Applied Math: More Useful Math Concepts

1. Identifying the Best Deal
It can be difficult to compare the final price of something at competing retailers when each states a different price and also has a sale or promotion that changes the price.

Example: Consider a situation where you want to buy a specific new tablet. Buy-Tech-Here has the tablet listed for $399, but they currently have a 25%-off sale on electronics. Meanwhile, at Gadgets-B-We, the tablet is listed for $359 with a 10%-off sale on items storewide and you have a Gadgets-B-We coupon for $25 off any purchase over $100. Which is the better deal?

To determine the final price of each, apply the discounts.

25% of $399 is , so, the final price at Buy-Tech-Here would be .

For Gadgets-B-We, 10% of $359 is .

In addition to this discount, you have $25 coupon, so the final price at Gadgets-B-We is .

Since $298.10 is less than $299.25, that means the best deal is at Gadgets-B-We.

Determining Relative Economic Value Graphically
When comparing the value of various alternatives, it can be helpful to plot a graph of each to more easily see how they compare. The options could result in a variety of functions that, once graphed, clearly show which offers the best value.

For instance, Company T is comparing deals offered from four different manufacturers on the cost of producing mechanical pencils.

The summary of the offers is listed in the table below:
 

 
It is not clear from the table which of the manufacturers would be best for a given order size without calculating the total cost of a certain order size for each.

However, looking at the graph depicting each of the offers over a range of order sizes, it is much more obvious which manufacturer offers the best deal for any given order size:

For instance, if the company wants to order 50,000 pencils, then Manufacturer A is the cheapest. On the other hand, if they want 700,000 pencils, then Manufacturer D has the best price. It is worth noting that you could order 1 million pencils for about the same price as 500,000, but that would mean that you need to store all of those pencils, which may not be feasible.

2. Percentage Change
Percentage change is calculated by dividing the absolute value of the difference between two values by one of those values as a reference and then multiplying the result by 100.

In equation form:

This is a method for gauging the relative size difference between the values in terms of the magnitude of those values. For example, it may seem obvious that the difference between $8 and $10 is more significant than between $998 and $1000, but in order to quantify the comparison, you can use the percentage change of each:

 
Even though the difference between each pair is the same, the percentage changes are very different! So, saving $2 dollars on a $10 purchase is much more significant than saving $2 on a $1000 purchase. Again, that may seem obvious, but it is useful to be able to compare them and verify your intuition with clear data.

3. Unit Cost
Often when shopping, it is best to compare the unit costs of items to be certain which has the best value.

The unit cost is the total price of an item divided by the units contained within that item.

Consider going to a store to buy bar soap. You see three brands that you like equally—Lux, Vulture, and Crank. The Young Herb soap is available in single 255-gram bars at $1.29 each, the Vulture soap comes in a 3-pack of 210-gram bars at $3.19 per pack, and the Crank soap comes in a 10-pack of 300-gram bars at $14.99 per pack.

To determine the best value, we can compare the unit cost, in this case the cost per gram.

 
Now we can see that the Vulture soap is the best value.