Question

A man goes to a traditional market to buy a steak for dinner. He finds a 12 -oz steak \((1 \mathrm{lbm}=16 \mathrm{oz})\) for \(\$ 5.50\) He then goes to the adjacent international market and finds a \(300-\mathrm{g}\) steak of identical quality for \(\$ 5.20 .\) Which steak is the better buy?

Short answer

Answer: The traditional market steak is the better buy, as it has a lower price per gram of $0.0162 compared to the international market steak's price per gram of $0.0173.

Step-by-step solution

Convert steak weights into the same unit

First, we will convert both steak weights into grams. Since the first steak's weight is given in ounces, we will convert it to grams using the conversion factor \(1\,\text{oz} \approx 28.35\,\text{g}\). 12-oz steak weight in grams: \(12\,\text{oz} \times 28.35\,\text{g/oz} = 339.6\,\text{g}\) Now we have both steak weights in grams: - Traditional market steak: \(339.6\,\text{g}\) - International market steak: \(300\,\text{g}\)

Calculate the price per gram for both steaks

Next, we will calculate the price per gram for both steaks by dividing the given steak prices by their respective weights in grams. Traditional market steak: Price per gram: \(\frac{\$5.50}{339.6\,\text{g}} \approx \$0.0162\,\text{g}\) International market steak: Price per gram: \(\frac{\$5.20}{300\,\text{g}} \approx \$0.0173\,\text{g}\)

Compare the price per gram and determine the better buy

Now, we will compare the price per gram of both steaks to determine which one is the better buy (the lower price per gram is the better buy). Traditional market steak: \(\$0.0162\,\text{g}\) International market steak: \(\$0.0173\,\text{g}\) Since the price per gram of the traditional market steak is lower than that of the international market steak, the traditional market steak is the better buy.

Key concepts

Dimensional Analysis

Dimensional analysis is a mathematical technique that is widely used to convert one unit of measurement to another. It's a process that applies conversion factors to transition between units, such as turning inches into centimeters or, in the context of our exercise, ounces into grams. This method not only aids in avoiding errors when dealing with different measurement systems but also ensures consistency and accuracy in calculations.

For instance, when converting the weight of a steak from ounces to grams, dimensional analysis involves multiplying the given amount by the appropriate conversion factor (\(1\text{ oz} \rightarrow 28.35\text{ g}\)). The process follows a very logical pattern; by setting up the conversion factor as a fraction and multiplying it by the original quantity, any unit can be converted to another, while maintaining the same value in different terms.

Conversion Factors

Conversion factors are the backbone of unit conversion. They act like mathematical 'bridges' that allow you to convert a quantity from one unit to another. A conversion factor is a ratio or fraction that represents the relationship between two different units—for example, the knowledge that 1 pound is equal to 16 ounces or that 1 ounce is approximately 28.35 grams.

When performing a conversion, it's vital to arrange the conversion factor so that the unit you want to get rid of (in our exercise, ounces) is in the denominator, and the unit you're converting to (grams) is in the numerator. This cancels out the original unit and leaves you with the desired unit. It's important to ensure the conversion factor is accurate to maintain the precision of results, especially in scenarios like cooking or scientific experiments, where measures must be precise.

Price per Unit Mass

Price per unit mass is an elemental concept in comparing costs between items sold in different quantities or units, such as comparing the cost of steaks sold by weight in grams or ounces. It entails calculating the cost for a standard unit of mass—typically grams or kilograms—allowing for a straightforward comparison, independent of differences in size or packaging.

In the given scenario, we calculated the price per gram for both steaks by dividing the total price by the steak's weight in grams. This calculation is beneficial when trying to find the best value for money, as it standardizes the prices to a per-gram cost, making it clear which option is cheaper when comparing products that may come in varying weights or sizes.