Question

A rock is made up by volume of \(32 \%\) coal, which has a specific gravity (expressed in grams per cubic centimeter) of \(1.20\). The rock contains \(29 \%\) granite with a specific gravity of \(2.60\). The rock also contains \(39 \%\) of an unknown mineral. If the specific gravity of the entire rock is \(1.4\), the unknown material has what approximate specific gravity? (A) \(0.43\) (B) \(0.54\) (C) \(0.67\) (D) \(0.81\)

Short answer

The unknown material has a specific gravity of approximately 0.67 (Answer C).

Step-by-step solution

Calculate the Contribution of Coal

To calculate the contribution of coal to the specific gravity of the rock, multiply its percentage by its specific gravity. This gives: \(0.32 \times 1.20 = 0.384\).

Calculate the Contribution of Granite

Similarly, for granite, multiply its percentage by its specific gravity: \(0.29 \times 2.60 = 0.754\).

Setup for the Unknown Mineral

Let \(x\) be the specific gravity of the unknown mineral. The unknown mineral contributes \(0.39 \times x\) to the overall specific gravity.

Setup the Equation for Overall Specific Gravity

The total specific gravity of the rock is the sum of the contributions from coal, granite, and the unknown mineral. Set up the equation: \(0.384 + 0.754 + 0.39x = 1.4\).

Solve for the Unknown Mineral's Specific Gravity

Simplify and solve the equation: \[0.384 + 0.754 + 0.39x = 1.4\]Subtract \(0.384\) and \(0.754\) from both sides: \[0.39x = 1.4 - 0.384 - 0.754\]Calculate this result: \[0.39x = 0.262\]Now solve for \(x\): \[x = \frac{0.262}{0.39} \approx 0.67\].

Key concepts

Understanding Specific Gravity

Specific gravity is a measure that compares the density of a substance to the density of water. This idea is particularly useful in geology and material science for understanding the composition of mixtures, such as rocks. When dealing with specific gravity, you're looking at how much denser one material is compared to water, which has a specific gravity of 1. For example, coal in the problem has a specific gravity of 1.20, meaning it's 1.20 times denser than water.
Specific gravity is usually expressed in grams per cubic centimeter, making it dimensionless. This simplifies calculations by focusing solely on the density ratios without worrying about unit conversions. Knowing the specific gravity helps quickly estimate the mass of materials and is critical in this problem to finding the unknown mineral's specific gravity.

Mastering Percentage Calculations

Percentage calculations are a mathematical process used to find a portion of a whole in terms of 100. In this context, understanding how to work with percentages is essential for determining the contribution of each component to the total specific gravity of the rock.
To find how much each part contributes, you multiply the percentage of each component by its specific gravity. For instance, coal makes up 32% of the rock's volume, and its specific gravity is 1.20, so its contribution is calculated as follows:

• Percentage as a decimal: 32% = 0.32
• Contribution to specific gravity: 0.32 \(\times\) 1.20 = 0.384

Repeat these steps for each component, remembering to convert the percentage into a decimal. This approach ensures all contributions are accurately incorporated into your calculations.

Setting Up and Using Equations

Equations are central tools in many mathematical and scientific problems, allowing professionals and students to find unknown values systematically. In the scenario here, an equation helps us find the unknown mineral's specific gravity. The equation reflects the sum of the parts' contributions equaling the whole rock's specific gravity.
The given equation from the solution is:

• 0.384 + 0.754 + 0.39\(x\) = 1.4

This equation incorporates all the calculated contributions from coal, granite, and the unknown mineral. Solving this helps you understand how each part affects the whole. It's crucial to balance equations correctly and pay attention to each term's placement. Practicing these techniques in various problems enhances proficiency in handling complex calculations.

Effective Problem-Solving Steps

Approachable problem-solving involves breaking down a complex task into manageable steps. Following a structured approach ensures efficiency and reduces stress when tackling challenging problems.
Here's an overview of structured problem-solving steps:

• Identify what you know: Determine known values and relationships. Here, you have specific gravities and percentages for coal and granite.
• Determine what you need to find: You were tasked with finding the unknown mineral's specific gravity.
• Set up necessary calculations: Translate known data into mathematical expressions, like contributions using percentages and specific gravities.
• Write and solve an equation: Ensure the equation encompasses everything you know, leading to solving for the unknown.
• Check your work: Look back to ensure all calculations were performed correctly and make logical sense.

Applying these steps can make tackling math problems more straightforward and methodical, leading to solutions that are both correct and comprehensive.