Problem 46 - Fundamentals of Mathematics

Question

Estimate each calculation using the method of rounding. After you have made an estimate, find the exact value and compare this to the estimated result to see if your estimated value is reasonable. Results may vary. (87.013)(21.07)

Step-by-Step Solution

Answer

The estimated value (1827) is very close to the exact value (~1834), making the estimate reasonable.

Step 1: Rounding the Numbers

First, round each number to the nearest whole number. 87.013 rounds to 87, and 21.07 rounds to 21.

Step 2: Calculating the Estimated Product

Multiply the rounded numbers to get an estimated result: \( 87 \times 21 \).Break it down as:\( 87 \times 21 = 87 \times (20 + 1) = 87 \times 20 + 87 \times 1 \).\( 87 \times 20 = 1740 \) and \( 87 \times 1 = 87 \).Therefore, the estimated product is \(1740 + 87 = 1827\).

Step 3: Calculating the Exact Product

Multiply the original numbers to find the exact product:\((87.013) \times (21.07)\). Using a calculator, this results in approximately 1833.99491.

Step 4: Comparing the Estimated and Exact Values

Compare the estimated product (1827) with the exact product (1833.99491).

Key Concepts

Estimation

Estimation is a valuable math skill that involves simplifying numbers to make calculations easier and faster. This method is particularly helpful when it is unnecessary to have an exact answer, or when verifying the reasonableness of an exact calculation. Estimation is often done through rounding.
When you round numbers, you adjust them to the nearest defined value based on the level of precision you need. In the example, 87.013 is rounded to 87 and 21.07 to 21. Rounding helps us reduce complex numbers to simpler ones, making mental math much more manageable. The goal of estimation is to get close to the actual result rather than aim for precision.
Estimation can be used in everyday situations, like quickly checking your bill at a store or planning time for activities. It gives you a reasonable guess, so you understand if a detailed calculation's result should be trusted.

Exact calculation

Exact calculation is the process of performing mathematical operations using the precise values without any form of rounding. For situations where accuracy is vital, exact calculations give the correct answer. In our example, the original numbers 87.013 and 21.07 are used to compute an exact result of approximately 1833.99491.
To obtain this number, one could employ a calculator or a computer, especially when dealing with decimals or large numbers, to avoid manual errors. Even though the processes might be longer or seem more complex, they ensure the results are as close to the actual value as possible.

• Always double-check your exact calculations when a precise outcome is critical.
• Ensure each step of the calculation is accurate, especially when using more complex operations.

Exact calculations are essential in scenarios like financial audits, scientific research, or engineering projects where decisions are based on these results.

Comparison of estimates

Comparing estimates to exact results helps evaluate the effectiveness and accuracy of the estimation. This step is crucial to see if our estimate was reasonably close to the actual number. In this exercise, the estimated product was 1827, while the exact one was approximately 1833.99491.
Upon comparing them, it is clear that the estimate is quite close to the exact result. Such comparisons reveal if our rounding strategy and estimation methods were efficient. If the estimate is far from the exact calculation, consider if the rounding was too rough or if calculation errors occurred.

• Regularly check your estimates with exact calculations to enhance skills.
• Use these comparisons to refine your estimation techniques over time.

Learning to make accurate estimates can save time and assist in quickly checking the validity of more precise calculations.

Mathematical reasoning

Mathematical reasoning involves using logical thinking to draw conclusions or make predictions based on calculations. It's the thought process behind both estimation and exact calculation. Reasoning helps identify when estimation or an exact calculation is more appropriate based on the context.
This concept is central when solving problems because it allows us to evaluate different methods' effectiveness. In our example, mathematical reasoning justifies rounding as a checking method while relying on the exact calculation for the precise result.

• Apply reasoning to determine when quick estimates suffice or when precision is crucial.
• Utilize logical steps whenever verifying or performing calculations to reinforce understanding.

Improving mathematical reasoning enhances your ability to approach and solve problems effectively, useful across various real-world tasks and professions.