Problem 19 - Technical Mathematics with Calculus

Question

Find: $$33.3 \% \text { of } 662 \mathrm{kg}$$

Step-by-Step Solution

Answer

33.3% of 662 kg is 220.386 kg.

Step 1: Understand Percentage

The term 'percentage' means a number or ratio expressed as a fraction of 100. To find 33.3% of a number, you convert the percentage to its decimal form by dividing by 100.

Step 2: Convert Percentage to Decimal

Convert 33.3% to a decimal by dividing by 100. This gives 0.333.

Step 3: Multiply by the Quantity

Multiply the decimal form of the percentage by the total quantity to find the portion. This means multiplying 0.333 by 662 kg.

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is a fundamental skill in mathematics and real-world applications. A percentage is essentially a number that represents a fraction of 100. To convert a percentage to a decimal, think of the percentage symbol as a prompt to divide by 100.

For instance, to convert 33.3% to a decimal, you simply divide 33.3 by 100. The mathematical operation looks like this: \( \frac{33.3}{100} = 0.333 \). This simple conversion is powerful because it allows you to carry out other mathematical operations with ease. It's like translating a language into a more universally understood form in the world of numbers.

Remember that moving the decimal point two places to the left is a quick method for division by 100, which is especially handy for percentages that are whole numbers. For mixed or decimal percentages like 33.3%, division is the most accurate method.

Fraction of 100

The idea of a 'fraction of 100' is at the core of understanding percentages. By definition, a percentage is the portion out of a total of 100, making it a specific kind of fraction. This concept translates to a visual understanding where you can imagine a whole item divided into 100 equal parts.

For example, when you have 33.3% of something, you are considering 33.3 parts out of those 100 parts. So, if 100% represents the entirety of something, like the total weight of a batch of sugar, then 33.3% would be just one-third of that whole because \( \frac{33.3}{100} \) simplifies closely to \( \frac{1}{3} \). The power of percentages lies in their ability to quantitatively compare parts to a whole in a universally standardized manner.

Mathematical Operations with Percentages

Performing mathematical operations with percentages can seem complicated, but it's quite straightforward once you've converted the percentage to a decimal. When asked to find a percentage of a quantity, say 33.3% of 662 kg, what you're essentially being asked to do is calculate a part of the whole. This involves multiplication.

Once you've converted 33.3% to its decimal form (0.333), you simply multiply this decimal by the total quantity to find the portion. The calculation is expressed as: \( 0.333 \times 662 \ \text{kg} \) which reveals the weight that represents 33.3% of the total. Multiplying by the decimal is the equivalent of taking that specific fraction of the 100 parts – in other words, it's like taking 33.3 parts out of a 100 of something.