The curious case of rounding

Scenario 1

Let’s assume, we have a farmer named Chirag. Chirag booked a spraying service for one acre. Our spraying rate per acre is 10000 cents and we are offering a discount of 10.66 percent.

So, he gets the following bill.

Spraying CostDiscountTotalFinal after Rounding off
10000 cents10.66%8934 cents89 $

He pays the bill before spraying, it’s prepaid.

Now,  when spraying happens on his farm, it’s actually 2 acres and not 1 acre that he told before. So, we give him another bill for his extra acre.

Spraying CostDiscountTotalFinal after Rounding off
10000 cents10.66%8934 cents89 $

So, How much does Chirag pay total for 2 acres?

89 + 89 = 178 $.

Scenario 2

But what if, Chirag had booked for 2 acres in the first go itself?

What would the bill look like?

Spraying CostDiscountTotalFinal after Rounding off
20000 cents10.66%17868 cents179 $

What?

179 $.

That’s one dollar more than the previous case.

Where did the magic happen?

It’s in rounding.

Photo by no one cares on Unsplash

Assume we have new denomination called unit where 10^4 units = 1 $

Now, if we use 10^4 as storage instead of cents, the problem will still persist but it will have negligible practical effect. Let’s see how. We will replay both scenarios that we discussed above again.

So now, 10^4 units = 1 $

Scenario 1:

When he booked 1 acre farm:

Spraying CostDiscountTotalFinal after Rounding off
1000000 units10.66%893400 units89 $

When he later booked for one more acre:

Spraying CostDiscountTotalFinal after Rounding off
1000000 units10.66%893400 units89 $

Total paid till now: 178 $

Scenario 2:

If he would have booked 2 acres:

Spraying CostDiscountTotalFinal after Rounding off
2000000 units10.66%1786800 units178 $

Here, in both cases, we have paid 178 $.

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