Problem 40 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. \(120 \%\)

Step-by-Step Solution

Answer

120% as a fraction is \(\frac{6}{5}\) or 1\(\frac{1}{5}\), and as a decimal is 1.2.

Step 1: Convert Percent to Fraction

To convert a percentage to a fraction, divide the percentage by 100. So, for 120%, it becomes \(\frac{120}{100}\).

Step 2: Simplify the Fraction

Simplify \(\frac{120}{100}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 20. This results in \(\frac{6}{5}\).

Step 3: Convert Improper Fraction to Mixed Number

The fraction \(\frac{6}{5}\) is an improper fraction. Divide 6 by 5, which equals 1 with a remainder of 1. Thus, \(\frac{6}{5}\) is equivalent to the mixed number 1\(\frac{1}{5}\).

Step 4: Convert Percent to Decimal

To convert a percentage to a decimal, divide by 100. For 120%, divide 120 by 100 to get 1.2.

Key Concepts

Convert Improper Fractions

An improper fraction is one where the numerator (top number) is greater than the denominator (bottom number). This can often be seen when converting percentages greater than 100% to fractions.To convert an improper fraction to a mixed number, you will need to divide the numerator by the denominator. For example, with the fraction \(\frac{6}{5}\):

• Divide: Divide 6 by 5, which equals 1 with a remainder of 1.
• Remainder as Fraction: Use the remainder (1) as the new numerator, keeping the original denominator (5).
• Mixed Number: Combine these to get the mixed number 1\(\frac{1}{5}\).

This step of converting is crucial as it helps simplify understanding or further calculations involving the fraction.

Simplifying Fractions

Simplifying fractions means expressing them in their simplest form, where the numerator and denominator have no common divisors other than 1. For example, while converting from a percent to a fraction, we got \(\frac{120}{100}\). Here's how to simplify it:

• Identify the Greatest Common Divisor (GCD): The GCD of 120 and 100 is 20.
• Divide Both Terms: Divide the numerator and the denominator by the GCD:
  • \( \frac{120 \div 20}{100 \div 20} = \frac{6}{5} \)

This fraction, \(\frac{6}{5}\), cannot be simplified further, so it is now in its simplest form. Simplification helps make fractions easier to work with and understand.

Percent to Decimal

Converting a percent to a decimal is a simple but essential skill in mathematics. A percentage is essentially a fraction with a denominator of 100, so to convert it to a decimal, all you have to do is move the decimal point two places to the left. To better illustrate this, take 120% as an example:

• Divide by 100: Simply divide the percentage number by 100 or, equivalently, move the decimal point two places to the left.
• 120% becomes 1.2 when converted into a decimal.

Decimals are often easier to use in calculations than percentages, especially for multiplication and division. Understanding this conversion can significantly enhance your efficiency in tackling math problems.