Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(125: 0.4=50: x\)

Short answer

The value of \(x\) is 0.16.

Step-by-step solution

Understand the Problem

We are given a proportion \(125:0.4 = 50:x\). This can also be written as a fraction equation \(\frac{125}{0.4} = \frac{50}{x}\). Our goal is to solve for \(x\).

Cross-Multiply

Cross-multiplication is a method where we multiply the means and the extremes. So, we will multiply 125 by \(x\) and 0.4 by 50: \[125 \times x = 0.4 \times 50\]

Calculate Product of 0.4 and 50

Multiply 0.4 by 50: \[0.4 \times 50 = 20\] Our equation is now \(125x = 20\).

Solve for x

To isolate \(x\), divide both sides of the equation by 125: \[x = \frac{20}{125}\] After division, \(x = 0.16\).

Round to Two Decimal Places

The calculated value of \(x\), 0.16, is already rounded to two decimal places, so no further action is needed here.

Key concepts

Cross-Multiplication

Cross-multiplication is a powerful technique for solving proportions. Proportions are equations that state two ratios are equal. When faced with a proportion, such as \( \frac{a}{b} = \frac{c}{d} \), we can find an unknown by using cross-multiplication. This involves the following steps:

• Multiply the numerator of the first fraction (\(a\)) by the denominator of the second fraction (\(d\)).
• Multiply the denominator of the first fraction (\(b\)) by the numerator of the second fraction (\(c\)).

These two products are equal: \( a \times d = b \times c \). This method simplifies solving proportional equations by eliminating fractions, allowing us to solve a simple equation instead. It's like setting a balance where the cross products are the supporting beams, ensuring equality.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our exercise, after using cross-multiplication, we obtained the equation \( 125x = 20 \). To isolate \(x\), we follow these steps:

• Determine which operation is needed to get \(x\) alone on one side of the equation. In this case, division is necessary because \(x\) is multiplied by 125.
• Divide both sides of the equation by 125. This leaves \(x\) on one side and the result of the division on the other.

Therefore, \( x = \frac{20}{125} \). Simplifying the result offers the answer \( x = 0.16 \). These simple algebraic steps help in breaking down complex problems into manageable steps and finding solutions effectively.

Math Problem Solving

Math problem solving involves several essential skills like analyzing the problem and applying the appropriate methods. In this exercise, we are given a proportion, which requires understanding the concept of ratios and equations. Here are some key problem-solving strategies:

• Carefully read and understand the problem; define what is given and what needs to be found.
• Choose the right mathematical methods and operations, such as cross-multiplication for proportions.
• Break down the solution into a series of logical steps, solving step-by-step to avoid errors.
• Double-check your calculations especially with operations like division to ensure accuracy, as with rounding results.

Effective problem solving is not just about finding answers but understanding processes and applying logical reasoning to a variety of mathematical scenarios.