Problem 17 - Prealgebra

Question

Use the percent proportion to solve each problem. Round to the nearest tenth. Find \(83 \%\) of 200

Step-by-Step Solution

Answer

83% of 200 is 166.0.

Step 1: Understand the Percent Proportion Formula

The percent proportion formula is \( \frac{part}{whole} = \frac{percent}{100} \). Here, "part" is the part that we are trying to find, "whole" is the total amount, and "percent" is the given percentage. In this problem, the "whole" is 200 and the "percent" is 83\%.

Step 2: Set Up the Proportion

Set up the proportion based on the given problem. We want to find the "part" (which is 83\% of 200), so let \( x \) represent the "part". The proportion can be set up as:\[ \frac{x}{200} = \frac{83}{100} \]

Step 3: Cross Multiply to Solve for x

Cross multiply to find the value of \( x \). This yields:\[ 100x = 200 \times 83 \] Calculate the product on the right side:\[ 100x = 16600 \]

Step 4: Solve for x

Divide both sides by 100 to solve for \( x \):\[ x = \frac{16600}{100} \] Simplify the division: \[ x = 166.0 \]

Step 5: Round to Nearest Tenth

The value of \( x \) calculated is already 166.0, which is already rounded to the nearest tenth. Thus, \( 83\% \) of 200 is \( 166.0 \).

Key Concepts

Calculating Percentages

Calculating percentages is a fundamental math skill that helps us understand proportions in real life. For instance, you might want to know what 83% of a number represents. To calculate a percentage of a number, you can use the formula:

• Percentage Value = (Percentage/100) × Whole Number

In simpler terms, this formula tells us that finding a percentage is the same as taking a fraction of the whole. Using our problem as an example, we want to find 83% of 200. So, it is the same as finding 83/100 of 200, which gives us the part of the whole that 83% represents.
Breaking this task into steps will help make it easier and ensure accuracy. Once you understand this, calculating percentages for various scenarios, such as discounts or grades, becomes second nature.

Cross Multiplication

Cross multiplication is a handy algebraic technique used to solve equations that involve fractions or ratios. When we "cross multiply," we're essentially getting rid of the fractions to solve for a variable.Let's say we have the proportion \( \frac{x}{200} = \frac{83}{100} \). Cross multiplication means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.

• This gives us the equation \( 100x = 83 \times 200 \).

This step is crucial because it transforms the proportion into a simple, solvable equation. Once cross-multiplied, we're left with a basic arithmetic problem that can be easily solved. This simple trick makes dealing with fractions and proportions a lot less intimidating.

Division

Division is one of the core operations in mathematics, used to split a number into equal parts. When solving equations involving percentages, division often comes in as the final step. After cross-multiplying to find our equation \( 100x = 16600 \), we use division to isolate the variable \( x \).

• To do this, divide both sides of the equation by 100, our coefficient of \( x \).

This operation gives us \( x = \frac{16600}{100} \). Performing this division provides the solution for \( x \), which represents 83% of 200. In this case, dividing 16600 by 100 results in 166.0. Division is essential in breaking down figures into comprehensible percentages, making it possible to deduce neat and rounded figures.

Rounding Decimals

Rounding decimals is a mathematical process that simplifies a number to make it easier to work with, especially when precision is only needed to a certain extent. It's common in percentage calculations to round decimals to a specific place, such as the nearest tenth.Consider our calculated value for \( x \), which is 166.0. It's already rounded to the nearest tenth. Rounding ensures that our figures are easy to read and work with, without overcomplicating things with unnecessary decimal places. Here's how rounding works:

• If the digit after the place you're rounding to is 5 or more, round up.
• If it's less than 5, keep the digit unchanged.

Being comfortable with rounding helps in everyday calculations like figuring out total costs when shopping or estimating project timelines. It's a skill that proves invaluable in both academic and real-world scenarios.