A pair of dice are thrown 50 times. What is the chance that 8 will be rolled 18 times and 6 will be rolled 9 times?

Step-by-step explanation:

1. Let each individual act will last for I minutes and each group act wil last for G minutes.

The overall show contains 12 individual act and 6 group acts.

Hence, the total time is 12I + 6G minutes.

According to the question,  12I + 6G = 120.

Similarly, as the final show contains 4 individual act and 1 group act, 4I + G = 30.

Comparing, the above two equation, with the given equations in the question, it can be said that, x is the time required for individual act and y is the time for a group act.

2. As per the above explanation, the equations are correct.

But, we had never been told that each individual act takes the same time or each group act takes same time. From that point of view, the equations are not correct.

3. If we will multiply the second equation with -6 and then add it with the first equation, then y will be eliminated.

The second equation will be transformed to -24x -6y = -180.

4. 12x + 6y - 24x - 6y = 120 - 180

or, -12x = -60

or, x = 5.

5. Putting x = 5, in the second equation we get, 20 + y = 30 or, y = 10.

6. If x = 5 and y = 10, then 4x + y = 20 + 10 = 30 and 12x + 6y = 60 + 60 = 120. Hence, the solutions are correct.

7. The co-ordinates of the intersecting point is (5, 10).

8. a) Each individual act requires 5 minutes.

b) Each group act requires 10 minutes.

9. a) The answers are reasonable. These are shown through solving the equations as well as graphing.

b) Negative answers cannot be acceptable here, as we are talking about time, which can not be negative.