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
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
Show 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.
Updated On: Jul 9, 2026
- \(\dfrac{1}{15}\)
- \(\dfrac{2}{15}\)
- \(\dfrac{1}{3}\)
- \(\dfrac{1}{5}\) \bigskip
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
The required probability is
\[
P(E)=\frac{\text{Number of favourable outcomes}}
{\text{Total number of outcomes}}.
\]
Here, we first solve the inequality and then count the multiples of \(3\) satisfying it.
Step 1: Solve the inequality. Given \[ x-\frac{26}{x}>25. \] Since \[ x\in\{1,2,\ldots,30\}, \] we have \[ x>0. \] Multiplying both sides by \(x\), \[ x^2-26>25x. \] \[ x^2-25x-26>0. \] Factorizing, \[ x^2-26x+x-26>0. \] \[ (x-26)(x+1)>0. \]
Step 2: Find the values of \(x\). Since \[ (x-26)(x+1)>0, \] the solution is \[ x26. \] But \[ x\in\{1,2,\ldots,30\}, \] therefore, \[ x>26. \] Hence, \[ x\in\{27,28,29,30\}. \]
Step 3: Select the multiples of \(3\). Among \[ 27,28,29,30, \] the multiples of \(3\) are \[ 27,\;30. \] Thus, the number of favourable outcomes is \[ 2. \]
Step 4: Compute the probability. Total possible choices from \[ \{1,2,3,\ldots,30\} \] are \[ 30. \] Therefore, \[ P(E) = \frac{2}{30} = \frac{1}{15}. \]
Step 5: Write the final answer. \[ \boxed{\frac{1}{15}} \]
Step 1: Solve the inequality. Given \[ x-\frac{26}{x}>25. \] Since \[ x\in\{1,2,\ldots,30\}, \] we have \[ x>0. \] Multiplying both sides by \(x\), \[ x^2-26>25x. \] \[ x^2-25x-26>0. \] Factorizing, \[ x^2-26x+x-26>0. \] \[ (x-26)(x+1)>0. \]
Step 2: Find the values of \(x\). Since \[ (x-26)(x+1)>0, \] the solution is \[ x26. \] But \[ x\in\{1,2,\ldots,30\}, \] therefore, \[ x>26. \] Hence, \[ x\in\{27,28,29,30\}. \]
Step 3: Select the multiples of \(3\). Among \[ 27,28,29,30, \] the multiples of \(3\) are \[ 27,\;30. \] Thus, the number of favourable outcomes is \[ 2. \]
Step 4: Compute the probability. Total possible choices from \[ \{1,2,3,\ldots,30\} \] are \[ 30. \] Therefore, \[ P(E) = \frac{2}{30} = \frac{1}{15}. \]
Step 5: Write the final answer. \[ \boxed{\frac{1}{15}} \]
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