https://r2.erweima.ai/mathbot/image/1725235574189233eopdg.jpg

Let's analyze the problem step-by-step. 

**Given Information:**

1. The geometric ratio between two numbers \(a\) and \(b\) is \(\frac{a}{b} = \frac{5}{2}\).
2. The sum of these two numbers is \(a + b = 28\).
3. We need to determine the value of \(J\) where:

\[
J = \frac{(a - 2b)(a - 3b)}{a + b - 30}
\]

### Step-by-Step Solution:

1. **Find the values of \(a\) and \(b\):**

   Given the ratio \(\frac{a}{b} = \frac{5}{2}\), we can express \(a\) in terms of \(b\):

   \[
   a = \frac{5}{2}b
   \]

   Substituting \(a = \frac{5}{2}b\) into the sum equation \(a + b = 28\):

   \[
   \frac{5}{2}b + b = 28
   \]

   Combine like terms:

   \[
   \frac{7}{2}b = 28
   \]

   Multiply both sides by 2 to solve for \(b\):

   \[
   7b = 56 \implies b = 8
   \]

   Now substitute \(b = 8\) back into the expression for \(a\):

   \[
   a = \frac{5}{2} \times 8 = 20
   \]

   So, \(a = 20\) and \(b = 8\).

2. **Calculate \(J\):**

   Substitute \(a = 20\) and \(b = 8\) into the formula for \(J\):

   \[
   J = \frac{(a - 2b)(a - 3b)}{a + b - 30}
   \]

   Compute each term:

   \[
   a - 2b = 20 - 2 \times 8 = 20 - 16 = 4
   \]

   \[
   a - 3b = 20 - 3 \times 8 = 20 - 24 = -4
   \]

   \[
   a + b - 30 = 20 + 8 - 30 = 28 - 30 = -2
   \]

   Substitute these into the expression for \(J\):

   \[
   J = \frac{(4)(-4)}{-2} = \frac{-16}{-2} = 8
   \]

So, the value of \(J\) is **8**.

### Conclusion:

The value of \(J\) is \(8\).

Would you like more details on any step, or do you have any questions?

### Related Questions to Expand Your Understanding:

1. How would the problem change if the geometric ratio was different?
2. What is the significance of the geometric ratio in this problem?
3. Can you derive the expression for \(J\) in a different way?
4. What would happen if the sum of \(a\) and \(b\) was a different number?
5. How do the steps change if \(a\) and \(b\) were different types of numbers, like fractions?

### Tip:

Always double-check each step of your solution to avoid any arithmetic or algebraic errors.

Explore detailed solutions to problems involving razones y proporciones (ratios and proportions). This problem covers key concepts including setting up and solving equations based on given ratios and proportions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation