Problem 23 - Algebra for College Students

Question

Solve each of the following for the indicated variable. \(I=\frac{100 M}{C}\) for \(C \quad\) (Intelligence quotient)

Step-by-Step Solution

Answer

\( C = \frac{100M}{I} \)

Step 1: Understand the Equation

The equation given is \( I = \frac{100M}{C} \), where you need to solve for \( C \). This means we want to express \( C \) in terms of the other variables, \( I \) and \( M \).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \( C \): \[ C imes I = 100M \] This step allows us to get rid of the denominator in the fraction.

Step 3: Solve for C

Now that we have \( C imes I = 100M \), solve for \( C \) by dividing both sides by \( I \): \[ C = \frac{100M}{I} \] This is the expression of \( C \) in terms of \( M \) and \( I \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value(s) of unknowns in mathematical statements. When solving equations, the main goal is to isolate the variable of interest on one side, making it the subject of the formula. This is typically done using operations like addition, subtraction, multiplication, or division to both sides of the equation.

In our given problem, the original equation is \( I = \frac{100M}{C} \), and we need to solve for \( C \).

• First, understand what the equation represents and what variable we want to find (in this case, \( C \)).
• Next, eliminate any fractions by multiplying through by the denominator. This gives a clearer path to isolating the variable.
• The final step usually involves performing the operation, such as dividing or multiplying both sides by a variable or constant, to solve for the desired variable.

By manipulating the equation, we discovered that \( C = \frac{100M}{I} \), showing that \( C \) is a function of \( M \) and \( I \). Practice solving different types of equations to strengthen your understanding and ability to approach similar problems.

Variables

Variables are essential to algebra and broader mathematical concepts. They represent unknown quantities that can change or vary, thus the name "variable." In our equation \( I = \frac{100M}{C} \), \( I \), \( M \), and \( C \) are all variables.

Each variable can represent quantities or characteristics that are not yet defined.

• \( I \) might represent the intelligence quotient, which is a measure of intelligence.
• \( M \) could stand for a person's mental age.
• \( C \) is what we are solving for, which might represent chronological age in this context.

To work with variables efficiently:

• Understand what each variable stands for in a given equation.
• Recognize how changes in one variable affect others in the equation.
• Learn to manipulate the equation to reconfigure it with any variable as the subject.

Variables provide the flexibility to model a range of scenarios in mathematics and beyond, representing conditions or elements we often try to solve.

Intelligence Quotient

The Intelligence Quotient, or IQ, is a significant concept often expressed in mathematical terms. Traditionally, as in our equation, it was calculated using the formula \( I = \frac{100M}{C} \). This simple yet powerful formula captures a ratio of mental to chronological age, multiplied by 100 to remove the decimal.

Breaking it down:

• \( I \) is the intelligence quotient; it provides a standardized measure.
• \( M \) represents mental age, reflecting the level at which an individual functions intellectually.
• \( C \) represents chronological age, indicating the actual age of the individual.

Understanding IQ:

• The formula shows the relationship between age and intellectual performance.
• Higher IQ scores typically suggest higher intellectual functioning.
• The formula uses basic algebra to provide insights into a person's cognitive abilities relative to their age group.

While modern tests use more complex methods, this equation provides foundational insight into how intelligence was once quantified.