Question

Solve the percent problem. 42 feet is \(50 \%\) of what length?

Short answer

The total length is 84 feet.

Step-by-step solution

Identify the part and percent

In this percent problem, we are given that 42 feet is 50% of the total length. So, we have the part as 42 feet and the percent as 50.

Insert values into the formula

To find the whole, we will insert the given values into the formula. In our case, this will look like: \( \frac{42}{Whole} = \frac{50}{100} \)

Rearranging the equation

By rearranging the equation to solve for the whole length, we multiply both sides of the equation by the denominator on the left side (i.e., 'Whole'). The equation now becomes: \( Whole = \frac{42 * 100}{50} \)

Solving for Whole

By carrying out the calculation on the right side of the equation, we find that: \( Whole = 84 feet \)

Key concepts

Solving Percent Equations

Understanding how to solve percent equations is fundamental in algebra, as it frequently appears in various mathematical contexts. When faced with a percent problem, the first step often involves identifying the part, the percent, and the whole. For instance, if we are given that 42 feet represents 50% of a certain length, we recognize that '42 feet' is our part and '50%' is our percent.

The next step involves setting up an equation to represent this relationship. To convert a percent to a decimal for equation solving, we divide it by 100. Here, 50% becomes 0.50. The equation then sets the part over the whole equals the percent over 100, expressed as \( \frac{Part}{Whole} = \frac{Percent}{100} \). In our exercise, we substitute the known values and get \( \frac{42}{Whole} = \frac{50}{100} \).

Next comes the process of rearranging the equation. This involves algebraic manipulation to isolate the variable ('Whole' in our case) on one side. Multiplying both sides of the equation by the 'Whole' and then by 100 gives us the actual number the percentage represents. Completing this calculation yields the whole length. By mastering the steps of solving percent equations, students can tackle a wide variety of problems involving percentages.

Percent Formula

The percent formula is a key tool in percentage problems and is vital for students to understand it thoroughly. The generic formula is as simple as \( \frac{Part}{Whole} = \frac{Percent}{100} \). This formula is the backbone of most percent problems and allows us to find the part, the whole, or the percent, given the other two quantities.

To apply the percent formula, follow these steps:

• Identify the percent (convert it to a decimal if necessary).
• Determine the part and the whole based on the problem's context.
• Use the percent formula to set up an equation.
• Rearrange the equation to solve for the unknown.

In our example, the unknown is the 'Whole' that 42 feet is 50% of. We use the formula to set up and solve the equation by filling in the known values and then perform the arithmetic. It's essential to pay attention to whether we are looking for the part, the whole, or the percent because that will dictate the direction we manipulate the equation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). In the context of solving percent problems, these expressions help us translate a word problem into a form we can solve mathematically.

For instance, in the given problem, '42 feet is 50% of what length?', we use algebraic expressions to represent the known elements and the unknown we're trying to find. Our initial expression based on the percent formula becomes \( \frac{42}{Whole} = \frac{50}{100} \), where 'Whole' is the variable representing the unknown length. We then manipulate this expression algebraically to isolate the variable and solve for it, as seen in the steps provided.

Algebraic expressions can be manipulated through various operations to simplify or solve them. This kind of manipulation is key to finding solutions to problems in algebra. Developing a strong foundation in understanding and working with algebraic expressions will greatly benefit students in tackling not just percent problems, but a wide array of mathematical challenges.