Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{3}{x}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The value of x that satisfies the given proportion is \(x=6\).

Step 1: Apply reciprocal property (Cross Multiplication)

Cross-multiply the fractions present in the equation. That means multiply the numerator of the first fraction with the denominator of the second and vice versa. From there, form a new equation as follows: \(3 * 2 = x * 1\).

Step 2: Simplify the equation

Simplify the left-hand side of the equation from \(3 * 2\) to get \(6 = x\).

Step 3: Check the solution

Substitute the solution for x back into the original equation, this gives \(\frac{3}{6} = \frac{1}{2}\). Since this equation is true, the solution is correct.

Key Concepts

Reciprocal Property

To understand how to solve a proportion using the reciprocal property, it's crucial to know what a reciprocal is. A reciprocal of a number is simply 1 divided by that number. When dealing with fractions, the reciprocal is obtained by swapping the numerator and the denominator. For example, the reciprocal of \( \frac{3}{x} \) is \( \frac{x}{3} \), and the reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \), which simplifies to 2.
Using the reciprocal property allows us to solve proportions more easily by cross-multiplying the terms involved. This property comes in handy when you have an equation like \( \frac{3}{x} = \frac{1}{2} \). Rather than finding the reciprocal directly, you apply cross multiplication to solve for \(x\). This method involves multiplying across the equals sign, which we'll cover next.

Cross Multiplication

Cross multiplication is a powerful technique used to solve equations involving two fractions set equal to each other. This method eliminates the fractions by multiplying the numerator of one fraction by the denominator of the other, effectively 'crossing' the terms.
When you have an equation like \( \frac{3}{x} = \frac{1}{2} \), you can perform cross multiplication by multiplying 3 (the numerator of the left fraction) by 2 (the denominator of the right fraction), and \(x\) (the denominator of the left fraction) by 1 (the numerator of the right fraction).

• The equation then becomes \( 3 \times 2 = x \times 1 \).
• After substituting these values, you simplify to get \(6 = x\).

This step is crucial as it clears the fractions, making it easier to solve for the unknown variable. Cross multiplication not only simplifies solving proportions but also keeps the equations balanced.

Checking Solutions

After calculating a potential solution, it is important to verify its accuracy. This step ensures that our answer solves the original proportion correctly. In our example, after finding that \( x = 6 \), we substitute it back into the original equation, \( \frac{3}{x} = \frac{1}{2} \).
To check:

• Replace \(x\) with 6 to get \( \frac{3}{6} = \frac{1}{2} \).
• Simplify \( \frac{3}{6} \) to check if it equals \( \frac{1}{2} \).
• Indeed, \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \), confirming that \(3 \times 6 = 18\) matches \(6 \div 12\).

This process is integral to solving proportions as it validates our solution. Verifying solutions helps build confidence in your math skills and ensures you have not made errors during calculation, such as arithmetic mistakes.