A box contains cards numbered from 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card is:
A square number

Answer

To find the probability that the number on the drawn card is a square number, we will follow these steps:

### Step 1: Determine the total number of cards
The cards are numbered from 11 to 123. To find the total number of cards, we can use the formula for counting the numbers in a range:

\[ \text{Total numbers} = \text{Last number} - \text{First number} + 1 \]

Substituting the values:

\[ \text{Total numbers} = 123 - 11 + 1 = 113 \]

### Step 2: Identify the square numbers between 11 and 123
Square numbers are the numbers that can be expressed as \(n^2\), where \(n\) is a whole number. We need to find the square numbers that fall within the range of 11 to 123.

Calculating the square roots:
- The smallest integer \(n\) such that \(n^2 \geq 11\) is \(4\) (since \(4^2 = 16\)).
- The largest integer \(n\) such that \(n^2 \leq 123\) is \(11\) (since \(11^2 = 121\)).

Now, we can list the square numbers from \(4^2\) to \(11^2\):
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)
- \(10^2 = 100\)
- \(11^2 = 121\)

Thus, the square numbers between 11 and 123 are: 16, 25, 36, 49, 64, 81, 100, and 121. This gives us a total of 8 square numbers.

### Step 3: Calculate the probability
The probability \(P\) of drawing a card with a square number is given by the formula:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{nf}{nt} \]

Where:
- \(nf\) = number of favorable outcomes (square numbers) = 8
- \(nt\) = total number of outcomes (total cards) = 113

Substituting the values:

\[ P = \frac{8}{113} \]

### Final Answer
The probability that the number on the drawn card is a square number is:

\[ \frac{8}{113} \]

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