Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{250}: 2=\frac{1}{150}: x\)

Short answer

x is approximately 3.33.

Step-by-step solution

Understand the Proportion

The given problem is a proportion with the equation \( \frac{1}{250}: 2 = \frac{1}{150}: x \). It means that the ratio of \( \frac{1}{250} \) to 2 is equal to the ratio of \( \frac{1}{150} \) to \( x \). We need to solve for \( x \).

Write the Proportion as a Fraction Equation

Convert the given proportion into a fractional equation: \( \frac{1}{250} \div 2 = \frac{1}{150} \div x \). This can be rewritten using the property of proportions: \( \frac{1/250}{2} = \frac{1/150}{x} \).

Cross-Multiply to Find a Single Variable Equation

Cross-multiply the terms to set up an equation: \( \frac{1}{250} \cdot x = 2 \cdot \frac{1}{150} \). This simplifies the equation to \( \frac{x}{250} = \frac{2}{150} \).

Simplify the Right Side Fraction

Simplify the fraction on the right: \( \frac{2}{150} = \frac{1}{75} \). This simplifies the equation to \( \frac{x}{250} = \frac{1}{75} \).

Solve for x by Cross-Multiplying

Cross-multiply to solve for \( x \): \( x \cdot 75 = 250 \). Thus, \( x = \frac{250}{75} \).

Calculate x

Calculate the value of \( x \): \( x = \frac{250}{75} \approx 3.33 \).

Key concepts

Fraction Equations

Fraction equations are mathematical expressions where two fractions are set equal to each other, often involving proportions. In a proportion, the equation is composed of two ratios. Understanding how to work with fraction equations is essential. You can use these equations to find an unknown variable when the condition of equality is met for two sets of quantities. Identifying components of the equation is important:

• The numerators represent parts of the whole.
• The denominators indicate the total or comparative parts in each section.

In our exercise, we can convert the given proportion \[ \frac{1}{250}: 2 = \frac{1}{150}: x \]into a fraction equation as follows:\[ \frac{1/250}{2} = \frac{1/150}{x} \]. Now, you have a clear path to solve for the unknown using cross-multiplication.

Cross-Multiplication

Cross-multiplication is a powerful technique used when solving fraction equations or proportions. This method simplifies calculations by eliminating the fractions through multiplication. It works by multiplying diagonally across the equal sign.Here’s how to use cross-multiplication:

• Multiply the numerator of the left fraction by the denominator of the right fraction.
• Multiply the denominator of the left fraction by the numerator of the right fraction.

The result will be a linear equation that is much easier to solve. For example, from\[ \frac{1/250}{2} = \frac{1/150}{x} \], we cross-multiply to get \[ 1 \cdot x = 2 \cdot \frac{1}{150} \].This results in \[ \frac{x}{250} = \frac{2}{150} \]. It transforms the problem into a simpler form where we can find the variable by further simplification.

Ratio Simplification

Simplifying ratios often simplifies solving equations involving proportions. Ratio simplification involves finding smaller integer values for both terms in the ratio, maintaining the same relationship between them.Here's how to simplify:

• Find a common factor for the terms in the ratio.
• Divide both terms by the greatest common factor.

For instance, in the equation\[ \frac{x}{250} = \frac{2}{150} \],the right-hand side can be simplified by dividing the numerator and denominator by their greatest common factor, which is 2 in this case.Therefore, the fraction becomes \[ \frac{1}{75} \].Now the equation simplifies to \[ \frac{x}{250} = \frac{1}{75} \]. Finally, by cross-multiplying once more, you can solve to get \[ x = \frac{250}{75} \approx 3.33 \]. This simplification helps resolve the variable more efficiently while maintaining accuracy.