If the tens digit d is replaced with one of the digits 0-9, what is the probability that the four-digit positive integer 12d4 is divisible by 12? Express as a common fraction.

to be divisible by 12, d has to be 2 or 7

Pr=2/10= ?

To find the probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.

First, let's determine the favorable outcomes. We need to find the digit that makes the number divisible by 12.

A number is divisible by 12 if it is divisible by both 3 and 4. Therefore, our favorable outcomes are the numbers that are divisible by both 3 and 4.

To be divisible by 4, the last two digits (4d) must form a number that is divisible by 4. The only possible options are 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96.

To be divisible by 3, the sum of all the digits (1 + 2 + d + 4) must be divisible by 3. Since we know that 1 + 2 + 4 = 7, we need to find which digit value (d) would make the sum (7 + d) divisible by 3.

The possible values for d that make (7 + d) divisible by 3 are 2, 5, and 8.

Since we have three favorable outcomes (40, 44, and 48) for each eligible value of d (2, 5, and 8), we have a total of 3 x 3 = 9 favorable outcomes.

Now, let's determine the total number of possible outcomes. We have 10 options for the digit d (0-9).

Hence, the total number of possible outcomes is 10.

The probability is obtained by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = Favorable outcomes / Total outcomes
Probability = 9 / 10

Therefore, the probability that the four-digit positive integer 12d4 is divisible by 12 is 9/10.