Problem 5 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$125 \%$$

Step-by-Step Solution

Answer

125% is \( \frac{5}{4} \), 1\( \frac{1}{4} \), and 1.25.

Step 1: Understand Percent

The term 'percent' means 'per hundred'. Therefore, to express 125% as a fraction, write it as \( \frac{125}{100} \), because 125% is the same as 125 per hundred.

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{125}{100} \). Both the numerator and the denominator can be divided by their greatest common divisor, which is 25. \( \frac{125 \div 25}{100 \div 25} = \frac{5}{4} \). So, 125% expressed as a simplified fraction is \( \frac{5}{4} \).

Step 3: Convert to a Mixed Number

Convert \( \frac{5}{4} \) to a mixed number. Since 5 divided by 4 is 1 with a remainder of 1, \( \frac{5}{4} \) can be written as 1\( \frac{1}{4} \).

Step 4: Express as a Decimal

To convert 125% to a decimal, divide 125 by 100. 125 divided by 100 is 1.25. Thus, 125% expressed as a decimal is 1.25.

Key Concepts

Simplifying Fractions

Fractions often start as larger numbers and can be simplified to make them easier to work with and understand. Simplifying a fraction means finding an equivalent fraction where the numerator and the denominator have no common divisors other than 1. This process makes the fraction as simple as possible.

For example, starting with the fraction \( \frac{125}{100} \), we want to simplify it. Both 125 and 100 can be divided by 25, which is their greatest common divisor (GCD). Dividing them both by 25, we get:

• \( \frac{125 \div 25}{100 \div 25} = \frac{5}{4} \)

The outcome \( \frac{5}{4} \) is the simplest form of \( \frac{125}{100} \). Always check if there’s a number larger than 1 that can divide both the numerator and denominator to ensure the fraction is fully simplified.

Converting Fractions to Mixed Numbers

Converting an improper fraction to a mixed number involves dividing the numerator by the denominator to find out how many whole numbers can be made, and what’s left over will remain as a fraction.

Take the fraction \( \frac{5}{4} \) as an example. This is called an improper fraction because the numerator (5) is larger than the denominator (4). To convert it:

• Divide 5 by 4, which equals 1 with a remainder of 1.
• This means you have 1 whole unit.
• The remainder becomes the numerator of the new fraction over the original denominator, turning it into 1\( \frac{1}{4} \).

This mixed number, 1\( \frac{1}{4} \), is easier to understand and visualize because it clearly shows there is a whole number with a part left over.

Decimal Conversion

Decimals are another way to represent fractions and percentages, often used in everyday arithmetic because they tie in well with our base-10 numbering system. Converting percentages to decimals simplifies calculations and data analysis.

To convert a percent like 125% into a decimal, you divide by 100. Here's why:

• The term 'percent' means per hundred, so dividing by 100 transitions from percent to pure decimal form.
• For 125%, this means \( \frac{125}{100} \).
• This division results in 1.25.

Thus, 125% is equivalent to the decimal 1.25. This format is highly useful, especially when performing further mathematical computations with the number.