A salesman has to choose between two schemes of remuneration.The first scheme has a fixed salary of rs 3700 and a commission of 2% on sales above rs 50000. The second scheme has no salary but offer commission only. The commission starting from 3% of sales for the first rs 50000 or part there of increase at the rate of 1 percentage point for every increase of rs 50000 or part there of sales upto a maximum of 20% of sales. what is the minimum value of sales above which he can prefer the second scheme?
it's unfortunate that the 50K break points are included in the step function. Note that
[x/50k] = 0 for x<50K, but 1 for x=50K
So, assuming whole dollars, we have adjust our floor function to be
[(x-1)/50000]+1
The first plan pays
3700 for 0 <= x <= 50000
3700 + .02(x-50000) for 50000 < x
The second plan pays
(.02 + 0.01([(x-1)/50000] + 1))*x
So, when are they equal?
Looks like the 100K point marks the break:
http://www.wolframalpha.com/input/?i=plot+y%3D3700+%2B+.02(x-50000),+y%3D(.02+%2B+0.01(floor%5B(x-1)%2F50000%5D+%2B+1))*x+for+50000+%3C%3D+x+%3C%3D+200000
To find the minimum value of sales above which the salesman would prefer the second scheme, we need to compare the total earnings for both schemes.
Let's start with Scheme 1:
Total earnings for Scheme 1 = Fixed Salary + Commission
Given:
Fixed Salary = Rs 3700
Commission = 2% on sales above Rs 50000
Now, let's consider Scheme 2:
Total earnings for Scheme 2 = Commission only
Commission for Scheme 2 is calculated as follows:
- Commission starts at 3% for the first Rs 50000 or part thereof.
- Commission increases by 1 percentage point for every increase of Rs 50000 or part thereof in sales.
- The maximum commission is limited to 20% of sales.
To find the minimum value of sales above which the second scheme is preferred, we need to compare the total earnings of both schemes. Let's assume the minimum value of sales required is "X."
For Scheme 1, the total earnings can be calculated as:
Total earnings for Scheme 1 = Fixed Salary + Commission
= Rs 3700 + (2% of (Sales - Rs 50000))
For Scheme 2, the total earnings can be calculated as:
Total earnings for Scheme 2 = Commission
= Commission % of Sales
Since we are trying to find the point at which the salesman would prefer Scheme 2, we need to set the total earnings for Scheme 2 greater than the total earnings for Scheme 1:
Total earnings for Scheme 2 > Total earnings for Scheme 1
Commission % of Sales > (Rs 3700 + (2% of (Sales - Rs 50000)))
Now, let's solve for the minimum sales value above which Scheme 2 is preferred:
Commission % of Sales > (Rs 3700 + (2% of (Sales - Rs 50000)))
3% of Sales > Rs 3700 + (2% of (Sales - Rs 50000))
0.03 * Sales > 3700 + 0.02 * (Sales - 50000)
Simplify the equation:
0.03 * Sales > 3700 + 0.02 * Sales - 1000
0.01 * Sales > 2700
Sales > 2700 / 0.01
Sales > Rs 270000
Therefore, the minimum value of sales above which the salesman would prefer the second scheme is Rs 270000.