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Chapter 6: Problem 364

In the following exercises, translate and solve. 200\(\%\) of what number is 50\(?\)

Short Answer

Expert verified
The number is 25.

Step by step solution

01

- Understand the Problem

The problem is asking to find a number such that 200% of this number equals 50.
02

- Translate the Problem into an Equation

Translate the word problem into a mathematical equation. Let the unknown number be represented by the variable \(x\). The equation based on the problem statement is:\[200\text{\textpercent} \times x = 50\]
03

- Convert Percentage to Decimal

Convert 200% into a decimal by dividing by 100. This gives us 2. The equation now becomes:\[2 \times x = 50\]
04

- Solve for x

To solve for \(x\), isolate \(x\) by dividing both sides of the equation by 2:\[x = \frac{50}{2} = 25\]
05

- Verify the Solution

Verify the solution by checking if 200% of 25 equals 50. Calculate 200% of 25:\[200\text{\textpercent} \times 25 = 2 \times 25 = 50\]Since this is true, the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translating Word Problems into Equations
Translating word problems into equations is a technique used to convert written information into a mathematical expression. It allows us to solve real-life scenarios using algebra. Here's a step-by-step approach to translate word problems into equations:
  • **Identify the unknown(s):** Look for what you need to find (often represented by variables like x, y). In the given problem, this unknown is the number we have to find.
  • **Determine the relationship:** Connect the numbers and operations based on the context of the problem. For example, '200% of what number is 50' translates to 200% times that number equals 50.
  • **Formulate the equation:** Use mathematical symbols to represent the problem as an equation. From the problem, we form the equation: \[ 200\text{\textpercent} \times x = 50 \]
This method helps simplify complex problems, making them easier to solve mathematically.
Converting Percentages to Decimals
Understanding and converting percentages to decimals is critical for solving many math problems, especially those involving percentages. Here's how to do it:
Percentages are a way of expressing numbers as parts of 100. To convert a percentage to a decimal:
  • **Step 1:** Remove the percent (\textpercent) sign: For instance, 200% becomes 200.
  • **Step 2:** Divide by 100: This helps convert the value into a decimal representation. For 200%, you would do:\[ 200 \div 100 = 2 \]
Once converted, the problem becomes easier to manipulate. For instance, our equation \[ 200\text{\textpercent} \times x = 50 \] turns into \[ 2 \times x = 50 \] using this method. This technique is widely used in various areas of mathematics.
Solving Linear Equations
Solving linear equations involves finding the value of the unknown variable that makes the equation true. Here's a basic guide to solving linear equations like the one in our example:
  • **Isolate the variable:** Arrange the equation to get the variable on one side and other numbers on the opposite side. For \[ 2 \times x = 50, \] you need to isolate x.
  • **Perform the necessary operations:** To isolate x, divide both sides of the equation by 2:\[ x = \frac{50}{2} = 25 \]
  • **Verify your solution:** Always re-check your solution by substituting the found value back into the original equation. From our example, if we substitute x = 25: \[ 2 \times 25 = 50 \] which is correct.
By following these steps, you can confidently solve most linear equations.

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Most popular questions from this chapter

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