A real estate agent is considering changing her cell phone p

A real estate agent is considering changing her cell phone plan. There are three plans to choose from, all of which involve a monthly service charge of $20. Plan A has a cost of $.38 a minute for daytime calls and $.18 a minute for evening calls. Plan B has a charge of $.48 a minute for daytime calls and $.14 a minute for evening calls. Plan C has a flat rate of $75 with 250 minutes of calls allowed per month and a charge of $.36 per minute beyond that, day or evening.

If the agent will use the service for daytime calls, over what range of call minutes will each plan be optimal?

Suppose that the agent expects both daytime and evening calls. At what point (i.e., percentage of total call minutes used for daytime calls) would she be indifferent between plans A and B?

Solution

The plan C costs$ 75 for 250 call minutes i.e. $ 0.30 per call minute for 250 call minutes.Beyond this, the call charges are $ 0.36 per additional call minute. If the real estate agent makes all the calls/ major bulk of the calls during the day, then plan C is optimal as $ 0.30 and $ 0.36 are both less than $ 0.38 or $ 0.48.

However, if a major bulk of the calls are in the evening, then the plan A or plan B may be optimal depending upon the total number of calls and the ratio of calls in the evening to the calls in the daytime.

If x and y are the number of calls during the daytime and the evening respectively, then (excluding the fixed rental of $ 20), the call charges as per the three plans are as under:

Plan A 0.38x + 0.18y

Plan B 0.48x + 0.14y

Plan C 75 [ if (x + y) 250 ] or, 75 + ( x + y - 250)* 0.36 = 0.36(x + y) -15 if (x + y) > 250

If the total call charges as per the plan A and B are to be equal, then we have  0.38x + 0.18y =  0.48x + 0.14y or, 0.10x = 0.04y or x = 2/5y. Thus if the percentage of calls during the daytime is (2/7)* 100 % = 28.57 % (approximately) of the total calls, the real estate agent would be indifferent to the Plan A and Plan B .

If the percentage of calls during the daytime to the total calls made by the real estate agent exceeds 28.57 %, then Plan A would be cheaper than Plan B. On the other hand, if percentage of calls during the daytime to the total calls made by the real estate agent is less than 28.57 %, then Plan B would be cheaper than Plan A.

In order to compare the Plan C with either of Plans A and B, we will need the ratio of calls in the dattime to the total number of calls as also the total number of calls made by the real estate agent.