Problem 5 - Prealgebra

Question

Find the value of \(k\) that makes \(6=\frac{48}{k}\) true. \(\mathbf{A} 6\) \(\quad\) \(\mathbf{B} 7\) \(\quad\) \(\mathbf{C} 8\) \(\quad\) \(\mathbf{D} 12\)

Step-by-Step Solution

Answer

The value of \(k\) is 8.

Step 1: Understand the Equation Format

We start with the given equation: \(6 = \frac{48}{k}\). This is a fraction equation where we need to find the value of \(k\) that makes this equation true.

Step 2: Multiply Both Sides by k

To eliminate the fraction, multiply both sides of the equation by \(k\). This will give us: \(6k = 48\).

Step 3: Solve for k

Now, divide both sides of the equation by 6 to solve for \(k\): \(k = \frac{48}{6}\).

Step 4: Simplify the Division

Calculate the result of the division: \(k = 8\).

Key Concepts

Solving Equations

Solving equations is like unlocking a puzzle. An equation is a mathematical statement where two expressions are equal, connected by an equals sign, indicating balance.
To find the unknown, you have to perform operations that keep the equation balanced on both sides.

• Identify the variable you need to solve for. In our case, it's k.
• Perform operations that isolate the variable on one side of the equals sign.
• Keep the equation balanced by doing the same operation on both sides.

In the example equation of \(6 = \frac{48}{k}\), we started by eliminating the fraction by multiplying both sides by k. This step moves from an equation where the variable is in a fraction, to one where the variable is in a simple multiplication, enabling us to find its value easily.
The key here is to always ask yourself what you can do to one side, can you do to the other side. This thought process keeps your solution logical and correct.

Fractions

Fractions represent a part of a whole and are an essential part of equations and everyday mathematics. They consist of a numerator (top number) and a denominator (bottom number).

• Understanding fractions helps in visualizing parts of a number.
• Being able to manipulate fractions is key, often needing to convert them to whole numbers or different fractions.

In our example, we began with a fraction, \(\frac{48}{k}\). Fractions in equations can initially seem complex, but your goal is to manage them effectively by eliminating or simplifying them.
By multiplying both sides by the denominator, you can "clear" the fraction, making it easier to solve. This step often transitions a problem from a complex fraction to a simple arithmetic operation.
Becoming comfortable with fractions and their manipulation will make solving a variety of mathematical problems much more accessible.

Basic Arithmetic Operations

Basic arithmetic operations are the fundamental tools of mathematics, including addition, subtraction, multiplication, and division. These operations form the bedrock on which solving equations rests.

• Multiplying connects numbers by repeated addition.
• Dividing splits a number into equal parts or finds out how many times one number fits into another.

In solving \(6k = 48\), we used division, which meant splitting 48 into six equal parts to "find" the value of k. This resulted in the calculation \(k = \frac{48}{6}\), giving us k as 8.
Arithmetic operations are pivotal not just in solving equations but in day-to-day practical scenarios too. Mastery of these operations will aid in more complex algebraic manipulations and problem-solving tasks.