Best Buys

The Unit Price Method

• Step 1: Identify the total cost and the total quantity for each item being compared.
• Step 2: Divide the cost by the quantity to find the price per single unit: $\text{Unit Price} = \frac{\text{Total Cost}}{\text{Total Quantity}}$
• Step 3: Compare the resulting unit prices; the lowest value indicates the best buy.

The Scaling Method

• Step 1: Find a common quantity that both items can easily scale to (e.g., if comparing 200g and 500g, scale both to 1000g).
• Step 2: Multiply the price of each item by the factor needed to reach that common quantity.
• Step 3: Compare the total costs for the same quantity to identify the cheaper option.

Quantity per Unit Currency

• Alternative approach: Divide the quantity by the cost to find how much product you get for 1 dollar: $\text{Quantity per Dollar} = \frac{\text{Total Quantity}}{\text{Total Cost}}$
• Decision Criteria: In this specific method, the highest value is the best buy, as it represents getting more product for the same amount of money.

• Weight vs. Volume: It is crucial to ensure you are comparing like with like. You cannot directly compare a price per kilogram with a price per liter without knowing the density of the substance.

• Gross Price vs. Unit Price: A lower gross price (total price) does not necessarily mean a better deal. A small jar of honey for 3 dollars might be more expensive per gram than a large jar for 10 dollars.

• Unit Consistency: Always check that the units are identical before performing calculations. If one item is in grams and the other in kilograms, convert both to the same unit (usually the smaller one) to avoid decimal errors.

• Rounding Precision: When calculating unit prices, carry the division to at least three or four decimal places. Often, the difference between two 'Best Buys' is less than a cent per unit, and premature rounding can lead to the wrong conclusion.

• Sanity Check: After finding the unit price, multiply it back by the original quantity to see if you get the original price. This verifies that you didn't accidentally divide the quantity by the cost.

• Contextual Awareness: In exam questions, look for 'hidden' costs or multi-buy offers (e.g., 'Buy one get one half price'). These must be factored into the total cost and total quantity before calculating the unit price.

• The 'Bulk' Assumption: A common misconception is that the larger container is always the best buy. Manufacturers sometimes price mid-sized items more competitively, or bulk items may have a 'convenience premium' attached.

• Division Order: Students frequently divide the quantity by the price ($Q/P$) instead of price by quantity ($P/Q$). While $Q/P$ is a valid comparison (Quantity per Dollar), the student must remember that in that case, the larger number is better, which is counter-intuitive to the 'lowest price' goal.

• Ignoring Waste: In real-world applications, a 'Best Buy' mathematically might not be a best buy practically if the quantity is so large that the product expires before it can be used.