Which is the better buy: 33% off the price of an item or 33% more quantity of that item for the same price?
At first students guess (neither deal tends to draw significantly more votes than the other initially) and then I ask them to try running the deals with some numbers. It takes some students awhile to figure out that they need to choose both a price and a quantity of an item to compare these deals. Students work in groups and the numbers that they choose are interesting. Some groups choose an item that they have bought for which they know the price and quantity. Others make up numbers that don't have a connection to real life. Other groups choose the numbers with simplicity in mind (like $1 for 1 gallon).
The next hurdle for some students is figuring out how to compare the deals. Comparing unit prices can be a good way of finding which is the best deal, but not all students come into the class remembering how to do that. Comparing unit prices (price per ounce, for example) is the most common strategy that students employ, but when I taught this lesson last week, one group used unit quantities (ounces per dollar, for example). They puzzled for awhile about whether the higher unit quantity or lower unit quantity was the better deal.
After we come to a consensus that 33% off the price is the better deal, I ask them if it is always the better deal. Since each group of students usually chooses different prices and different quantities and they all calculate that 33% off the price is the better deal, some students conclude that it is always the best deal. A short discussion about whether many examples prove that the idea is correct is followed by me challenging students to think of how they could prove that the discounted price is always better than more quantity for the same price. A few students will realize that they need to create a variable for price and a variable for quantity and do the calculations with the variables rather than their particular price and particular quantity.
This lesson prompted me to determine whether it is always better to get the same quantity at a discounted price for any discount amount - not just 33%. Here is the way I figure that out:
Question:
Is it better to get 1⁄x off the price of an item or 1⁄x more quantity of that item for the same price? (where x > 1)
Solution:
Let p = price and q = quantity.
First consider getting 1⁄x off the price of an item for the same quantity:
New price = p - (1⁄x) p = p - (p⁄x) = (px - p)⁄x = [p(x - 1)]⁄x
Unit Price (price per quantity) = ([p(x - 1)]⁄x) ÷ q = (p⁄q )((x - 1)⁄x)
Is it better to get 1⁄x off the price of an item or 1⁄x more quantity of that item for the same price? (where x > 1)
Solution:
Let p = price and q = quantity.
First consider getting 1⁄x off the price of an item for the same quantity:
New price = p - (1⁄x) p = p - (p⁄x) = (px - p)⁄x = [p(x - 1)]⁄x
Unit Price (price per quantity) = ([p(x - 1)]⁄x) ÷ q = (p⁄q )((x - 1)⁄x)
Next consider getting 1⁄x more quantity of an item for the same price:
New quantity = q + (1⁄x) q = q + (q⁄x) = (qx + q)⁄x = [q(x + 1)]⁄x
Unit Price (price per quantity) = p ÷ ([q(x + 1)]⁄x) = (p⁄q )(x⁄(x+1))
How should these unit prices be compared? The lower unit price is the better deal.
So we are comparing (p⁄q )((x - 1)⁄x) to (p⁄q )(x⁄(x+1))
Since each expression contains (p⁄q ), we can simplify and just compare ((x - 1)⁄x) to (x⁄(x+1)).
Getting a common denominator results in comparing ([x2 - 1]⁄[x(x+1)]) to (x2⁄[x(x+1)]).
Since both fractions have the same denominator, the smaller fraction will be the one with the smaller numerator.
So now we can compare (x2 - 1) to x2.
(x2 - 1) is less than x2 for all x.
Therefore, getting 1⁄x off the price of an item is always a better deal than getting 1⁄x more quantity of an item for the same price!
