Problem 14 - Fundamentals of Mathematics

Question

Estimate each value. After you have made an estimate, find the exact value. Results may vary. \((\) Section 8.3) \(43+39+89+92\)

Step-by-Step Solution

Answer

Estimated sum: 260. Exact sum: 263.

Step 1: Break Down the Problem

The problem involves estimating and then calculating the exact sum of the numbers: 43, 39, 89, and 92. Our goal is to first estimate the total and then add the numbers precisely.

Step 2: Make an Estimation

Round each number to the nearest ten for an estimation: - 43 rounds to 40 - 39 rounds to 40 - 89 rounds to 90 - 92 rounds to 90 Now you calculate the estimated sum: 40 + 40 + 90 + 90 = 260.

Step 3: Calculate the Exact Sum

Add the numbers exactly as they are given without rounding: - First, add 43 and 39: 43 + 39 = 82 - Then, add 89 and 92: 89 + 92 = 181 - Finally, add both intermediate results together: 82 + 181 = 263.

Key Concepts

Rounding Numbers

Rounding numbers is a crucial skill in mathematics that simplifies the process of estimation. You decide which place value, such as tens, hundreds, etc., to round to, helping ease computation in your head.
When you round a number, here's what you generally do:

• Look at the digit to the right of the desired place value.
• If that digit is 5 or more, increase the rounding digit by 1.
• If it's less than 5, keep the rounding digit the same.
• Make all digits to the right of the rounding digit zero.

In the given exercise, the numbers 43, 39, 89, and 92 are rounded to the nearest ten: 43 becomes 40, 39 becomes 40, 89 becomes 90, and 92 becomes 90. This process allows for quick mental estimates of the sum, making rounding a valuable tool in everyday calculations.

Exact Sum Calculation

Once you have a rough estimate, it's crucial to find the precise answer by calculating the exact sum. Unlike estimation, exact calculations consider every digit to achieve a correct result.

Here's how you find the exact sum of the numbers 43, 39, 89, and 92:

• First, add 43 and 39 to get 82. This step involves direct addition, respecting all digits involved.
• Next, add 89 and 92, resulting in 181. Again, precision in aligning numbers is key.
• Finally, combine those intermediate results: 82 added to 181 equals 263.

By following these steps, you ensure the final sum is correct and not distorted by rounding, reflecting the exact total of the original numbers.

Mathematical Problem-Solving

Mathematical problem-solving is a structured process involving several steps to tackle questions efficiently. It often begins with understanding the problem, determining a strategy, estimating, executing calculations, and reviewing results.

In the example exercise, problem-solving is clearly demonstrated by:

• Breaking down the task into estimating and calculating exact sums separately. This structured approach simplifies complex problems into manageable parts.
• Employing rounding for estimation, which helps quickly verify if the exact sum seems reasonable. Estimation serves as a useful check.
• Executing detailed calculations to verify precision. This reinforces the practice of cross-checking estimates against exact results, vital for accuracy.

By carefully employing these problem-solving strategies, learners can develop skills to handle more challenging mathematical tasks with confidence and precision.