Supply and Demand

Question

Natali has decides she wants to run a daycare. Her monthly expenses include $800 for rent of the building and $400 for insurance. She will provide each child with breakfast (~$2) and lunch (~$3.50) each day as well. She plans to charge $25 dollars a day for each child. Assume there are 20 days in a month (Monday through Friday, 4 weeks). Find the monthly cost as well as monthly revenue. How many kids would Natali need to watch each month to break even? How many kids would Natali need to watch to each month to earn at least $1500 a month?

Answer 1

First, note that both daily and monthly values are given, but the problem asks for results in monthly terms.

 

Costs:

   $800    rent/month

   $400    insurance/month

   $1100x   meals/month   [where x is the number of kids;

                    this is 20*($2+$3.50)]

 

TOTAL Monthly Costs:   $1100x + $1200

 

Revenue:

   $500x    charges/month  [this is 20*$25 for each student]

 

TOTAL Monthly Revenue:     $500x

 

To break even, monthly costs = monthly revenue.  Find x:

    $110x + $1200  =  $500x

         $1200 = $390x                [subtract $110x from both sides]

        3.08 = x

            Natali will break even with 4 students    [always round up to a whole student]

 

Check:

  For 3 students, 

             Costs = $110*3 + $1200 = $1530

             Revenue = $500*3 = $1500      [not quite enough]

  For 4 students,

             Costs = $110*4 + $1200 = $1640

             Revenue = $500*4 = $2000      [$360 profit]

 

To earn at least $1500 a month,

             Revenue - Costs ≥ $1500

             $500x  - ( $110x  + $1200 )   ≥  $1500

             $390x ≥ $2700

                 x ≥ 6.92

     Natalie will earn more than $1500 per month with at least 7 students.

 

You may want to calculate the actual values for 6 and for 7 students to check this.

Answer 2

Let c = # children attending

 

Monthly Expenses:  800 + 400 + 20(2c + 3.5c)

                            = 1200 + 110c

 

Monthly Revenue:  20(25c) = 500c

 

To break even set the expenses equal to revenue and solve for c

500c = 1200 + 110c

390c = 1200

c = 3.077

With 3 children she will be just under the break even point.

At 4 children she will have a small profit.

 

check:

3 children:

revenue = 500(3) = $1500

expenses = 1200 + 110(3) = 1200 + 330 = $1530

Loss of $30 per month

 

4 children:

revenue = 500(4) = $2000

expenses = 1200+110(4) = 1640

profit = $360 per month

*****************************

To earn at least $1500 per month:

Revenue - Expenses ≥ 1500

 

500c - (1200+110c) ≥ 1500

500c - 1200 - 110c ≥ 1500

390c - 1200 ≥ 1500

390c ≥  2700

c ≥ 2700/390

c ≥ 6.9

To earn at least $1500 per month she will

need to provide daycare to at least 7 children