Question

Apples cost \(\$ 0.79\) per pound. What conversion factor is appropriate to express the cost of 5.3 lb of apples?

Short answer

The appropriate conversion factor is \(0.79 \frac{\$}{lb}\), and the cost of 5.3 lb of apples is $4.187.

Step-by-step solution

Identify the given information

We are given that apples cost $0.79 per pound and we want to find the cost for 5.3 pounds of apples.

Write down the conversion factor

The conversion factor is the cost of apples per pound, which in this case is $0.79 per pound. We can write it as \(0.79 \frac{\$}{lb}\).

Multiply using the conversion factor

Now we need to multiply 5.3 lb by the conversion factor to find the total cost. \[5.3\ lb \times 0.79\ \frac{\$}{lb}\]

Cancel the units and simplify

We can cancel the "lb" units in the multiplication and then multiply 5.3 by 0.79 to get the total cost: \[5.3 \times 0.79 = 4.187\] So, the cost of 5.3 pounds of apples is $4.187.

Key concepts

Unit Conversion

Unit conversion is the process of converting a measured quantity from one unit of measurement to another without changing the actual value of the quantity. Understanding unit conversion is crucial in chemistry and everyday life, allowing us to compare quantities, adapt recipes, or understand distances in different units.

For instance, if we know that apples cost \(0.79 per pound, and we wish to purchase 5.3 pounds, we need to convert the quantity of pounds to the equivalent monetary value. This process involves multiplying the given quantity by the appropriate conversion factor to attain the desired unit outcome. A conversion factor is essentially a ratio equal to one, but written in a way that allows a unit to be converted to another unit. In our example, the conversion factor is \)0.79 per pound, which serves as a bridge from the unit of pounds and transforms it into dollars.

Dimensional Analysis

Dimensional analysis, sometimes known as the factor-label method or unit analysis, is a powerful tool in problem-solving that involves units and measurements. This method uses conversion factors to move from one unit to another, ensuring that the end result has the correct unit.

In dimensional analysis, units are treated as algebraic quantities that can cancel each other out when they appear in both numerator and denominator. To perform dimensional analysis, we first need to identify the relevant conversion factor—in our example, $0.79 per pound of apples. With this, we set up an equation where the initial unit is multiplied by the conversion factor, ensuring that the units we want to remove (pounds) cancel out, leaving us with the desired final unit (dollars).

In this context, the equation \[5.3\ lb \times 0.79\ \frac{\$}{lb}\] represents the practical use of dimensional analysis, allowing us to accurately calculate the cost of 5.3 pounds of apples.

Stoichiometry

Stoichiometry is a section of chemistry that involves calculating the quantities of reactants and products in chemical reactions. The principles of stoichiometry are based on the law of conservation of mass, which states that in a chemical reaction, matter is neither created nor destroyed.

To solve stoichiometry problems, we need to establish relationships between the different entities involved in a reaction. This often requires unit conversions and dimensional analysis to ensure that all the quantities are expressed in the same units and correctly represented in terms of moles, volumes, or masses.

Although our example of determining the cost of apples does not involve a chemical reaction, the concept of stoichiometry is similar in that it deals with proportional relationships. Here, we consider the proportion of dollars to pounds to find the total cost. In more complex stoichiometry problems within chemistry, we use coefficients from balanced chemical equations and the mole concept to relate the amounts of various substances that participate in or are formed from a reaction.