Problem 81 - Algebra 1: Concepts and Skills

Question

Write the percent as a fraction or as a mixed number in simplest form. $$ 128 \% $$

Step-by-Step Solution

Answer

The simplest form of the given percentage, \(128\%\), as a fraction is \(\frac{32}{25}\), and as a mixed number it is \(1\frac{7}{25}\).

Step 1: Convert the Percent to a Fraction

The percentage \(128\%\) can be rewritten as the fraction \(\frac{128}{100}\). This is achieved by placing the percentage number over 100, since percent means 'per hundred'.

Step 2: Simplify the Fraction

Now the goal is to simplify the fraction \(\frac{128}{100}\) to its simplest form. First, locate the greatest common factor (GCF) of the numbers 128 and 100. The GCF is 4. Divide both the numerator and the denominator by 4 to simplify the fraction. This results in \(\frac{32}{25}\).

Step 3: Convert to Mixed Number

Convert \(\frac{32}{25}\) to a mixed number by performing the division. The quotient is the whole number part, and the remainder makes the fractional part. Dividing 32 by 25 gives a quotient of 1 and a remainder of 7. Hence, \(\frac{32}{25}\) as a mixed number is \(1\frac{7}{25}\).

Key Concepts

Simplify Fractions

Learning how to simplify fractions is an important mathematical skill. It makes working with fractions much easier. A fraction is simplified when the numerator (top number) and denominator (bottom number) have no common factors other than 1.
To simplify a fraction, follow these steps:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.

For example, in the solution provided, the fraction \(\frac{128}{100}\) was simplified by finding the GCF, which was 4. By dividing both 128 and 100 by 4, we get \(\frac{32}{25}\). This gives the fraction in its simplest form.

Convert Percents to Mixed Numbers

Converting a percent to a mixed number involves a few clear steps. A mixed number contains both a whole number and a fraction.Firstly, convert the percent to a fraction by placing it over 100. For example, \(128\%\) becomes \(\frac{128}{100}\). Simplify this fraction if needed.
Once in its simplest form, here \(\frac{32}{25}\), convert to a mixed number:

• Divide the numerator by the denominator.
• The whole number result is the integer part of the mixed number.
• The remainder is the new numerator of the fractional part.
• The denominator remains the same.

For \(\frac{32}{25}\), divide 32 by 25 to get 1 with a remainder of 7. Therefore, the mixed number is \(1 \frac{7}{25}\). This process helps to clearly show how many complete wholes we have along with the leftover fraction.

Greatest Common Factor

The greatest common factor (GCF) of two numbers is the largest number that can divide both without leaving a remainder. Finding the GCF is essential in simplifying fractions, as it helps to reduce them to their simplest form.To find the GCF:

• List the factors of each number.
• Identify the largest factor common to both lists.

For example, when simplifying \(\frac{128}{100}\), the factors of 128 are 1, 2, 4, 8, 16, 32, 64, 128 and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. The largest factor that appears in both lists is 4.
This means 4 is the GCF, and using it to divide both the numerator and denominator leads to the fraction's simplest form, \(\frac{32}{25}\). Understanding how to find the GCF is a valuable skill that eases fraction simplification.