Problem 16 - Prealgebra

Question

Find the percent of each number mentally. $$12 \frac{1}{2} \% \text { of } 32$$

Step-by-Step Solution

Answer

12 \(\frac{1}{2}\%\) of 32 is 4.

Step 1: Separate Percentage into Components

The percentage given is \(12 \frac{1}{2}\%\). This is a mixed number which can be broken down into two parts: \(12\%\) and \(\frac{1}{2}\%\).

Step 2: Calculate 12% of 32

To find \(12\%\) of \(32\), convert \(12\%\) to a decimal by dividing by 100. Thus, \(12\% = 0.12\). Multiply \(0.12\) by \(32\) to find \(12\%\) of \(32\):\[ 0.12 \times 32 = 3.84 \]

Step 3: Calculate \(\frac{1}{2}\%\) of 32

Convert \(\frac{1}{2}\%\) to a decimal by dividing \(0.5\%\) by 100, which gives \(0.005\). Multiply \(0.005\) by \(32\) to find \(\frac{1}{2}\%\) of \(32\):\[ 0.005 \times 32 = 0.16 \]

Step 4: Sum the Results

Add the results from Step 2 and Step 3 to find \(12 \frac{1}{2}\%\) of \(32\):\[ 3.84 + 0.16 = 4.00 \]

Key Concepts

Percentages

Understanding percentages is fundamental when dealing with real-world problems, whether it's calculating discounts, interest rates, or statistics. The word "percent" means "per hundred," so when we talk about finding a percent of a number, we are essentially finding how many parts per hundred of that number.

Here are some key points:

• Percent to Decimal Conversion: Convert a percentage to a decimal by dividing it by 100. For example, to convert 12% to a decimal, you calculate: \[ 12\% = \frac{12}{100} = 0.12 \]
• Calculating Percentages: Once you have the decimal form, multiply it by the number you are interested in. If you want to find 12% of 32, you compute: \[ 0.12 \times 32 = 3.84 \]

Breaking the percentage into easy-to-process components, like in a mixed number, helps simplify the calculation process, especially useful when doing mental math.

Decimal Conversion

Converting fractions and percentages into decimals is a critical skill in mathematics. Decimals make calculations easier and more intuitive, particularly when dealing with non-integer values in sums or products.

Here’s how it works:

• Simple Conversion: Converting a fraction like \( \frac{1}{2} \) into a decimal involves simple division: \[ \frac{1}{2} = 0.5 \]
• Fractional Percentage: When you have a fractional percentage such as \( \frac{1}{2}\% \), you first convert it to a decimal by dividing by 100: \[ 0.5\% = \frac{0.5}{100} = 0.005 \] Then, multiply by the number you're calculating the percentage of. In this case, it would be: \[ 0.005 \times 32 = 0.16 \]

Decimals make it quick and predictable to complete multiplicative operations, minimizing mental effort by simplifying numbers into standard forms.

Mixed Numbers

A mixed number combines a whole number with a fractional part, such as \(12 \frac{1}{2}\). This form is useful when dealing with numbers that are slightly above a whole number, and can simplify both interpretation and calculation.

To handle mixed numbers:

• Separate Components: Break the mixed number into its components to perform operations separately. For example, \(12 \frac{1}{2}\%\) is broken into \(12\%\) and \(\frac{1}{2}\%\).
• Independent Calculations: Solve each component independently and then combine the results. Thus, for \(12 \frac{1}{2}\%\) of 32, you find both \(12\%\) and \(\frac{1}{2}\%\) of 32, then add them for the final result.

Handling mixed numbers by breaking them down avoids confusion and streamlines the calculation process, particularly in percentage-based problems.