Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{4}: 16=\frac{1}{8}: x\)

Short answer

\(x = 8\)

Step-by-step solution

Understand the Proportion

The problem involves solving the proportion \(\frac{1}{4}:16 = \frac{1}{8}:x\). This can be rewritten as an equation: \(\frac{1}{4} \div 16 = \frac{1}{8} \div x\). Our goal is to solve for \(x\).

Set Up the Equation

Write the equation as follows: \(\frac{1}{4} \cdot x = 16 \cdot \frac{1}{8}\). This is because the cross products in a proportion are equal.

Simplify the Right Side

Calculate \(16 \cdot \frac{1}{8} = 2\). Simplifying the right side using multiplication gives us \(2\).

Rewrite the Equation and Solve for x

Now our equation is \(\frac{1}{4} \cdot x = 2\). To solve for \(x\), multiply both sides by \(4\) to get \(x = 2 \cdot 4\).

Calculate the Value of x

Finally, calculate \(x = 2 \cdot 4 = 8\). The value of \(x\) is \(8\).

Key concepts

Understanding Proportions

Proportions are a way to show that two ratios are equal. In simpler terms, they compare two fractions or ratios to see if they have the same value. For example, the problem we are dealing with is set up like this: \( \frac{1}{4}:16 = \frac{1}{8}:x \). These two ratios are proportional, meaning they hold the same value when simplified. It's important to note that with proportions, we can use a cross-multiplication method to solve them. Cross-multiplication involves multiplying the top number of one ratio by the bottom number of the other ratio and vice versa. This method helps in verifying if the ratios are indeed equal or to find the unknown value when one part of the proportion is missing, like the \(x\) in our current problem.

• Proportions help us find an unknown part of a ratio.
• They use equal fractions to compare different quantities.
• Cross-multiplication is a key tool used to solve these problems.

Exploring Equations

Equations are mathematical statements that show two expressions are equal. In our exercise, we originally have the proportion \( \frac{1}{4} : 16 = \frac{1}{8} : x \), which can be rewritten into an equation format. Transforming a proportion into an equation involves setting up both parts of the proportion as equal products, like this: \( \frac{1}{4} \cdot x = 16 \cdot \frac{1}{8} \).The beauty of equations is that they provide a way to solve for unknown variables, like our \(x\). In algebra, we use various operations like addition, subtraction, multiplication, and division to isolate the unknown variable on one side of the equation and solve for it. This approach helps provide a clearer pathway to our answer.

• Equations are foundational in finding unknowns.
• Turning proportions into equations makes it easier to solve.
• Operations can be used to balance and simplify these equations.

Effective Mathematical Problem-Solving

Mathematical problem-solving is about understanding the problem, planning how to solve it, executing your plan, and then reviewing your solution. In our case, solving the equation derived from the proportion is an excellent example.Firstly, we identified what was being asked: finding \(x\) in the proportion \( \frac{1}{4}:16 = \frac{1}{8}:x \). The plan involved transforming the proportion into an equation and then using algebra to isolate and solve for \(x\). Execution of the plan included setting up the equation \( \frac{1}{4} \cdot x = 2 \), and calculating through multiplication to find \(x = 8\). Finally, reviewing confirms that putting \(8\) back into the proportion keeps it equal, verifying our solution.

• Successful problem-solving starts with a clear understanding.
• Planning and executing mathematical steps is crucial.
• Verification ensures the solution is correct.