Question

Translate to an equation and solve. What number is \(12 \frac{1}{2} \%\) of \(440 ?\)

Short answer

Answer: 55

Step-by-step solution

Convert the percentage to a decimal

To convert the percentage \(12 \frac{1}{2} \%\) to a decimal, we first convert the mixed number to an improper fraction and then, divide the numerator by the denominator and finally divide the result by 100. \(12\frac{1}{2} = \frac{24 + 1}{2} = \frac{25}{2}\) Now, convert the fraction to a decimal: \(\frac{25}{2} = 12.5\) So, \(12 \frac{1}{2} \%\) as a decimal is: \(12.5 \div 100 = 0.125\)

Multiply the decimal by the given number

Now that we have the decimal representation of the percentage, we can find \(12 \frac{1}{2} \%\) of \(440\) by multiplying the decimal by the given number: \(0.125 \times 440 = 55\) So, \(12 \frac{1}{2} \%\) of \(440\) is \(\boxed{55}\).

Key concepts

Decimals

Decimals are a way of representing numbers that are not whole but consist of a whole part and a fractional part separated by a decimal point. For example, in converting a percentage such as \(12 \frac{1}{2}\%\) to a decimal, you need to understand what a decimal is.

• Decimals are effectively fractions written in a different form.
• They offer a convenient way to express any non-whole number using powers of ten.

To convert a mixed number percentage into a decimal form, start by changing the mixed number to an improper fraction. For example, \(12 \frac{1}{2}\%\) converts to \(\frac{25}{2}\) because \(12\) whole parts plus \(\frac{1}{2}\) equals \(\frac{24 + 1}{2}\).
Next, divide the numerator by the denominator to get \(12.5\), and finally divide by \(100\) to convert the percentage to a decimal: \(12.5 \div 100 = 0.125\).
Understanding decimals is essential in this process as it sets the stage for using them in arithmetic operations like multiplication, especially when dealing with percentages.

Multiplication

Multiplication is one of the fundamental operations of arithmetic, and learning how to use it with decimals is crucial. When you need to find a certain percentage of a number, like \(12 \frac{1}{2}\%\) of \(440\), multiplication is the key operation.

• First, ensure that the percentage is properly converted to a decimal. In this context, it means turning \(12 \frac{1}{2}\%\) into \(0.125\).
• Then, multiply this decimal value by the number in question to find the percentage of that number.

The multiplication process here is straightforward: you simply multiply \(0.125\) by \(440\). This operation can be solved either manually or using a calculator. Here's the operation:
\(0.125 \times 440 = 55\).
This result is the final answer, showing that \(12 \frac{1}{2}\%\) of \(440\) is \(55\). Multiplication in such scenarios requires a sound understanding of how decimals work and being comfortable with the arithmetic operation itself.

Percentages

Percentages are a way of expressing a number as a fraction of 100, which makes it easier to compare values. The term "percent" derives from Latin meaning "per hundred," hence 50 percent is simply \(\frac{50}{100}\) or \(0.5\) in decimal form. Understanding percentages allows individuals to interpret data quickly and assess proportions intuitively.

• A percentage, like \(12 \frac{1}{2}\%\), is first converted to decimal form before it can be used in calculations.
• This involves treating the percentage as a fraction of 100, making it ready for multiplication.

Once a percentage is expressed as a decimal, it easily lends itself to various mathematical operations, especially multiplication, for applications such as calculating discounts, interest rates, and countless other day-to-day calculations.
For instance, converting \(12 \frac{1}{2}\%\) to \(0.125\) allows straightforward use in finding specific proportions of numbers. This transformation highlights how percentages, fractions, and decimals are interconnected, each serving crucial roles in math and practical applications.