Problem 21 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$87.5 \%$$

Step-by-Step Solution

Answer

87.5% as a fraction is \(\frac{7}{8}\) and as a decimal is 0.875.

Step 1: Express as a Fraction

First, express 87.5% as a fraction. Start by writing it over 100, because percent means per hundred: \[87.5\% = \frac{87.5}{100}\].

Step 2: Eliminate the Decimal

Since there is a decimal in the numerator, multiply both the numerator and the denominator by 10 to eliminate it: \[\frac{87.5 \times 10}{100 \times 10} = \frac{875}{1000}\].

Step 3: Simplify the Fraction

Now, simplify \(\frac{875}{1000}\) by finding the greatest common divisor (GCD) of 875 and 1000, which is 125. Divide both the numerator and the denominator by 125: \[\frac{875 \div 125}{1000 \div 125} = \frac{7}{8}\]. Thus, 87.5% as a fraction in simplest form is \(\frac{7}{8}\).

Step 4: Express as a Decimal

Convert 87.5% directly to a decimal by dividing by 100: \[87.5 \div 100 = 0.875\]. So, the decimal representation of 87.5% is 0.875.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. Think of a fraction like a slice of pizza. If you have a pizza cut into 8 slices and you eat 3, then you’ve eaten 3 out of 8 parts, or \(\frac{3}{8}\).
Fractions consist of two parts:

• The top number is called the numerator. It tells you how many parts are being considered.
• The bottom number, the denominator, tells you the total number of equal parts the whole is divided into.

When converting a percentage to a fraction, remember that 'percent' means per hundred, so you start with the fraction \(\frac{\text{percent value}}{100}\). This makes it easy to see that percent is always referring to a part of 100.
For example, \(87.5\%\) means 87.5 parts out of 100.

Decimal Conversion Made Easy

Decimal conversion is the process of transforming a fraction into a decimal. This is often helpful when you need a different way to visualize a number or need to use it in calculations.

• To convert a percentage like 87.5% to a decimal, simply divide by 100.
• This operation shifts the decimal point two places to the left, which is an intuitive way to understand decimal moves in powers of ten.

So, 87.5% becomes 0.875 as a decimal. Although this calculation is simple, keeping track of decimal places is crucial to getting the correct answer.
Converting between fractions and decimals provides flexibility in mathematical calculations and can make solving problems easier.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form by eliminating any common factors from the numerator and the denominator. Here’s how it works:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both parts of the fraction by this GCD to reduce it.

In our example of \(\frac{875}{1000}\), the GCD is 125. Dividing both the numerator and denominator by 125 simplifies the fraction to \(\frac{7}{8}\).
This process not only helps in making numbers smaller and more manageable but also in maintaining their value, enhancing understanding. Simplified fractions are easier to interpret and apply in both everyday contexts and mathematical computations.