Problem 66 - Chemistry: An Atoms-Focused Approach

Question

A U.S. dime weighs \(2.5 \mathrm{g} .\) What is the U.S. dollar value of exactly \(1 \mathrm{kg}\) of dimes?

Step-by-Step Solution

Answer

Answer: $40

Step 1: Convert 1 kg to grams

First, we need to convert 1 kg to grams. There are 1000 grams in 1 kg, so our conversion is: 1 kg = 1000 g

Step 2: Calculate the number of dimes in 1 kg

Next, we will divide the total number of grams (1000 g) by the weight of a single dime (2.5 g) to find the total number of dimes in 1 kg. Total number of dimes = \(\frac{1000 \mathrm{g}}{2.5 \mathrm{g}}\) = 400 dimes

Step 3: Calculate the U.S. dollar value of 400 dimes

Now that we know there are 400 dimes in 1 kg, we can multiply the amount by the value of a single dime to determine the total value. A U.S. dime is worth $0.10: Total value = 400 dimes x \(0.10 = \)40 So, the U.S. dollar value of exactly 1 kg of dimes is $40.

Key Concepts

Mass and Weight Measurement

When it comes to mass and weight measurement, understanding the conversion between different units is crucial. Mass typically measures the amount of matter in an object, whereas weight is the force exerted by gravity on that mass. In most scientific contexts, the metric system is used, where mass is commonly measured in grams (g) or kilograms (kg).

Understanding how to convert between these units is simple. For example, to convert kilograms to grams, one needs to know that:

• 1 kilogram (kg) equals 1000 grams (g).

In our example, converting 1 kg to grams involves multiplying by 1000, thus giving us 1000 grams. This basic knowledge helps in understanding how many items of a certain mass can fit into a larger mass value. Being proficient with these conversions makes solving mass-related problems straightforward and less error-prone in chemistry and everyday contexts.

Currency Value Calculation

In many problems, including those in chemistry, calculating currency value is necessary, especially when items have a monetary aspect. In our example, knowing the value of a U.S. dime is essential to finding the total value of a specific mass of dimes. A U.S. dime is valued at $0.10.

To solve this type of problem, follow these steps:

• First, calculate how many individual units (dimes here) are within the total mass you are working with.
• Then, multiply the number of units by the value of each unit.

In the given example, once we've determined there are 400 dimes in 1 kg, we can multiply 400 by $0.10 to find the total dollar value, which equals $40. This simple multiplication helps you understand the concept of scaling a single unit's value up to accommodate larger quantities.

Problem-Solving in Chemistry

Problem-solving in chemistry often involves applying mathematical principles to convert units and calculate values accurately. The process requires not only understanding chemistry concepts but also being comfortable with mathematical operations.

In the original exercise, several chemistry problem-solving skills come into play:

• First, unit conversion is used to switch between mass units (kg to g).
• Next, division is employed to ascertain the number of smaller units (dimes) in a given mass.
• Finally, multiplication helps find the total value of these units in currency.

Grasping these steps is key. By breaking down each problem into smaller components - like converting mass, counting items, and calculating value - you can tackle more complex problems efficiently. Understanding each step builds confidence, leading to a better grasp of chemistry and practical mathematics.