A game of numbers has cards marked with 11, 12, 13, ……, 40. A card is drawn at random. Find the probability that the number on the card drawn is :
(i) A perfect square

Answer

To solve the problem of finding the probability that the number on a randomly drawn card is a perfect square, we can follow these steps:

### Step 1: Identify the range of numbers on the cards
The cards are marked with numbers from 11 to 40.

**Hint:** List all the numbers from 11 to 40 to see the total count.

### Step 2: Count the total number of cards
The total numbers from 11 to 40 can be calculated as:
\[ \text{Total numbers} = 40 - 11 + 1 = 30 \]

**Hint:** Use the formula for counting numbers in a range: \( \text{Last number} - \text{First number} + 1 \).

### Step 3: Identify the perfect squares in the range
We need to find the perfect squares between 11 and 40. The perfect squares are:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)

Thus, the perfect squares in this range are 16, 25, and 36.

**Hint:** Recall that perfect squares are numbers that can be expressed as \( n^2 \) where \( n \) is an integer.

### Step 4: Count the number of perfect squares
From the identified perfect squares, we see there are 3 perfect squares: 16, 25, and 36.

**Hint:** Simply count the perfect squares you found in the previous step.

### Step 5: Calculate the probability
The probability \( P \) of drawing a card with a perfect square is given by the formula:
\[ P(\text{perfect square}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Substituting the values we found:
\[ P(\text{perfect square}) = \frac{3}{30} \]

### Step 6: Simplify the probability
Now simplify \( \frac{3}{30} \):
\[ P(\text{perfect square}) = \frac{1}{10} \]

### Final Answer
The probability that the number on the card drawn is a perfect square is \( \frac{1}{10} \).

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