Question

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest positive product, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $n - m$ equals :

• A. 9
• B. 11
• C. 8
• D. 10

Answer 1

The Correct Option is D

To solve this problem, we are given that the sum of two positive integers is 24. We need to find the probability that their product is not less than \(\frac{3}{4}\) times their greatest positive product, and then find the difference \(n - m\) where the probability is \(\frac{m}{n}\) in simplest form.

Step 1: Let's denote the two numbers as \(x\) and \(y\). Given \(x + y = 24\), we can express \(y\) as \(24 - x\). So, the product \(P\) of \(x\) and \(y\) is:

\(P = x(24 - x) = 24x - x^2\)

Step 2: The product \(x(24 - x)\) is a quadratic function in terms of \(x\). The maximum product can be found by finding the vertex of this parabola, which is given by:

\(x = -\frac{b}{2a} = \frac{24}{2} = 12\)

So, the maximum product occurs when \(x = 12\) and \(y = 24 - 12 = 12\). The greatest product is:

\(P_{\text{max}} = 12 \times 12 = 144\)

Step 3: We need to calculate the condition:

\(x(24 - x) \geq \frac{3}{4} \times 144 = 108\)

Simplifying, this is:

\(24x - x^2 \geq 108\) \(x^2 - 24x + 108 \leq 0\)

Step 4: Solving the quadratic inequality:

\(x^2 - 24x + 108 = 0\)

Using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{24 \pm \sqrt{24^2 - 4 \times 1 \times 108}}{2}\) \(x = \frac{24 \pm \sqrt{576 - 432}}{2} = \frac{24 \pm 12}{2}\)

This gives:

\(x = 18 \quad \text{and} \quad x = 6\)

Step 5: The inequality \(x^2 - 24x + 108 \leq 0\) is valid for:

\(6 \leq x \leq 18\)

Step 6: Count the number of integer solutions for \(x\):

• If \(x = 6\), then \(y = 18\).
• If \(x = 18\), then \(y = 6\).

The integers from 6 to 18 inclusive provide the pairs:

\((6,18), (7,17), \ldots, (18,6)\)

These are 13 pairs (from 6 to 18 inclusive), so \(x\) has 13 integer solutions.

Step 7: Calculate the probability:

There are 23 total pairs where \(x+y=24\), with \(x\) ranging from 1 to 23. Hence, the probability is:

\(\frac{13}{23}\)

Therefore, \(m = 13\) and \(n = 23\), giving \(n - m = 23 - 13 = 10\).

Thus, the correct answer is 10.

Answer 2

Given \( x + y = 24 \), \( x, y \in \mathbb{N} \), the greatest product occurs at:

\[ x = y = 12 \implies \text{Maximum Product} = 144. \]

Step 1: Define the condition:

\[ xy \geq \frac{3}{4} \cdot 144 \implies xy \geq 108. \]

Step 2: List favorable pairs:

\[ (13, 11), (12, 12), (14, 10), (15, 9), (16, 8), (17, 7), (18, 6), (6, 18), (7, 17), (8, 16), (9, 15), (10, 14), (11, 13). \]

Step 3: Total cases and favorable cases:

There are \( 13 \) favorable cases out of \( 23 \) total cases.

\[ \text{Probability} = \frac{13}{23}. \]

Step 4: Calculate:

\[ m = 13, \quad n = 23 \implies n - m = 10. \]

Final Answer:

\[ \boxed{10.} \]