Problem 18 - Prealgebra

Question

Use the percent proportion to solve each problem. Round to the nearest tenth. 36 is \(72 \%\) of what number?

Step-by-Step Solution

Answer

36 is 72% of 50.

Step 1: Understanding the Percent Proportion

The percent proportion formula states that part/whole equals percent/100. Here, 36 is the part, and 72% is the percent we know. We need to find the whole.

Step 2: Set Up the Equation

According to the percent proportion, we set up the equation as follows: \( \frac{36}{x} = \frac{72}{100} \). This equation states that 36 is to the unknown number \( x \) as 72 is to 100.

Step 3: Cross Multiply

To eliminate the fractions, cross multiply the terms in the proportion: \(36 \times 100 = 72 \times x\). This simplification step allows us to solve for \(x\) without dealing with fractions.

Step 4: Solve for the Unknown

Now, calculate the cross multiplication: \(3600 = 72x\). To isolate \(x\), divide both sides by 72: \(x = \frac{3600}{72}\).

Step 5: Calculate the Result

Perform the division to find the whole number: \(x = 50\).

Step 6: Final Check and Rounding

Since we are rounding to the nearest tenth, and since 50 is already a whole number, the answer remains 50. Double-check the division to ensure accuracy is preserved even though there was no need to round further.

Key Concepts

Understanding Cross Multiplication

Cross multiplication is a valuable mathematical technique used to solve equations involving fractions or ratios, like in percentage problems. It simplifies the process of solving equations that have the form \( \frac{a}{b} = \frac{c}{d} \). To cross-multiply, you multiply the numerator of each fraction by the denominator of the opposite fraction.

Imagine you have a proportion like \( \frac{36}{x} = \frac{72}{100} \). You cross-multiply to connect the fractions without actually dealing with them directly. You'll get:

• Multiply 36 by 100: \( 36 \times 100 \)
• Multiply 72 by \( x \): \( 72 \times x \)

Cross multiplication gives you an equation without fractions: \( 3600 = 72x \). It is an effective step as it brings everything into a form where you can directly solve for the unknown variable. This technique speeds up calculations and reduces errors when finding missing values in related terms.

Steps to Solving Equations

Solving equations is all about finding the value of the unknown variable that makes a mathematical statement true. Once you've cross-multiplied and have an equation like \( 3600 = 72x \), you're a step closer to finding out what \( x \) actually is.

To isolate \( x \), you need to perform operations that will leave \( x \) alone on one side of the equation:

• In our equation, \( 3600 = 72x \), divide both sides by 72 to solve for \( x \): \( x = \frac{3600}{72} \)

Now, perform the division to find out what \( x \) equals:

• The division \( 3600 \div 72 \) simplifies to \( 50 \)

Since solving equations requires patience and accuracy, always double-check your work to ensure no mistakes have been made. This is especially important when dealing with more complex equations or when the numbers aren't as straightforward. Normally, solving equations involves a series of steps including simplification, addition or subtraction, and division or multiplication, depending on where the variable is located.

The Art of Rounding Numbers

Rounding numbers is a mathematical process to simplify numbers, making them easier to work with or communicate. It involves adjusting the value of a number to a specific degree of precision. Here, we are asked to round to the nearest tenth, which involves looking at the number to one decimal place.

Suppose we've calculated a number like 50. We need to round this to the nearest tenth. But since the number is already whole, there isn't any decimal part to alter. Yet, understanding rounding helps when dealing with decimals:

• Identify the digit at the tenths place and examine the digit immediately following it.
• If the digit is 5 or more, increase the tenths digit by one and drop all digits further to the right.
• If less than 5, leave the tenths digit as is and eliminate subsequent digits.

Rounding numbers is important for approximations and is often required in situations like financial calculations, scientific data, and other fields requiring precise, yet streamlined, figures. Remember that the goal of rounding is to reflect a value that is as close as possible to the original number within your required precision.