GEOLOGY ROCKS: A geologist and has a laboratory full of rocks. He knows that each rock weighs exactly one pound (+1), and he would like to figure out how many rocks are in each crate. He places crates and rocks on a scale until they are balanced. Using his math skills, he is able to reason how many rocks are in each crate without having to look inside. Write an equation. Can you use mental math to see how many rocks are in each crate for each scenario?

Question

First questions
Two crates and two small rocks four small rocks and one crate

Second questions
four crates and two small rocks 18 small rocks

Third questions
Two crates and 12 small rocks three crates

Fourth questions
Two crates and eight small rocks twelve small rocks and one crate

Answer 1

To solve for the number of rocks in each crate, we can use variables to represent the unknowns. Let's assign the variable \( C \) to represent the number of rocks in each crate, and \( R \) will represent the number of rocks (1 pound each) used.

Now, we can create equations based on the information provided in your scenarios. Let's address each question.

First Question:

Equation Setup:

• Left Side: 2 crates + 2 small rocks → \( 2C + 2 \)
• Right Side: 4 small rocks + 1 crate → \( 4 + C \)

Equation: \[ 2C + 2 = 4 + C \]

Solving the Equation: \[ 2C - C = 4 - 2\ C = 2 \] Thus, each crate contains 2 rocks.

Second Question:

Equation Setup:

• Left Side: 4 crates + 2 small rocks → \( 4C + 2 \)
• Right Side: 18 small rocks → \( 18 \)

Equation: \[ 4C + 2 = 18 \]

Solving the Equation: \[ 4C = 18 - 2\ 4C = 16\ C = 4 \] Thus, each crate contains 4 rocks.

Third Question:

Equation Setup:

• Left Side: 2 crates + 12 small rocks → \( 2C + 12 \)
• Right Side: 3 crates → \( 3C \)

Equation: \[ 2C + 12 = 3C \]

Solving the Equation: \[ 12 = 3C - 2C\ C = 12 \] Thus, each crate contains 12 rocks.

Fourth Question:

Equation Setup:

• Left Side: 2 crates + 8 small rocks → \( 2C + 8 \)
• Right Side: 12 small rocks + 1 crate → \( 12 + C \)

Equation: \[ 2C + 8 = 12 + C \]

Solving the Equation: \[ 2C - C = 12 - 8\ C = 4 \] Thus, each crate contains 4 rocks.

Summary of Results:

1. Question 1: 2 rocks per crate
2. Question 2: 4 rocks per crate
3. Question 3: 12 rocks per crate
4. Question 4: 4 rocks per crate

Answer 2

Think of 4x - 1 = 15. What value of X would make 4x - 1 equal to 15?

Answer 3

To solve the equation \( 4x - 1 = 15 \) for \( x \), follow these steps:

1. Add 1 to both sides to isolate the term with \( x \): \[ 4x - 1 + 1 = 15 + 1 \] This simplifies to: \[ 4x = 16 \]

2. Divide both sides by 4 to solve for \( x \): \[ x = \frac{16}{4} \] This simplifies to: \[ x = 4 \]

Thus, the value of \( x \) that makes \( 4x - 1 \) equal to 15 is \( x = 4 \).