discount calculation problem

tuncob

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Oct 17, 2021
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This is a problem regarding calculating the discount amount and applying it to the register. But there are different regulations ( or ideas ) on how to calculate the discount amount since there is sales tax involved.

Customers come to a restaurant. After they have their meal, they ask for the check and the waiter brings the check to the table as you see below.

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $0.00
SALES TAX (%10) : $15.00
---------------------------------
AMOUNT DUE : $215.00


When one of the customers was about to pay, the owner saw them and realized that they were his friends. The owner wanted to give them a special discount and told the waiter to charge only $100 dollars.


How do we arrange the check, so the AMOUNT DUE would be $100? Here is the trick:

CASE 1:
Some people say that the discount should be applied to the taxable subtotal first, then whatever is left ( if any ) should be applied to the nontaxable subtotal. In this case the new check should look as you see below;


* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $104.55*
SALES TAX (%10) : $4.55 ( calculated from $45.05 )
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 1 ), I must apply a $104.55 discount to the taxable amount.

CASE 2:
Some people say that the discount should be applied to the nontaxable subtotal first, then whatever is left ( if any ) should be applied to the taxable subtotal. In this case the new check should look as you see below;

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $109.09*
SALES TAX (%10) : $9.09 (calculate from $90.91)
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 2 ), I must apply a $50 discount to the nontaxable amount and $59.09 discount to the taxable subtotal.

CASE 3:
Some people say that the discount should be distributed to both taxable and nontaxable amounts equally based on their weight on the check. In this case the new check should look as you see below;

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $106.98*
SALES TAX (%10) : $6.98 (calculate from $69.77)
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 3 ), I must apply a total of $106.98 discount. ( equally distributed between taxable subtotal $80.23 and nontaxable subtotal $26.74)


Here is the question :

I could not figure out a way to go backwards towards calculating the discount amount. How do I find the discount amount which is different for the three cases above?
 
This is a problem regarding calculating the discount amount and applying it to the register. But there are different regulations ( or ideas ) on how to calculate the discount amount since there is sales tax involved.

Customers come to a restaurant. After they have their meal, they ask for the check and the waiter brings the check to the table as you see below.

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $0.00
SALES TAX (%10) : $15.00
---------------------------------
AMOUNT DUE : $215.00


When one of the customers was about to pay, the owner saw them and realized that they were his friends. The owner wanted to give them a special discount and told the waiter to charge only $100 dollars.


How do we arrange the check, so the AMOUNT DUE would be $100? Here is the trick:

CASE 1:
Some people say that the discount should be applied to the taxable subtotal first, then whatever is left ( if any ) should be applied to the nontaxable subtotal. In this case the new check should look as you see below;


* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $104.55*
SALES TAX (%10) : $4.55 ( calculated from $45.05 )
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 1 ), I must apply a $104.55 discount to the taxable amount.

CASE 2:
Some people say that the discount should be applied to the nontaxable subtotal first, then whatever is left ( if any ) should be applied to the taxable subtotal. In this case the new check should look as you see below;

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $109.09*
SALES TAX (%10) : $9.09 (calculate from $90.91)
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 2 ), I must apply a $50 discount to the nontaxable amount and $59.09 discount to the taxable subtotal.

CASE 3:
Some people say that the discount should be distributed to both taxable and nontaxable amounts equally based on their weight on the check. In this case the new check should look as you see below;

* * * * * R E C E I P T * * * * *
---------------------------------
FOOD ITEM A $50.00T
FOOD ITEM B $50.00T
FOOD ITEM C $50.00T
FOOD ITEM D $50.00
---------------------------------
TAXABLE SUBTOTAL : $150.00
NON-TAXABLE SUBTOTAL : $50.00
DISCOUNT : $106.98*
SALES TAX (%10) : $6.98 (calculate from $69.77)
---------------------------------
AMOUNT DUE : $100.00


*I found by trial and error that in order for me to have a $100 check ( by case 3 ), I must apply a total of $106.98 discount. ( equally distributed between taxable subtotal $80.23 and nontaxable subtotal $26.74)


Here is the question :

I could not figure out a way to go backwards towards calculating the discount amount. How do I find the discount amount which is different for the three cases above?
Okay - I will show you - for the first case.

Suppose AFTER discount - the total taxable part of the bill is 'x'. Then:

x + 0.10*x + 50 = 100

1.1 * x = 50

x = 50/1.1 =45.455 = 45.46

check:

taxable bill = 45.46

tax = 45.46 * 0.1 = 4.546

Total bill = 45.46 + 4.55 + 50 = 100.01 [you can take care of that 1 ¢ by reducing taxable or non-taxable amount]

No Trial&Error needed. Similarly, model other cases. Do not forget to check your model.

Please tell us if you did not understand any of the 'steps' above.
 
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