Find the value of p if
(14p-4): (8p-1) = (3p+8): (9p+5) -

Options

A. 1
B. 0.5
C. 0.75
D. none of these

Answer

To find the value of \( p \) in the equation

\[ (14p - 4) : (8p - 1) = (3p + 8) : (9p + 5), \]

we can set up a proportion and solve for \( p \).

### Step 1: Set up the equation
We can write the proportion as:

\[ \frac{14p - 4}{8p - 1} = \frac{3p + 8}{9p + 5}. \]

### Step 2: Cross-multiply
Cross-multiplying gives us:

\[ (14p - 4)(9p + 5) = (3p + 8)(8p - 1). \]

### Step 3: Expand both sides
Now we will expand both sides of the equation.

**Left Side:**
\[ 14p \cdot 9p + 14p \cdot 5 - 4 \cdot 9p - 4 \cdot 5 = 126p^2 + 70p - 36p - 20 = 126p^2 + 34p - 20. \]

**Right Side:**
\[ 3p \cdot 8p + 3p \cdot (-1) + 8 \cdot 8p + 8 \cdot (-1) = 24p^2 - 3p + 64p - 8 = 24p^2 + 61p - 8. \]

### Step 4: Set the equation to zero
Now we set the two sides equal to each other:

\[ 126p^2 + 34p - 20 = 24p^2 + 61p - 8. \]

Subtract \( 24p^2 + 61p - 8 \) from both sides:

\[ 126p^2 - 24p^2 + 34p - 61p - 20 + 8 = 0. \]

This simplifies to:

\[ 102p^2 - 27p - 12 = 0. \]

### Step 5: Factor or use the quadratic formula
To solve for \( p \), we can use the quadratic formula:

\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]

where \( a = 102 \), \( b = -27 \), and \( c = -12 \).

### Step 6: Calculate the discriminant
First, we calculate the discriminant:

\[ b^2 - 4ac = (-27)^2 - 4 \cdot 102 \cdot (-12). \]
\[ = 729 + 4896 = 5625. \]

### Step 7: Substitute into the quadratic formula
Now substitute back into the formula:

\[ p = \frac{27 \pm \sqrt{5625}}{2 \cdot 102}. \]

Since \( \sqrt{5625} = 75 \):

\[ p = \frac{27 \pm 75}{204}. \]

### Step 8: Calculate the two possible values for \( p \)
Calculating the two values:

1. \( p = \frac{27 + 75}{204} = \frac{102}{204} = \frac{1}{2} \).
2. \( p = \frac{27 - 75}{204} = \frac{-48}{204} = \frac{-16}{68} = \frac{-4}{17} \).

### Step 9: Conclusion
Since \( p \) must be a positive value in this context, we take:

\[ p = \frac{1}{2}. \]