Problem 94 - Elementary Algebra

Question

$$ \text { Solve: } 0.75=\frac{k}{60} $$

Step-by-Step Solution

Answer

k = 45

Step 1: Identify the given equation

The given equation to solve is: \( 0.75 = \frac{k}{60} \).

Step 2: Isolate the variable

To solve for \( k \), multiply both sides of the equation by 60 to isolate \( k \): \( 60 \times 0.75 = \frac{k}{60} \times 60 \) This results in: \( 60 \times 0.75 = k \)

Step 3: Calculate the result

Perform the multiplication on the left side: \( 60 \times 0.75 = 45 \). So, \( k = 45 \).

Key Concepts

Isolating the variable

In any linear equation, the goal is to find the value of the unknown variable. Here, the unknown variable is denoted by \( k \).
The process of isolating the variable involves rearranging the equation such that the variable stands alone on one side of the equation.
For example, in the equation \( 0.75 = \frac{k}{60} \), the term \( \frac{k}{60} \) represents the unknown variable \( k \) divided by 60.
Isolating the variable means we want \( k \) on its own, without any other numbers attached to it on one side of the equation.
To do this, you need to perform the same operation on both sides of the equation. This keeps the equation balanced.
Here, we multiply both sides by 60 to cancel out the 60 in the denominator, making \( k \) stand alone.
Once this is achieved, you can easily find the value of \( k \).

Multiplication

Multiplication is one of the fundamental arithmetic operations and is key to solving many linear equations.
In our exercise, we use multiplication to isolate the variable \( k \).
The equation starts with \( 0.75 = \frac{k}{60} \). Multiplying both sides by 60 will cancel out the denominator on the right-hand side.
Mathematically, it looks like this:
\( 60 \times 0.75 = 60 \times \frac{k}{60} \).
The 60s on the right side cancel each other out, simplifying the equation to \( 60 \times 0.75 = k \).
Thus, multiplication helps in isolating \( k \) and simplifies our equation significantly.

Solving for a variable

Once the variable is isolated, the next step is usually straightforward: solving for the variable.
In the given example, after multiplication, the equation simplifies to \( k = 60 \times 0.75 \).
You just need to perform the multiplication to find the value of \( k \).
So, \( 60 \times 0.75 = 45 \).
This means \( k = 45 \).
Solving for a variable involves isolating the variable and simplifying the equation to find its exact value.
It makes use of basic arithmetic operations like addition, subtraction, multiplication, and division.
Each step should keep the equation balanced and ultimately reveal the unknown variable.