Problem 158 - Elementary Algebra

Question

Without solving the problem "What is \(20 \%\) of 300?" think about what the solution might be. Should it be a number that is greater than 300 or less than 300 ? Explain your reasoning.

Step-by-Step Solution

Answer

20% of 300 is less than 300 because a portion of 100 is smaller than the whole number.

Step 1: Understand the Problem

We need to determine if 20% of 300 will be greater than or less than 300.

Step 2: Concept of Percentage

A percentage represents a portion out of 100. So, 20% means 20 out of 100 parts.

Step 3: Apply the Concept to the Given Number

When we apply 20% to 300, we are effectively considering 20 parts out of 100 parts for the number 300.

Step 4: Compare the Proportions

Since 20 out of 100 represents a fraction less than the whole, taking 20% of 300 will give a result that is a fraction of 300.

Step 5: Conclusion

Therefore, 20% of 300 should be less than 300 because a fraction (20%) of any number is smaller than the number itself.

Key Concepts

Percentage

To understand what a percentage is, think of it as a way to express a part of a whole number. The word 'percent' literally means 'per hundred,' so when we talk about 20%, we mean 20 out of every 100 parts. Using the exercise we have, where we need to find 20% of 300, we are essentially converting a part of the total number of 300.

To calculate percentages, you can use the formula:
\[ \text{Percentage of a number} = \frac{\text{Percentage}}{100} \times \text{Total number} \].
For our example:
\[ 20\text{\text{%}} \text{ of } 300 = \frac{20}{100} \times 300 = 60 \]
So, 20% of 300 is 60.

Proportions

A proportion is essentially a statement that two ratios are equal. Ratios compare two quantities by division. In our example, when we talk about 20 parts out of 100, we are speaking of a ratio of 20:100.

Let's break it down into simpler terms. If you have 300 apples and you want to find out what 20% would look like in terms of apples, you can consider 20 out of every 100 apples. So,
\[ \text{Ratio} = \frac{20}{100} = 0.2 \]
Then multiply this ratio by the total number of apples (300) to get
\[ 0.2 \times 300 = 60 \]
This means that 20:100 proportion maintains the same ratio when applied to any total number, effectively making 60 apples out of 300.

Fraction

Fractions represent a part of a whole and are generally written as one number over another, separated by a slash. For example, 20% is equivalent to the fraction \( \frac{20}{100} \) or simplified to \( \frac{1}{5} \).

Fractions perform a similar function to percentages but in a different form. When you calculate 20% of 300, you can convert the percentage to a fraction like so:
\[ 20\text{\text{%}} = \frac{20}{100} = \frac{1}{5} \].
Next, use this fraction to find the part of 300:
\[ \frac{1}{5} \times 300 = 60 \].
Therefore, when working with fractions, you are essentially dividing the quantity into equal parts, based on the given fraction.

Understanding these concepts helps you not only solve the problem “What is 20% of 300?” but also equips you with the fundamental skills to tackle similar percentage, proportion, and fraction problems in the future.