Problem 15 - Fundamentals of Mathematics

Question

Estimate each value using the method of rounding. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. $$ 87,612+2,106 $$

Step-by-Step Solution

Answer

Estimated value is 90,000; exact value is 89,718. They are close, with a difference of 282.

Step 1: Round Each Number

First, let's round each number to the nearest thousand. \( 87,612 \) rounds to \( 88,000 \) because the digit in the hundreds place (6) is greater than 5. Similarly, \( 2,106 \) rounds to \( 2,000 \) because the digit in the hundreds place (1) is less than 5.

Step 2: Perform the Estimated Addition

Now, add the rounded numbers together. \( 88,000 + 2,000 = 90,000 \). This gives us an estimated sum.

Step 3: Calculate the Exact Value

Next, add the original numbers to find the exact value. \( 87,612 + 2,106 = 89,718 \).

Step 4: Compare Estimated and Exact Values

Finally, compare the estimated value and the exact value. The estimated value is \( 90,000 \) and the exact value is \( 89,718 \). The estimation is very close to the exact value with a difference of \( 282 \).

Key Concepts

Estimation

Estimation is a valuable skill that helps simplify complex calculations by providing a quick approximation. The process often involves rounding numbers to make mental math more manageable. For instance, in this exercise, the numbers 87,612 and 2,106 were rounded to the nearest thousand before adding. This simplification allows us to estimate the sum rapidly.

• 87,612 rounded to 88,000 because the hundreds digit, 6, is greater than 5.
• 2,106 rounded to 2,000 because the hundreds digit, 1, is less than 5.

So, estimation in addition not only saves time but also gives us a general idea about the sum, which in this case is 90,000.

Exact Value Comparison

While estimation is handy, knowing the exact value is crucial for accuracy. After making an estimate, calculating the exact value helps validate the estimation. It's important to remember that estimation is just that: an approximation. But exact value provides precision.
For the exercise in question:

• The exact calculation is 87,612 + 2,106, which equals 89,718.
• When comparing the exact value with the estimated 90,000, the difference is subtle—just 282.

This comparison shows that our estimate was reasonably close, highlighting that rounding can give reliable insights.

Step-by-step Solutions

Breaking down a problem into step-by-step solutions can be incredibly beneficial in learning and understanding mathematical concepts. In this context, the exercise was divided into clear steps to promote clarity. Let's review them quickly:
- **Step 1:** Round each number. - **Step 2:** Add the rounded numbers to estimate the sum. - **Step 3:** Add the original numbers to get the exact sum. - **Step 4:** Compare both sums. These structured steps are effective because they isolate different phases of the problem. By managing each task separately, students can focus on specific concepts, such as rounding or addition. This approach helps deepen their understanding and encourages confidence when tackling similar problems in the future.