Question

What conversion factor could be used to determine how many pounds of apples could be bought for \(\$ 25.00 ?\)

Short answer

To determine how many pounds of apples could be bought for $25.00, use the conversion factor \( \frac{1 \, pound}{\$2.00} \) (assuming the price per pound of apples is $2.00). Divide $25.00 by the price per pound: \( Number \, of \, pounds \, of \, apples = \frac{\$25.00}{\$2.00} = 12.5 \, pounds \). So, 12.5 pounds of apples can be bought with $25.00.

Step-by-step solution

Determine the price per pound of apples

In this step, we need to find out the price per pound of apples from a given source. This could be provided by the teacher or alternatively looked up at the grocery store. Let's assume that the price per pound of apples is $2.00.

Calculate the conversion factor

Now that we know the price per pound of apples, we can calculate the conversion factor. The conversion factor is the amount of money we have (\(25.00) divided by the price per pound of apples (\)2.00 per pound). To find this, we will use the formula: \[ Conversion \, factor = \frac{Amount \, of \, money}{Price \, per \, pound} \]

Calculate the number of pounds of apples

Now, we will use the conversion factor obtained in Step 2 to calculate the number of pounds of apples that can be purchased for $25.00. The formula for this is: \[ Number \, of \, pounds \, of \, apples = Amount \, of \, money \, × \, Conversion \, factor \]

Plug in the values and calculate the result

Plugging in the values obtained in Steps 1 and 2 into the formula we get: \[ Number \, of \, pounds \, of \, apples = \$25.00 \times \frac{1 \, pound}{\$2.00} \] \[ Number \, of \, pounds \, of \, apples = \frac{\$25.00}{\$2.00} \] \[ Number \, of \, pounds \, of \, apples = 12.5 \, pounds \] So, a person can buy 12.5 pounds of apples with $25.00.

Key concepts

Determining Price Per Pound

Understanding the price per pound of an item is a fundamental skill in everyday life, especially when trying to make economical decisions while shopping. This unit price gives us insight into the best deals and helps manage our budget efficiently.

To determine the price per pound, you divide the total cost by the number of pounds. For example, if a bag of apples costs \$10 and weighs 5 pounds, the price per pound would be \$2 (\( \frac{\$10}{5 \text{ pounds}} = \text{\$2 per pound} \)). Often, prices per pound are readily available at grocery stores or markets, but it's good practice to calculate it yourself to ensure you're getting the best value for your money.

When the price per pound is not explicitly stated, you may need to perform this calculation by considering the total weight and cost of the product. Always look for the per-unit costs before making a decision; doing so can lead to significant savings over time, a skillful move for budget-conscious individuals.

Calculating Quantity with Conversion Factor

A conversion factor is a useful tool for translating one unit into another, and this concept is not only pivotal in everyday transactions like determining how many pounds of apples can be bought with a certain amount of money, but also in various fields like engineering and science.

To calculate quantity using a conversion factor, start by identifying the two units you're converting between. In our exercise example, we're converting dollars to pounds of apples. The conversion factor is derived from the ratio of how many units of one equals the other (\( \frac{\text{Amount of money}}{\text{Price per pound}} \) ).

Once you have the conversion factor, multiply the amount you have by the conversion factor to get the quantity you want. For instance, with \$25 and a conversion factor of \$2 per pound (\( \frac{1 \text{ pound}}{\text{\$2}} \)), you can purchase \$25 times \$1/2 per pound (\( \text{\$25} \times \frac{1 \text{ pound}}{\text{\$2}} = 12.5 \text{ pounds} \) ), giving us 12.5 pounds of apples. It's a straightforward approach that can be applied to many scenarios, such as converting lengths, volumes, and even currencies.

Problem-Solving in Chemistry

In the field of chemistry, problem-solving often involves conversion factors and dimensional analysis, which are techniques similar to those used in our apple-buying example. Chemists frequently need to convert between units such as grams to moles, liters to milliliters, or atoms to molecules.

A conversion factor in chemistry is usually based on a relationship defined by equalities, such as Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms} = 1 \text{ mole} \)) or the molar mass of a substance (e.g., the molar mass of water, H2O, is approximately 18 g/mol).

To solve problems, it is crucial to set up a conversion that cancels out unwanted units and leaves you with the desired unit. For example, converting grams to moles would involve dividing the mass of the substance by the molar mass of the substance. This allows chemists to calculate quantities needed for reactions, to determine the concentration of solutions, and to perform many other critical calculations.

Whether you're a student managing your finances or a scientist conducting experiments, mastering the concept of conversion factors is essential in navigating through a variety of quantitative problems with ease and precision.