Problem 3 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$12 \frac{1}{2} \%$$

Step-by-Step Solution

Answer

\(12 \frac{1}{2}\% = \frac{1}{8}\) as a fraction and \(0.125\) as a decimal.

Step 1: Convert Percent to Fraction

To convert a percentage to a fraction, we divide by 100. For \(12 \frac{1}{2} \%\), first convert the mixed number to an improper fraction: \(12 \frac{1}{2} = \frac{25}{2}\). Next, divide by 100 by multiplying: \(\frac{25}{2} \times \frac{1}{100} = \frac{25}{200}\).

Step 2: Simplify the Fraction

Simplify \(\frac{25}{200}\) by finding the greatest common divisor of 25 and 200, which is 25. Divide both the numerator and the denominator by 25 to get \(\frac{1}{8}\).

Step 3: Convert Fraction to Decimal

To convert the simplified fraction \(\frac{1}{8}\) to a decimal, divide the numerator by the denominator: \(1 \div 8 = 0.125\).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them means making the fraction as simple as possible. This involves rewriting it so that the numerator and the denominator have no common factors other than 1. To simplify the fraction \(\frac{25}{200}\), we need to identify the greatest common divisor (GCD) of 25 and 200. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In this example, the GCD is 25. We divide both the numerator and the denominator by 25:

• \(\frac{25}{25} = 1\)
• \(\frac{200}{25} = 8\)

By performing this division, the fraction simplifies to \(\frac{1}{8}\). This means \(\frac{1}{8}\) is the simplest form of \(\frac{25}{200}\), as no other common factor exists except 1.

Converting Mixed Numbers

Mixed numbers can be a bit tricky, but with a little practice, they become easy to manage. A mixed number consists of a whole number and a fraction, like \(12 \frac{1}{2}\). To deal with them effectively, we often convert them to improper fractions. This makes calculations and conversions a breeze.Here's how to convert the mixed number \(12 \frac{1}{2}\) to an improper fraction:

• First, multiply the whole number (12) by the denominator of the fraction (2), which gives us: \(12 \times 2 = 24\).
• Next, add the numerator of the fraction (1) to this product: \(24 + 1 = 25\).
• Finally, place the result over the original denominator: \(\frac{25}{2}\).

This conversion makes subsequent calculations simpler, such as dividing by 100 when converting percentages to fractions.

Decimal Conversion

Converting fractions to decimals is a common task in mathematics. It involves dividing the numerator (the top number) of a fraction by its denominator (the bottom number). This results in a decimal representation of the fraction.For example, let's convert the fraction \(\frac{1}{8}\) to a decimal:

• We divide 1 by 8 to get 0.125.
• This process can be performed manually by long division, or it can be done using a calculator for quicker results.

Decimals are very useful because they provide a clearer representation of fractional values, often making it easier to interpret and use numbers in various mathematical contexts.