Problem 70 - 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. $$ (1.07)(13.89) $$

Step-by-Step Solution

Answer

The estimated value is 14, and the exact value is approximately 14.87.

Step 1: Round the Numbers

First, round each number to the nearest whole number. 1.07 rounds to 1, and 13.89 rounds to 14.

Step 2: Multiply Rounded Numbers

Next, multiply the rounded numbers: \(1 \times 14 = 14\). This gives an estimated value of the product.

Step 3: Calculate the Exact Value

Now, calculate the exact product of the original numbers: \(1.07 \times 13.89 = 14.8683\).

Step 4: Compare Estimated and Exact Values

Compare the estimated value to the exact value: The estimate is 14, whereas the exact product is 14.8683. The difference is 0.8683, showing the estimation is fairly close but slightly lower than the exact value.

Key Concepts

Estimation Techniques

In mathematics, estimation involves finding a number that is close enough to the exact value for some purpose. Typically, this involves rounding numbers to make calculations easier and faster. Rounding is a common technique used in estimation. Here, we round off numbers to their nearest whole number or decimal point. In the given problem, the numbers 1.07 and 13.89 are rounded to 1 and 14, respectively.

• This makes the arithmetic simpler, especially when done mentally.
• For mathematical multiplication, the estimate is useful to get a quick overview of what the product should look like.
• Good estimates can prevent calculation mistakes when quick decisions are needed.

However, remember that estimation doesn't always provide an accurate answer. However, it leads us towards a value that is sufficiently close for many practical scenarios. The primary goal is to keep numbers manageable without using a calculator.

Exact Values Comparison

After estimating, it’s important to calculate the exact value and make comparisons. This aids in understanding how close the estimation was to the actual number.

• First, you perform the exact mathematical multiplication of the given numbers. Here, you multiply 1.07 by 13.89 to obtain the result 14.8683.
• Comparing the exact and estimated values will help identify the difference or error in estimation.
• In the example, the exact value is 14.8683, and the estimated product is 14, giving a difference of 0.8683.

This comparison can help improve future estimations by showing you the margin of error you are working within. It provides insight into how trustworthy a quick estimate might be, especially in contexts requiring accurate judgments.

Mathematical Multiplication

Mathematical multiplication is a fundamental arithmetic operation involving the product of numbers. The process can often be complex with long decimals. For example, multiplying 1.07 by 13.89 requires careful calculation, especially without a calculator.

• First, align the numbers properly based on their decimal points before multiplying.
• Proceed with regular multiplication steps to find the product, ensuring decimal places are correctly aligned for the final result.
• Finally, the exact product for this multiplication is 14.8683, indicating precision needed in computations.

Understanding and performing precise calculations are critical when the tolerance for error is minimal. Solidifying these skills helps not only in academics but also in real-life financial, scientific, and technical decisions where accuracy is paramount.