Let's solve this problem step by step.
1. Determine the range of numbers on the cards:
The numbers on the cards range from 7 to 40. So the lowest number is 7, and the highest number is 40.
2. Calculate the total number of cards:
The total number of cards is given by the number of integers from 7 to 40, inclusive.
[tex]\[ \text{Total number of cards} = 40 - 7 + 1 = 34 \][/tex]
3. Count the multiples of 7 in the given range:
Next, we should identify how many numbers between 7 and 40 are multiples of 7.
The multiples of 7 in this range are: 7, 14, 21, 28, and 35. We can count these to find the number of multiples:
[tex]\[ 7, 14, 21, 28, 35 \][/tex]
This results in 5 multiples of 7.
4. Find the probability of selecting a card with a number that is a multiple of 7:
The probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[ \text{Probability} = \frac{\text{Number of multiples of 7}}{\text{Total number of cards}} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Probability} = \frac{5}{34} \][/tex]
So, the probability that a randomly selected card has a number that is a multiple of 7 is:
[tex]\[
\boxed{\frac{5}{34}}
\][/tex]
Therefore, the correct answer is (D) [tex]\( \frac{5}{34} \)[/tex].
