Problem 58 - 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. \(74 \%\) of 21.93

Step-by-Step Solution

Answer

The estimate is 16.5, and the exact value is 16.23. The estimate is reasonable.

Step 1: Understanding the Problem

We need to estimate 74% of 21.93 by rounding the numbers to make the calculation easier. Then, we find the exact value and compare it to our estimate.

Step 1: Round the Numbers

First, round the percentage and the number to more convenient values for estimation. We can round 74% to 75%, and round 21.93 to 22.

Step 2: Estimate the Calculation

Calculate 75% of 22. Remember that 75% is equivalent to \( \frac{3}{4} \). So, \( 75\% \) of 22 can be estimated as \( 22 \times \frac{3}{4} = 16.5 \).

Step 3: Calculate the Exact Value

To find the exact value, calculate 74% of 21.93 by using the formula \( 74\% \times 21.93 = \frac{74}{100} \times 21.93 \). This results in \( 16.2282 \).

Step 4: Compare Estimate and Exact Value

The estimated value is 16.5 and the exact value is approximately 16.23. Both values are close, indicating that the estimate is reasonable.

Key Concepts

Estimations

When faced with a problem requiring calculations, estimations can serve as useful tools to quickly understand the scope of your answer and verify if your results are reasonable. The process typically involves simplifying the numbers through a technique like rounding before performing straightforward mental calculations.
By rounding numbers to simpler values, computations become less cumbersome. For instance, if you need to estimate 74% of 21.93, you might choose to round 74% to 75% and 21.93 to 22. This transforms the problem into calculating 75% of 22, a simpler task.

• Round percentages and numbers to the nearest value that makes calculations easier.
• Use round figures to perform quick mental math, making your estimation process both efficient and effective.

Estimations, while not perfectly accurate, provide a valuable insight into whether a solution is in the right ballpark. This approach offers a preliminary check against which you can compare your exact calculations later.

Percentages

Percentages are all about understanding the parts of a whole. In many cases, you simplify problems by transforming percentages into fractions or decimals. This can simplify both mental calculations and precise calculations, making the process more manageable.
To find a percentage of a number, you can transform the situation into a multiplication problem. For example, calculating 75% of a number can be interpreted as finding three-quarters of it—effectively multiplying the number by the fraction \( \frac{3}{4} \).
Here's how you can handle percentages with ease:

• Convert the percentage into a fraction or a decimal. For percentages like 0-100, this means dividing by 100.
• Use the resulting fraction or decimal to multiply with your original number, producing the desired result.

Understanding percentages will not only help in estimation but also in verifying the precision of your computations when conducting exact calculations.

Exact Calculations

Exact calculations allow you to determine the precise value of expressions and problems without approximation. While this approach may involve more complex computations, it ensures accuracy and is usually necessary for critical applications.
In the given exercise, after estimating, the exact value of 74% of 21.93 needs to be found to compare it with our estimated result. To perform exact calculations, convert the percentage into its decimal form by dividing by 100—74% becomes 0.74. Then, multiply by 21.93:
\[ 0.74 \times 21.93 = 16.2282 \]
Here’s why exact calculations matter:

• Precision: Provides a detailed and accurate output, minimizing errors.
• Verification: Allows cross-referencing with estimations to check for reasonableness.
• Critical Decision-Making: Especially in financial transactions or scientific research, exact numbers lead to sound decisions.

While estimations give you a quick idea, exact calculations ensure your answers are precise and reliable.