Problem 3 - Prealgebra

Question

Find the percent of each number mentally. $$33 \frac{1}{3} \% \text { of } 27$$

Step-by-Step Solution

Answer

33 1/3% of 27 is 9.

Step 1: Understand the Percent Value

The problem asks for 33 1/3% of 27. We first need to understand what 33 1/3% represents. It is equivalent to the fraction \( \frac{1}{3} \).

Step 2: Convert the Number to a Fraction

Since 33 1/3% is equivalent to \( \frac{1}{3} \), finding 33 1/3% of any number means finding \( \frac{1}{3} \) of that number. Here, our number is 27.

Step 3: Divide the Number by 3

To find \( \frac{1}{3} \) of 27, we divide 27 by 3. This gives us \( 27 \div 3 = 9 \).

Step 4: Solution and Verification

Thus, 33 1/3% of 27 is 9. You can verify this by multiplying back: \( 9 \times 3 = 27 \), which confirms our calculation is correct.

Key Concepts

Percentages

Percentages are a way to express a number as a fraction of 100. Think of them as "parts per hundred". When you see a percentage, it tells you how many parts out of 100 something is. For instance, 50% means 50 parts out of 100, or simply half. This makes percentages incredibly useful for comparisons and understanding proportions easily.

To mentally calculate percentages, especially something like 33 1/3%, it is helpful to convert them into fractions or other simple numbers that are easy to work with. This conversion can simplify the calculation and make it something you can do on the fly without needing a calculator. So, remember: converting percentages to fractions is a useful trick to simplify mental math.

Fractions

Fractions represent a part of a whole and are expressed as one number (the numerator) over another (the denominator). In the case of percentages, converting a percentage to a fraction helps simplify calculations.

For example, 33 1/3% is not commonly found in everyday percentage calculations, but by recognizing that it translates to the fraction \( \frac{1}{3} \), you can handle it with ease.

Converting percentages to fractions allows you to perform multiplications and divisions more intuitively - a particularly helpful tactic in mental math situations. Once you have a percentage as a fraction, you can multiply the fraction with any number to find the percentage of that number.

Division

Division is the process of splitting a number into equal parts. It is a fundamental arithmetic operation that you can use to calculate fractions and percentages. In the exercise, we used division to find \( \frac{1}{3} \) of 27.

Here's how it works: you divide the total number by the denominator of the fraction. In this case, it means performing the division \( 27 \div 3 \). This operation results in 9, which is \( \frac{1}{3} \) of 27 as calculated without requiring a calculator.

Understanding how division works with fractions and percentages is essential. By breaking down a number as needed, you can mentally calculate what percentage or fraction of a whole number is quickly and correctly. This skill enhances your ability to work with numbers effectively in everyday scenarios.