Problem 15 - Prealgebra

Question

Find the percent of each number mentally. $$16 \frac{2}{3} \% \text { of } 42$$

Step-by-Step Solution

Answer

The answer is 7.

Step 1: Convert Percent to Fraction

The first step is to convert \(16 \frac{2}{3}\%\) into a fraction. We know that \(1\%\) as a fraction is \(\frac{1}{100}\). Therefore, \(16 \frac{2}{3}\%\) is equivalent to \(\frac{16\frac{2}{3}}{100}\). Let's simplify \(16 \frac{2}{3}\) to an improper fraction first, resulting in \(\frac{50}{3}\). So, \(16 \frac{2}{3}\% = \frac{50}{3} \times \frac{1}{100} = \frac{50}{300}\). Simplifying \(\frac{50}{300}\) gives \(\frac{1}{6}\).

Step 2: Calculate the Fraction of 42

Now that we have \(\frac{1}{6}\), we need to find \(\frac{1}{6}\) of 42. To do this, multiply 42 by \(\frac{1}{6}\). The calculation is as follows: \(42 \times \frac{1}{6} = \frac{42}{6} = 7\).

Step 3: Conclusion

After completing the calculation, we find that \(16 \frac{2}{3}\%\) of 42 is equal to 7.

Key Concepts

Fractions

Fractions are a fundamental concept in mathematics that represent a part of a whole. They consist of a numerator and a denominator. The numerator is the top number that indicates how many parts we have, while the denominator is the bottom number that tells us into how many equal parts the whole is divided.

For example, the fraction \(\frac{1}{2}\) means one part out of two equal parts. This is particularly useful when dealing with percentages because percentages are essentially fractions of 100.

• To convert a percentage to a fraction, divide the percentage value by 100. This gives a clear fraction that can be simplified.

Understanding fractions makes working with percentages much easier, especially when converting between these forms during mental calculations.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the fraction represents a value greater than or equal to one whole.

For example, \(16 \frac{2}{3}\) can be expressed as the improper fraction \(\frac{50}{3}\). This conversion is done through multiplication and addition:

• Multiply the whole number part by the denominator of the fractional part.
• Add the numerator of the fractional part to this product to get the new numerator.

Thus, \(16 \times 3 + 2 = 50\), making our improper fraction \(\frac{50}{3}\).
Improper fractions are useful for calculations as they simplify the multiplication process compared to mixed numbers.

Mental Math

Mental math is an incredible skill that enables you to solve mathematical problems in your mind without the need for paper or a calculator. It involves simplifying problems, breaking them down into smaller parts, and using patterns and logic.

When working with percentages and fractions, mental math is especially useful. For instance, knowing that \(16 \frac{2}{3}\%\) simplifies to \(\frac{1}{6}\) makes it easier to calculate \(\frac{1}{6}\) of 42 mentally:

• First, recognize the simplified fraction that represents the percentage.
• Then, multiply this fraction with the given number using straightforward arithmetic.
• Divide 42 by 6, resulting in 7, which can be done quickly in your head.

Practicing mental math with fractions and percentages enhances your ability to perform calculations swiftly and accurately with minimal tools.