Problem 13 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{3}{4}=\frac{8}{3 c} $$

Step-by-Step Solution

Answer

The solution to the given proportion is \( c = \frac{32}{9} \).

Step 1: Apply The Reciprocal Property

Switch both fractions using the reciprocal property to get: \( \frac{4}{3} = \frac{3c}{8} \).

Step 2: Solve for c

Multiply both sides by 8 to isolate \( c \) on one side of the equation: \( \frac{4}{3} \times 8 = 3c \) which simplifies to \( c = \frac{32}{9} \).

Step 3: Check the solution

Substitute the calculated value of \( c \) back into the original equation to ensure it holds true: \( \frac{3}{4}=?\frac{8}{ 3 \times \frac{32}{9}} \) which simplifies to \( \frac{3}{4} = \frac{3}{4} \). Hence the solution is valid.

Key Concepts

Reciprocal Property

Understanding the reciprocal property can be very handy when solving proportions. A reciprocal of a fraction is essentially flipping the numerator and the denominator. For example, if you have the fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). In the context of proportions, applying the reciprocal property allows you to swap both fractions in the equation without changing the balance of the equation.

When you have a proportion like \( \frac{3}{4} = \frac{8}{3c} \), you can apply the reciprocal property to get \( \frac{4}{3} = \frac{3c}{8} \). This switch helps simplify solving for the unknown variable. Always remember though, reciprocals are only applicable in fraction format, and both sides must be flipped to maintain equality. This property is crucial, as it often makes solving for variables more straightforward.

Solving Equations

Solving equations involving proportions is all about finding the unknown variable. Once you've applied the reciprocal property and simplified your equation if needed, the next step is to isolate the variable. This often means getting the variable term alone on one side of the equation.

In our example, after switching to \( \frac{4}{3} = \frac{3c}{8} \), multiply both sides by 8, resulting in \( 8 \times \frac{4}{3} = 3c \). This reduces the equation to \( \frac{32}{3} = 3c \), which can be divided by 3 to solve for \( c \). Doing so gives us \( c = \frac{32}{9} \).

Finding the right steps to isolate the variable might take some practice, but always ensure each step maintains the equation's equality. The goal is to simplify step by step until the variable stands alone.

Check Solutions

Checking your solutions is a vital step to verify the accuracy of your answer. After solving for \( c \) as \( \frac{32}{9} \), substitute it back into the original equation to ensure it holds true.

Returning to our original proportion \( \frac{3}{4} = \frac{8}{3c} \), plug \( c = \frac{32}{9} \) to get \( \frac{8}{3 \times \frac{32}{9}} \). Simplifying this gives \( \frac{8}{\frac{96}{9}} \), further simplifying to \( \frac{3}{4} \), which matches the other side of the equation exactly.

By validating that both sides of the equation remain equal with the found value of \( c \), you confirm your solution is correct. Always perform this step as it acts like a safety check to catch any possible errors made during your calculations.