Problem 87 - Fundamentals of Mathematics

Question

For the following problems, try to add the numbers mentally. $$ \begin{array}{r} 82 \\ 18 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 100.

Step 1: Break Down the Numbers

First, break down both numbers into tens and ones. The number 82 can be broken down into 80 and 2, while the number 18 can be broken down into 10 and 8.

Step 2: Add the Tens

Add the tens first: 80 (from 82) and 10 (from 18). This gives you 90.

Step 3: Add the Ones

Now add the ones: 2 (from 82) and 8 (from 18). This gives you 10.

Step 4: Combine the Sums

Finally, add the sums from the tens and ones together. The sum of the tens is 90, and the sum of the ones is 10. Adding these gives you a total of 100.

Key Concepts

Breaking Down Numbers

When performing mental math, breaking down numbers is a helpful strategy. This method involves splitting each number into two parts: the tens and the ones. For example, if you have the number **82**, you can break it down into **80** (which is the tens) and **2** (which is the ones). Similarly, the number **18** can be split into **10** (the tens) and **8** (the ones).

This technique simplifies the process of addition because dealing with smaller, round numbers is often easier. By breaking down numbers, you turn a seemingly complex addition problem into smaller, more manageable parts. This prepares you for the next step of the process.

Adding Tens

Once you've broken down the numbers, the next step is to add the tens. Work with just the tens, simplifying the calculations. In our example, after breaking down **82** and **18**, you focus on **80** and **10**.

Adding these two numbers is straightforward:

• Add **80** plus **10**.
• This results in **90**.

Focusing on tens first allows you to clear one part of the problem and sets a strong foundation for adding the ones. This approach minimizes potential errors and helps maintain clarity as you handle larger sums.

Adding Ones

After handling the tens, the next step is to add the ones. In our example, the ones are **2** and **8**. This step involves adding simpler, smaller numbers, which can be done quickly even without writing it down.

• Start by adding **2** plus **8**.
• This gives a sum of **10**.

By dealing with the ones separately, you stay organized and can focus solely on adding smaller numbers accurately. Adding smaller numbers boosts confidence and ensures you're on the right path before combining all parts of the problem.

Combining Sums

At this point, you've added both the tens and the ones separately. The last crucial step is to combine these sums. This involves taking the sum from the tens and adding it to the sum from the ones. In our example:

• The sum of the tens was **90**.
• The sum of the ones was **10**.
• Combine these sums: **90** plus **10**.
• This results in **100**.

Combining sums is the final touch that completes the mental math process. All previous steps—breaking down numbers, adding tens, and adding ones—lead to this moment. With practice, this step becomes seamless and efficient, reinforcing the power of the mental math strategy.