How to manipulate taxes/tips calculation using percentage (Pre-algebra)

If there is a question that asked you to solve for the total amount paid at a dinner, there are at least 2 methods to solve the same problem.  If the bill for the dinner is $110, I am going to tip the waiter/waitress based on the entire amount of the bill with 15%, the student can do it 2 different ways:

1.  Entire amount paid = amount of tip (taking 15% of the amount of the dinner) + amount of dinner

Entire amount paid = 15% of $110 + $110 = $16.50 + $110 = $126.50
 
or

2.  Entire amount paid = (100%  + 15%) x amount of dinner

Entire amount paid = (100% + 15%) x $110 = 115% x $110 = $126.50

NOTE:  In real life some people tips the waiter/waitress based on the entire bill, while others tip based on the pretax amount.  Depending on the question and how it is worded, you have to figure out whether you are tipping based on the entire amount of the bill or the pretax amount.

So why is it that we can get to the answer using 2 different ways?  The explanation is easy.  Students typically solve the question using method #1 because that is the "natural" way that they were taught.  Students know how to find a percentage of a bill, in this case figuring out the amount of tip.  Then once the student figures out the amount of tip, he/she will just add the amount of the tip to the actual bill to come up with the total amount spent.

Another way that the students can be taught is to do everything in 1 step.   We know that the bill is $110, and that constitutes 100% of the amount.  Since we are adding a 15% tip to the amount of the bill, now the student should be looking for the 100% + 15% = 115% of the bill.  When the student evaluate 115% of the bill, they will end up with the exact same amount spent in entirety.

Another variation of this type of problem is when we are trying to buy something and there is a discount associated with it.  Let us give another hypothetical example.  In this example, we are buying a baseball bat for $30.  Your local sporting store is giving all the athletes a 7% discount. 

Again, there are 2 ways to come up with the final cost of the baseball bat (before taxes are considered).

1.  Final cost of the baseball bat = Cost of baseball bat - 7% discount of baseball bat

Final cost of the baseball bat = $30 - 7% x $30 = $30 - $2.10 = $27.90

or

2.  Final cost of the baseball bat = (100% - 7%) x cost of the baseball bat

Final cost of the baseball bat = (100% - 7%) x $30 = 93% x $30 = $27.90

As you can see there are many ways to get to the final answer for each of the problem.  Just like most math problems, there is no one solution that can fit all the problems.  There are many ways to arrive at the same answer for any given problem.  As long as the student are comfortable and understand each of the method, he/she should be encourage to use the method that is most comfortable to them.

If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.

Michael Huang
Mathnasium of Glen Rock/Ridgewood
236 Rock Road
Glen Rock, NJ 07452
Tel:  201-444-8020
glenrock@mathnasium.com