Problem 21 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the cross product property. Check your solution. $$ \frac{5}{3 d}=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution to the proportion \( \frac{5}{3 d}=\frac{2}{3} \) is \( d = \frac{5}{2} \)

Step 1: Cross Multiplication

To begin with, cross-multiply which means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. We get: \( 5 * 3 = 2 * 3d \)

Step 2: Simplify and Solving

We simplify the above equation. This gives us: 15 = 6d. Next, divide each side by 6 (coefficient of d) to isolate d. Thus, \( d = \frac{15}{6} \)

Step 3: Simplify Further

On simplifying our answer further, by cancelling out the common factor which is 3. \( d = \frac{5}{2} \).

Step 4: Check

Check the solution by substituting \( d = \frac{5}{2} \) back into the original equation. \( \frac{5}{3*\frac{5}{2}} = \frac{2}{3} \). Simplifying this, we do indeed find \( \frac{2}{3} = \frac{2}{3} \), confirming that our solution is correct.

Key Concepts

Proportion

A proportion is a statement that two ratios, or fractions, are equal. In this exercise, we have the proportion \( \frac{5}{3d} = \frac{2}{3} \). Proportions are used frequently in mathematics to express the equality of two ratios, especially when comparing different quantities. To solve a proportion, cross multiplication can be a very useful tool. This involves multiplying the numerator of one fraction by the denominator of the other. Similarly, you multiply the denominator of the first fraction by the numerator of the second. Using the cross product property helps us to clear fractions from the equation and proceed with solving for the unknown variable. In our example, cross multiplying the given proportion yields the equation: \( 5 \times 3 = 2 \times 3d \). This is the crucial first step to solving the proportion.

Solving Equations

Once we have the equation from cross multiplication, our task turns to solving for the variable. An equation is essentially a mathematical statement that asserts the equality of two expressions. Here, our equation from cross multiplication is \( 15 = 6d \). To find the solution for \( d \), we need to isolate the variable. This involves performing inverse operations to both sides of the equation, which helps in 'undoing' the operation done to the variable. In our situation, dividing both sides by the coefficient of \( d \), which is 6, yields \( d = \frac{15}{6} \). This step is crucial because it simplifies the expression and helps us get closer to finding the exact value of \( d \).

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. This makes them more readable and manageable for further calculations. When simplifying fractions, we often look for the greatest common factor (GCF) of the numerator and the denominator. For our fraction \( \frac{15}{6} \), the GCF is 3. Dividing both the numerator and the denominator by 3 simplifies the fraction to \( \frac{5}{2} \). This means that the simplest form of our answer for \( d \) is \( \frac{5}{2} \). Finally, after simplifying, it is always recommended to check your work by substituting back into the original proportion. By doing this, you verify that the simplification and solution are correct, ensuring that \( \frac{5}{3 \times \frac{5}{2}} \) indeed equals \( \frac{2}{3} \). As verified, simplifying fractions correctly maintains the true relationship presented in the proportion.