Problem 20 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the cross product property. Check your solution. $$ \frac{4}{2 w}=\frac{7}{3} $$

Step-by-Step Solution

Answer

The solution to the proportion \(\frac{4}{2w} = \frac{7}{3}\) is \(w = 0.857\).

Step 1: Cross Multiply

Cross multiplication implies multiplying the first term of the first ratio (4) by the second term of the second ratio (3) and the second term of the first ratio (2w) by the first term of the second ratio (7). This gives \(4 \times 3 = 2w \times 7\). Which simplifies to \(12 = 14w\).

Step 2: Solve for w

To resolve for w, divide both sides of the equation by 14. This gives \(w = 12/14\) which simplifies further to \(w = 0.857\).

Step 3: Verify the Solution

To check the solution, substitute w = 0.857 into the original equation \(\frac{4}{2w} = \frac{7}{3}\). This gives \(\frac{4}{2 \times 0.857} = \frac{7}{3}\) which simplifies to \(\frac{4}{1.714} = \frac{7}{3}\). Both sides reduce to approx 2.333, confirming that the solution \(w = 0.857\) holds true for the original ratio.

Key Concepts

Cross Product Property

In algebra, the cross product property comes in handy when solving proportions, such as in the case of \(\frac{4}{2 w}=\frac{7}{3}\). This property states that in a proportion made of two ratios, the product of the extremes (the first number of the first ratio and the second number of the second ratio) equals the product of the means (the second number of the first ratio and the first number of the second ratio).

Put simply, if we have \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\). Utilizing this property makes it easy to solve for an unknown variable because multiplying across the equal sign simplifies the equation into a more straightforward format that can be solved with basic algebraic operations.

Cross Multiplication

Cross multiplication is a method used to solve proportions, which is a depiction of the cross product property. When applied to the exercise \(\frac{4}{2 w}=\frac{7}{3}\), cross multiplication instructs us to multiply across the equation diagonally. This yields \(4 \times 3 = 2w \times 7\), leading to \(12 = 14w\).

The next step is to isolate the variable by dividing both sides by 14 to get \(w = \frac{12}{14}\). Simplifying this fraction gives us \(w = 0.857\), which is our potential solution. The process of cross multiplication turns a proportion into a linear equation, making it easier to find the value of an unknown quantity.

Verifying Solutions in Algebra

Once an equation is solved, it's critical to verify the solution to ensure accuracy. Verifying solutions is a process where the computed value is substituted back into the original equation to check if the equation holds true. For the given exercise, we verify that \(w = 0.857\) is correct by substituting it into the original proportion \(\frac{4}{2w} = \frac{7}{3}\).

Upon substitution, it simplifies to \(\frac{4}{1.714} = \frac{7}{3}\), which can be approximated as \(2.333 = 2.333\), confirming the equality. If both sides of the original equation yield the same result after the substitution, the solution is confirmed to be correct. This crucial step ensures our solution is not just a mathematical anomaly but a valid answer to the proportion problem.