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

The first scale shows 4 boxes and 2 weights on the left side, on the right side there are 18 weights

Answer 1

To set up the equation based on the provided scenario, let's denote the number of rocks in each crate as \( x \).

1. Left Side: There are 4 crates, which each contain \( x \) rocks. Since each rock weighs 1 pound, the total weight from the 4 crates is \( 4x \). Additionally, there are 2 weights on the left side, which we can assume weigh 1 pound each, totaling \( 2 \) pounds. Therefore, the total weight on the left side is: \[ 4x + 2 \]

2. Right Side: There are 18 weights, and since each weight is 1 pound, the total weight on the right side is: \[ 18 \]

Since the scale is balanced, we can equate the two sides: \[ 4x + 2 = 18 \]

Now we can solve this equation for \( x \):

1. Subtract 2 from both sides: \[ 4x = 16 \]

2. Divide both sides by 4: \[ x = 4 \]

Thus, there are 4 rocks in each crate.

Answer 2

The second scale shows two boxes and 12 weights on the left side, the right side has three weights

Answer 3

Let's set up the equation for the second scenario similarly as we did before.

1. Left Side: There are 2 crates, and let \( x \) represent the number of rocks in each crate. Since each rock weighs 1 pound, the total weight from the 2 crates is \( 2x \). There are also 12 weights on the left side, each weighing 1 pound, totaling \( 12 \) pounds. Therefore, the total weight on the left side is: \[ 2x + 12 \]

2. Right Side: There are 3 weights on the right side, which add up to: \[ 3 \]

Setting the two sides equal since the scale is balanced: \[ 2x + 12 = 3 \]

Now we can solve this equation for \( x \):

1. Subtract 12 from both sides: \[ 2x = 3 - 12 \] \[ 2x = -9 \]

2. Divide both sides by 2: \[ x = -\frac{9}{2} \]

In this case, \( x = -4.5 \), which doesn't make practical sense in the context of counting rocks because you cannot have a negative number of rocks in a crate.

This indicates that the setup of the scenario is not physically possible or that the weights on the left must be adjusted for the system to balance correctly. If we expect whole rocks, we should verify the number of weights on both sides to match conditions that allow for valid positive integers for \( x \).

Answer 4

The third scale shows 2 crates and eight weights on the left side and the right side shows 12 weights and 1 crate

Answer 5

Let's analyze the third scenario step by step and set up the equation.

1. Left Side: There are 2 crates, and let \( x \) represent the number of rocks in each crate. The total weight from the 2 crates is \( 2x \). There are also 8 weights on the left side, each weighing 1 pound, contributing a total of 8 pounds. Therefore, the total weight on the left side is: \[ 2x + 8 \]

2. Right Side: There is 1 crate and 12 weights. The weight from the crate is \( x \) (because it contains \( x \) rocks), and the total weight from the 12 weights is: \[ 12 \] Thus, the total weight on the right side is: \[ x + 12 \]

Now, we can set up the equation since the scale is balanced: \[ 2x + 8 = x + 12 \]

Let's solve for \( x \):

1. First, subtract \( x \) from both sides: \[ 2x - x + 8 = 12 \] \[ x + 8 = 12 \]

2. Next, subtract 8 from both sides: \[ x = 12 - 8 \] \[ x = 4 \]

Thus, there are 4 rocks in each crate in this scenario as well.

Answer 6

define expression in math

Answer 7

In mathematics, an expression is a combination of numbers, variables, operators, and functions that represents a value or quantity. Expressions can involve constants (like 2, 5, or 100), variables (like \(x\), \(y\), or \(z\)), and mathematical operations (such as addition, subtraction, multiplication, and division).

Key Components of an Expression:

1. Constants: Fixed values (e.g., \(3\), \(-5\), \(7.2\)).
2. Variables: Symbols that represent unknown or variable quantities (e.g., \(x\), \(a\), \(n\)).
3. Operators: Symbols indicating the operation to perform (e.g., \(+\) for addition, \(-\) for subtraction, \(\times\) for multiplication, \(\div\) for division).
4. Functions: Common functions (like \(sin\), \(cos\), \(log\), etc.) can also be a part of expressions.

Examples of Expressions:

• Algebraic Expression: \(3x + 2\)

  • Here, \(3x\) represents three times a variable \(x\), plus 2.

• Numerical Expression: \(5 + 8 - 3\)

  • A combination of constants with no variables.

