Problem 10 - Prealgebra

Question

Solve each of the following problems. What percent of 80 is \(20 ?\)

Step-by-Step Solution

Answer

20 is 25% of 80.

Step 1: Understanding the Problem

The problem is asking what percentage the number 20 is of the number 80. It requires us to find a percentage value.

Step 2: Set Up the Percentage Equation

To find what percent one number (20) is of another (80), use the formula for percentage: \[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \% \]Here, the 'Part' is 20 and the 'Whole' is 80.

Step 3: Substituting into the Formula

Replace 'Part' and 'Whole' in the formula with the numbers from the problem: \[ \text{Percentage} = \left( \frac{20}{80} \right) \times 100 \% \]

Step 4: Calculating the Fraction

Calculate \( \frac{20}{80} \). This simplifies to \( \frac{1}{4} \) since 20 divided by 80 is 0.25.

Step 5: Converting the Fraction to a Percentage

Now multiply \( 0.25 \) by 100 to convert it to a percentage:\[ 0.25 \times 100 = 25 \% \]

Step 6: Conclusion

We find that 20 is 25% of 80 by using the percentage calculation method.

Key Concepts

Fraction Simplification

Simplifying fractions is a key step in solving many math problems, including percentage calculations. To simplify a fraction, you want to reduce it to its smallest form, where the numerator and denominator have no common factors other than 1. For example, in the exercise given, we started with the fraction \( \frac{20}{80} \).

• First, identify the greatest common divisor (GCD) of the numerator and the denominator.
• Here, the GCD of 20 and 80 is 20, since 20 is the largest number that divides both 20 and 80 without a remainder.
• Next, divide both the numerator and the denominator by this GCD: \( \frac{20 \div 20}{80 \div 20} = \frac{1}{4} \).

Simplifying fractions makes calculations easier and is particularly helpful in percentage problems. It transforms fractions to their simplest form, making further arithmetic operations less complex.

Basic Arithmetic

Basic arithmetic is a cornerstone in prealgebra and forms the basis for more advanced mathematical concepts. For the problem at hand, we utilize basic arithmetic to discern what percentage 20 is of 80.

• We begin by creating a fraction to represent the problem: \( \frac{20}{80} \).
• Simplification transforms it into \( \frac{1}{4} \).
• From here, the next step involves converting this fraction to a percentage. First, recognize that a percentage is simply a fraction out of 100.
• To convert \( \frac{1}{4} \) to a percentage, multiply it by 100: \( \frac{1}{4} \times 100 = 25 \% \).

This process, starting from creating a fraction, simplifying, then performing multiplication, demonstrates fundamental arithmetic operations like division and multiplication.

Prealgebra Concepts

Prealgebra equips students with the tools needed to tackle more complex mathematical problems. It involves concepts such as fractions, decimals, and percentages, which are often interrelated. In the exercise discussed, understanding prealgebra principles is key.A primary focus is understanding how to move between fractions and percentages. A percentage is essentially a fraction with a denominator of 100. Therefore, if we identify that \( \frac{1}{4} \) represents the problem's scenario, expressing it as a percentage involves converting it to a form out of 100.

• Multiplying by 100 is a common technique used to switch a fraction into a percentage easily.
• Consistently, operations like simplifying or multiplying fractions require a basic grasp of number operations, including divisibility and multiplication.

Thus, prealgebra provides a stepping stone to engage with more intricate mathematical challenges, by instilling a solid foundation in handling numeric transformations.