Problem 39 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$56 \%$$

Step-by-Step Solution

Answer

56% as a fraction is \( \frac{14}{25} \) and as a decimal is \( 0.56 \).

Step 1: Understand the Percentage

Identify the given percentage, which is 56% in this case. To convert it into a fraction, we start by understanding that percentage means 'per hundred'.

Step 2: Convert Percentage to Fraction

Write the percentage as a fraction over 100. So, 56% becomes \( \frac{56}{100} \).

Step 3: Simplify the Fraction

Simplify \( \frac{56}{100} \) by finding the greatest common divisor (GCD) of 56 and 100, which is 4. Divide both the numerator and the denominator by 4 to get \( \frac{14}{25} \).

Step 4: Convert Percentage to Decimal

Convert the percentage to a decimal by dividing the percentage by 100. Therefore, 56% as a decimal is \( 0.56 \).

Key Concepts

Fraction Conversion

When we talk about converting percentages to fractions, we're really connecting two ways of expressing numbers. A percentage tells us how much out of 100, and this forms the basis of its conversion into a fraction.
Let's take a percentage, like 56%.

Since percentage means per hundred, 56% can be written as a fraction of 56 out of 100. This gives us the fraction:

• \( \frac{56}{100} \)

The fraction \( \frac{56}{100} \) shows us the ratio of 56 to 100. From here, you can perform further operations such as simplifying the fraction.
This conversion step is crucial because it lays the groundwork for simplifying and working with both fractions and decimals.

Decimal Conversion

Converting a percentage into a decimal is just as essential as converting it into a fraction. To convert a percentage to a decimal, you start by considering what 'percent' really means. It literally translates to 'per hundred'. This means when you have 56%, you are essentially working with the value of 56 out of 100.
To transform this into a decimal, you divide the percentage by 100. This changes our percentage into a decimal form:

• 56% becomes \( 0.56 \)

Decimals offer a different way of looking at numbers, especially when you're dealing with arithmetic or financial calculations.
Decimals enable more straightforward operations such as multiplication and addition without needing to convert back to fractions.

Simplifying Fractions

Simplifying fractions is a way of making them easier to read and work with. It's about finding a simpler fraction that is equivalent to the one you started with.
Let's continue with our earlier example. After writing 56% as \( \frac{56}{100} \), the goal is to simplify it to a smaller equivalent fraction.- First, identify any common factors shared by the numerator (56) and the denominator (100).- The greatest common divisor (GCD) helps find the common factor. Here, GCD is 4.- Divide both numerator and denominator by this GCD to simplify the fraction:

• \( \frac{56}{100} \div \frac{4}{4} = \frac{14}{25} \)

Thus, \( \frac{14}{25} \) is the simplified form of the fraction. Simplification ensures that everyone working with the fraction deals with the smallest, most manageable number set possible while maintaining the same value.