Cards marked with numbers 9,10,.......99 are placed in a bag. What is the probability that card drawn is a perfect square.

Options

A. `7//91`
B. `9//90`
C. `91//7`
D. `9//91`

Explanation

To solve the problem step by step, we need to find the probability that a card drawn from a bag containing cards numbered from 9 to 99 is a perfect square.

### Step 1: Identify the range of numbers
The cards are numbered from 9 to 99.

### Step 2: Count the total number of cards
The total number of cards can be calculated as follows:
- The smallest number is 9.
- The largest number is 99.
- To find the total count, we use the formula for counting integers in a range:
\[ \text{Total numbers} = \text{Largest number} - \text{Smallest number} + 1 = 99 - 9 + 1 = 91 \]

### Step 3: Identify the perfect squares in the range
Next, we need to find the perfect squares between 9 and 99. The perfect squares are:
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)

The next perfect square, \(10^2 = 100\), is outside our range. Thus, the perfect squares between 9 and 99 are: 9, 16, 25, 36, 49, 64, and 81.

### Step 4: Count the number of perfect squares
From our list, we see that there are 7 perfect squares: 9, 16, 25, 36, 49, 64, and 81.

### Step 5: Calculate the probability
The probability \(P\) of drawing a perfect square can be calculated using the formula:
\[ P(\text{perfect square}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{91} \]

### Step 6: Simplify the probability
To simplify \(\frac{7}{91}\):
- Both the numerator and denominator can be divided by 7:
\[ \frac{7 \div 7}{91 \div 7} = \frac{1}{13} \]

### Final Answer
Thus, the probability that a card drawn is a perfect square is \(\frac{1}{13}\).

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