A number x is chosen at random from the set {1, 2, 3,…., 100}. Define event: A = the chosen number x satisfies `(x-20)/(x-40) ge 2`. Then P(A) is

Options

A. `1/4`
B. `1/5`
C. `1/8`
D. `1/10`

Answer

To solve the problem, we need to find the probability \( P(A) \) where event \( A \) is defined by the condition:

\[ \frac{x - 20}{x - 40} \geq 2 \]

### Step 1: Solve the inequality

We start by manipulating the inequality:

\[ \frac{x - 20}{x - 40} \geq 2 \]

Multiplying both sides by \( x - 40 \) (noting that we need to consider the sign of \( x - 40 \) later):

\[ x - 20 \geq 2(x - 40) \]

Expanding the right side:

\[ x - 20 \geq 2x - 80 \]

Rearranging gives:

\[ -20 + 80 \geq 2x - x \]

\[ 60 \geq x \]

or

\[ x \leq 60 \]

### Step 2: Consider the cases for the sign of \( x - 40 \)

We need to consider two cases based on the sign of \( x - 40 \):

1. **Case 1:** \( x - 40 > 0 \) (i.e., \( x > 40 \))
2. **Case 2:** \( x - 40 < 0 \) (i.e., \( x < 40 \))

#### Case 1: \( x > 40 \)

In this case, the inequality \( \frac{x - 20}{x - 40} \geq 2 \) simplifies to:

\[ x - 20 \geq 2(x - 40) \]

As derived above, we found that \( x \leq 60 \). Therefore, for this case, we have:

\[ 40 < x \leq 60 \]

This means \( x \) can take values \( 41, 42, 43, \ldots, 60 \).

#### Case 2: \( x < 40 \)

In this case, the inequality reverses when we multiply by \( x - 40 \):

\[ x - 20 \leq 2(x - 40) \]

This simplifies to:

\[ x - 20 \leq 2x - 80 \]

Rearranging gives:

\[ 60 \leq x \]

However, this contradicts our assumption that \( x < 40 \). Therefore, there are no valid solutions in this case.

### Step 3: Determine the valid values of \( x \)

From Case 1, the valid values of \( x \) are:

\[ 41, 42, 43, \ldots, 60 \]

To find the total number of valid values, we calculate:

\[ 60 - 41 + 1 = 20 \]

### Step 4: Calculate the probability

The total number of possible outcomes when choosing \( x \) from the set \( \{1, 2, \ldots, 100\} \) is 100. The number of favorable outcomes is 20.

Thus, the probability \( P(A) \) is:

\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{20}{100} = \frac{1}{5} \]

### Final Answer

\[ P(A) = \frac{1}{5} \]

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