Karan needs to choose between two gym plans. He can either pay a / $150$150dollar sign, 150 joining fee and a / $10$10dollar sign, 10 monthly fee, or he can pay a / $50$50dollar sign, 50 joining fee and a / $30$30dollar sign, 30 monthly fee. On what month will the cumulative costs of the plans be equal, and what will those total costs be?

We can setup two different equations for the total cost of the gym for each month.

1) For $150 oining fee and $10 monthly fee, we can make

y=150+10x ... equation (1)

where x is the number of months and y is the total cost.

2) For $50 oining fee and $30 monthly fee, we can make

y=50+30x ... equation (2)

We need to find the number of months when cumulative costs of the plans be equal.

So, we need to set both equations equal and solve for x.

50+30x = 150+10x

Subtracting 50 from both sides, we get

50-50+30x = 150-50+10x

30x = 100 + 10x.

Subtracting 10 from both sides, we get

30x-10x = 100 + 10x-10x.

20x = 100

Dividing both sides by 20, we get

20x/20 = 100/20

x=5.

Plugging x=5 in first equation, we get

y=150+10 (5) = 150+50 = $200.

Therefore, after 5 month the cumulative costs of the plans be equal and those total costs be will be $200.