Problem 30 - Algebra 1: Concepts and Skills

Question

Solve the percent problem. 55 years is what percent of 20 years?

Step-by-Step Solution

Answer

55 years is 275% of 20 years.

Step 1: Understand the Percent Equation

The general form of a percent equation is \(\frac{Part}{Whole} = \frac{Percent}{100}\). In this case, the 'part' is 55 years, and the 'whole' is 20 years. The percent is what we are trying to find.

Step 2: Set Up the Equation

Based on the problem and the general form of the percent equation, we get the equation as follows: \(\frac{55}{20} = \frac{Percent}{100}\)

Step 3: Solve the Equation

Cross-multiply to solve for the unknown: Percent = \(\frac{55}{20} * 100\)

Step 4: Calculate the Percent

Calculate the value of Percent by multiplying \(\frac{55}{20}\) by 100. It results in 275.

Key Concepts

Understanding the Percent Equation

When approaching percent problems, it's essential to understand the fundamental structure of the percent equation. This equation typically takes the form of \(\frac{Part}{Whole} = \frac{Percent}{100}\) where the

• **Part** represents the amount you are questioning,
• **Whole** is the total or maximum possible amount, and
• **Percent** indicates the fraction of the whole represented by the part.

In the context of the exercise, 55 years is considered the "part," while 20 years is the "whole." We want to find out what percentage the part (55 years) is of the whole (20 years). To discover the unknown percentage, we set up our percent equation based on this logical breakdown. Understanding this equation's form is crucial for accurately solving various percentage problems.

Mastering Cross-Multiplication

Cross-multiplication is a simple algebraic technique used to solve equations involving fractions, particularly useful in percent problems. When given an equation like \(\frac{55}{20} = \frac{Percent}{100}\), cross-multiplication helps find the missing value by eliminating the fractions. The process involves multiplying diagonally across the equal sign, so:

• Multiply the numerator of the first fraction (55) with the denominator of the second fraction (100).
• Multiply the denominator of the first fraction (20) with the numerator of the second fraction (Percent).

The resulting equation from cross-multiplication is:\[55 \times 100 = 20 \times Percent\]This simplifies to\[5500 = 20 \times Percent\]for solving the unknown Percent. This method ensures the equation balances, facilitating an easier path to solving it by simple division in the following steps.

Solving for Percentage

The final step in solving our percent problem involves isolating the unknown percent to determine its value. Once we have established a clear equation after cross-multiplication:\[5500 = 20 \times Percent\]We solve for Percent by dividing both sides of the equation by 20. By performing this calculation:\[Percent = \frac{5500}{20}\]we find that \[Percent = 275\].This means that 55 years is 275 percent of 20 years. It's essential to understand that a percentage greater than 100% indicates that "part" is more than the "whole," a common reality depending on the context of the problem. Understanding how to manipulate the equation and execute these calculations accurately lets you tackle various percent problems with confidence.