming each sum, arrive at 149. By becoming familiar with the straightforward process of adding integers, handling increasingly complex operations becomes more manageable.

Place Value in Addition

Understanding place value is key to performing addition correctly. Numbers are organized by their position, with each place representing a power of ten. This understanding helps us accurately add or subtract numbers.

When adding numbers like 12 and 137, place value guides us:

• Ones place: The rightmost digits in both numbers, which for 12 and 137 are 2 and 7, respectively.
• Tens place: Moving one position to the left, we find 1 (in 12) and 3 (in 137).
• Hundreds place: Only 137 has a digit here, which is 1.

Start by adding the digits in the ones place, then proceed to the tens place. If any added values require carrying over, remember that ten in a lower place becomes one in the next higher place.

For instance, while adding 12 and 137:

• Ones place: 2 + 7 = 9
• Tens place: 1 + 3 = 4
• Hundreds place: 1 (as no hundreds in 12)

By consistently applying place value, you will ensure accuracy in addition, laying a strong foundation for tackling more advanced math problems.