• Polynomial Expression: \(4x^2 - 2x + 1\)

  • A sum of terms with a variable raised to whole number powers.

• Rational Expression: \(\frac{2x + 3}{x - 1}\)

  • A fraction where the numerator and denominator are both algebraic expressions.

Characteristics:

• No Equals Sign: Expressions differ from equations, which state that two things are equal (e.g., \(2 + 3 = 5\)).
• Can Be Simplified: Expressions can often be simplified or factored.

In summary, a mathematical expression can represent a wide variety of values and relationships and is fundamental in creating equations, functions, and models in mathematics.

Answer 8

Define equation

Answer 9

An equation is a mathematical statement that asserts the equality of two expressions by using the equals sign (\(=\)). It consists of two sides: the left-hand side and the right-hand side, each of which can be an expression made up of numbers, variables, operators, and possibly functions.

Key Features of an Equation:

1. Equality: An equation states that the values of the two expressions are equal. This is the core concept of an equation; it expresses a relationship of equality between two mathematical entities.

2. Components:

   • Expressions: Each side of the equation contains expressions that can include constants (fixed numbers), variables (symbols representing unknown values), and operators (addition, subtraction, multiplication, division).
   • Operators: Common operators found in equations include \(+\) (addition), \(-\) (subtraction), \(\times\) (multiplication), \(\div\) (division), and exponentiation (e.g., \(x^2\)).

3. Solutions: An equation often requires finding the value(s) of the variable(s) that make the equation true. These values are called the solutions of the equation.

Examples of Equations:

• Linear Equation: \[ 2x + 3 = 7 \]

  • This equation can be solved to find the value of \(x\) that makes it true.

• Quadratic Equation: \[ x^2 - 5x + 6 = 0 \]

  • This is a polynomial equation of degree two that can have two solutions.

• Rational Equation: \[ \frac{x + 1}{x - 2} = 5 \]

  • This involves a ratio of expressions and can also be solved for \(x\).

• Exponential Equation: \[ 2^x = 16 \]

  • In this case, we solve for \(x\) to find the exponent that makes the equation true.

Characteristics:

• Not Just Numeric: Equations can involve variables. For instance, \(y = 3x + 2\) is a linear equation expressing a relationship between \(x\) and \(y\).
• Dynamic: Equations can represent a broad range of relationships and can be manipulated using algebraic techniques to find solutions or transform them into different forms.

Summary:

An equation represents a precise relationship between two expressions, and solving equations is a fundamental part of mathematics, enabling the formulation of models in various fields such as physics, engineering, and economics.

Answer 10

Order of Operations

Answer 11

The Order of Operations is a set of rules that determines the sequence in which mathematical expressions should be evaluated to ensure accurate and consistent results. The acronym PEMDAS is often used to help remember this order:

1. P - Parentheses: Evaluate expressions inside parentheses first.
2. E - Exponents: Calculate powers and square roots next.
3. MD - Multiplication and Division: These operations are performed from left to right. They have the same level of priority.
4. AS - Addition and Subtraction: Lastly, perform addition and subtraction, also from left to right. They have the same level of priority.

Example of Order of Operations:

Consider the expression: \[ 3 + 6 \times (5 + 4) \div 3 - 7 \]

1. Parentheses: First, solve the expression inside the parentheses: \[ 5 + 4 = 9. \] The expression now looks like: \[ 3 + 6 \times 9 \div 3 - 7 \]

2. Multiplication and Division: Next, perform the multiplication and division from left to right:

   • First, calculate the multiplication: \[ 6 \times 9 = 54. \] The expression now looks like: \[ 3 + 54 \div 3 - 7. \]
   • Next, calculate the division: \[ 54 \div 3 = 18. \] The expression now looks like: \[ 3 + 18 - 7. \]

3. Addition and Subtraction: Finally, from left to right:

   • First, add: \[ 3 + 18 = 21. \]
   • Then, subtract: \[ 21 - 7 = 14. \]

So, the final answer is: \[ 14 \]

Alternative Mnemonics:

Some other mnemonics used to remember the order of operations include:

• BODMAS: Brackets, Orders (another term for exponents), Division, Multiplication, Addition, Subtraction.
• BIDMAS: Brackets, Indices (another term for exponents), Division, Multiplication, Addition, Subtraction.

Important Note:

Always perform operations in the order specified to avoid ambiguity and errors in calculations. When in doubt, using parentheses to clarify the order of operations can help ensure that expressions are evaluated correctly.