Question

Translate to an equation and solve. 605 is \(2.5 \%\) of what number?

Short answer

Answer: The unknown number is 24200.

Step-by-step solution

Understand the problem

We are given that 605 is 2.5% of an unknown number. Let's call this unknown number "x". We will set up an equation to represent this relationship between the percentage and the unknown number.

Set up the equation

To set up the equation, we'll use the fact that "2.5% of x" is equal to 605. In mathematical terms, we can write this as: \[0.025x=605\] Where 2.5% has been converted into a decimal by dividing by 100.

Solve for x

To find the value of x, we need to isolate it by dividing both sides of the equation by 0.025: \[\frac{0.025x}{0.025}=\frac{605}{0.025}\] This simplifies to: \[x=24200\]

State the conclusion

So, 605 is 2.5% of 24200. The unknown number we were looking for is 24200.

Key concepts

Translating Word Problems into Equations

Translating word problems into equations is a crucial skill in mathematics, allowing us to represent real-world situations mathematically. When faced with a word problem, the first step is to carefully read and understand what is being asked. Identify known and unknown quantities, keywords, and phrases that indicate mathematical operations. For example:

• "Is" often translates to equals (=).
• "Of" usually indicates multiplication.
• Percentages involve converting them into decimals or fractions for easy mathematical manipulation.

In our exercise, we have the phrase "605 is 2.5% of what number?" Translating this into an equation, "is" becomes "=", and "of" suggests multiplication. Next, represent 2.5% as a decimal by dividing it by 100, which gives 0.025. Therefore, the equation is 0.025 times a number (x) equals 605, or mathematically, \(0.025x = 605\). Translating correctly ensures you set the right mathematical foundation to solve the problem.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true. A linear equation is an algebraic statement where the highest power of the variable is one. In this context, we had the linear equation \(0.025x = 605\).
To isolate the variable \(x\), perform operations that 'undo' the equation's structure to find its value. This usually involves:

• Eliminating coefficients by dividing or multiplying both sides of the equation by the same number.
• Using addition or subtraction to isolate terms involving the unknown.

For \(0.025x = 605\), divide both sides by 0.025. This operation cancels out the 0.025 on the left, leaving \(x = \frac{605}{0.025}\). Calculating this gives \(x = 24200\). By methodically applying operations to both sides, you arrive at the solution, solving the equation and revealing \(x\)'s value.

Basic Algebra Concepts

Basic algebra involves understanding fundamental operations and principles that apply to equations and variables. These include knowing how to manipulate numbers and expressions to solve problems effectively. Key concepts include:

• Using variables to represent unknowns or quantities that can change.
• Performing arithmetic operations on both sides of an equation for balance.
• Using percentage calculations, often requiring conversion to decimals or fractions.

In this problem, we use the variable \(x\) for an unknown number. Understanding that 2.5% equates to a decimal (0.025), we integrate percentages into algebra seamlessly. The process involves rearranging the equation to isolate and find \(x\). Basic algebra teaches that preserving equality by performing the same operation on both sides keeps equations valid. Successfully applying these concepts helps solve problems with efficiency and clarity, a skill that is essential in progressing through algebra and other mathematics fields.