Question

If a number \(x\) is to be chosen randomly from the set \[ \{1,2,3,\ldots,30\}, \] then the probability of getting an \(x\) that is a multiple of \(3\) such that \[ \left(x-\frac{26}{x}\right)>25 \] is

• A. \(\dfrac{1}{15}\)
• B. \(\dfrac{2}{15}\)
• C. \(\dfrac{1}{3}\)
• D. \(\dfrac{1}{5}\) \bigskip

Hint:

When a variable is known to be positive, inequalities involving fractions can be simplified safely by multiplying both sides by the variable. After solving the inequality, apply any additional restrictions such as divisibility conditions.

Answer 1

The Correct Option is A

Concept: Solve the inequality over the given integer set, then count favorable numbers (multiples of 3) and divide by total.

Step 1: Solve \(x - \frac{26}{x} > 25\). Multiply by \(x>0\): \(x^2 - 25x - 26 > 0\). Factor: \((x-26)(x+1)>0\). Solution: \(x<-1\) or \(x>26\). Given \(x\in\{1,...,30\}\), \(x\in\{27,28,29,30\}\).

Step 2: Count multiples of 3. Among 27,28,29,30: 27 and 30 are multiples of 3. Favorable = 2.

Step 3: Probability = \(\frac{2}{30} = \frac{1}{15}\).

Step 4: Write the final answer. \(\boxed{\frac{1}{15}}\)