If ` p + q = 10 and pq = 5` then the numerical value of ` (p)/(q) + (q)/( p)` will be

Options

A. A)16
B. B)20
C. C)22
D. D)18

Answer

To solve the problem, we need to find the numerical value of \(\frac{p}{q} + \frac{q}{p}\) given that \(p + q = 10\) and \(pq = 5\).

### Step 1: Rewrite the expression
We start with the expression \(\frac{p}{q} + \frac{q}{p}\). This can be rewritten using a common denominator:
\[ \frac{p}{q} + \frac{q}{p} = \frac{p^2 + q^2}{pq} \]

**Hint:** Remember that \(\frac{a}{b} + \frac{c}{d}\) can be combined into a single fraction using a common denominator.

### Step 2: Find \(p^2 + q^2\)
To find \(p^2 + q^2\), we can use the identity:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Substituting the known values:
\[ p + q = 10 \quad \text{and} \quad pq = 5 \]
we get:
\[ p^2 + q^2 = (10)^2 - 2 \times 5 \]

**Hint:** Use the square of a binomial to simplify \(p + q\).

### Step 3: Calculate \(p^2 + q^2\)
Now, calculate:
\[ p^2 + q^2 = 100 - 10 = 90 \]

**Hint:** Ensure you perform the arithmetic operations carefully.

### Step 4: Substitute back into the expression
Now substitute \(p^2 + q^2\) and \(pq\) back into our rewritten expression:
\[ \frac{p^2 + q^2}{pq} = \frac{90}{5} \]

**Hint:** Simplifying fractions can often lead to the final answer.

### Step 5: Simplify the fraction
Now, simplify:
\[ \frac{90}{5} = 18 \]

**Hint:** Always double-check your division for accuracy.

### Conclusion
Thus, the numerical value of \(\frac{p}{q} + \frac{q}{p}\) is \(18\).

**Final Answer:** 18