Problem 18 - Technical Mathematics with Calculus

Question

Find: $$15.3 \% \text { of } 326 \mathrm{mi}$$

Step-by-Step Solution

Answer

15.3% of 326 mi is approximately 49.88 mi.

Step 1: Convert Percentage to Decimal

To find a percentage of a number, first convert the percentage to a decimal. To convert 15.3% to a decimal, divide by 100: \(15.3 \div 100 = 0.153\).

Step 2: Multiply Decimal by the Number

Multiply the decimal equivalent of the percentage by the number to find the part. In this case, multiply 0.153 by 326 mi: \(0.153 \times 326 \text{ mi} = 49.878 \text{ mi}\).

Step 3: Round to Appropriate Units

If needed, round the result to the nearest appropriate unit. For distance, it's common to round to two decimal places, or to the nearest mile if precision is not vital: \(49.878 \text{ mi} \approx 49.88 \text{ mi}\).

Key Concepts

Converting Percentages to Decimals

The conversion of percentages to decimals is a foundational skill in mathematics that is used across various fields. To convert a percentage to a decimal, simply divide the percentage by 100. This essentially moves the decimal point two places to the left. For example, converting 15.3% into a decimal involves dividing by 100:

\[\begin{equation}15.3 \% = 15.3 \div 100 = 0.153\end{equation}\]
Understanding this process is crucial for calculations in finance, statistics, and more. It serves as a stepping stone to more complex mathematical operations such as multiplying decimals.

Multiplying Decimals

Multiplying decimals may seem daunting, but it follows the same principles as multiplying whole numbers. After converting a percentage to a decimal, multiplying it by a number gives you the portion of the number represented by that percentage. In our exercise, the decimal 0.153 is multiplied by 326 mi, which is written mathematically as:

\[\begin{equation}0.153 \times 326 \text{ mi} = 49.878 \text{ mi}\end{equation}\]
Note that when multiplying, the number of decimal places in the result should be the sum of the decimal places in the factors. However, the result may need rounding for practical use. Multiplying decimals is used in financial calculations, measurement, and scientific data processing.

Real-World Math Applications

Understanding percentage calculations has significant implications in the real world. For instance, calculating tips, discounts, taxes, and interest rates all involve percentages.

In our exercise, finding 15.3% of 326 mi could represent calculating a proportional distance in route planning or assessing fuel consumption for a trip. The ability to perform such calculations is indispensable in fields like engineering, economics, and daily life. As technology advances, numeracy remains an essential skill, enabling both professionals and consumers to make informed decisions based on mathematical reasoning.

Unit Conversion

Unit conversion is another critical mathematical skill, particularly in scientific and engineering contexts where different measurement systems are used. In the context of our exercise, distances might need to be converted from miles to kilometers or vice versa depending on the country's standard units of measurement.

For a rough conversion from miles to kilometers, one might multiply the distance in miles by 1.60934. This skill is crucial for international travel, scientific research, cooking, and more. Precise unit conversion ensures accuracy in measurements and data comparison across different systems.