Problem 70 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. \(\frac{k}{2}=8\)

Step-by-Step Solution

Answer

The value of 'k' is 16.

Step 1: Understanding the Equation

When you see \(\frac{k}{2}=8\), it means that when 'k' is divided by 2, the result is 8. This needs to be converted into a simpler form.

Step 2: Rearranging the Equation

To solve for 'k', we want to isolate it on one side of the equation. We can achieve this by performing the inverse operation. Thus, multiply both sides by 2.

Step 3: Solving the Equation

Multiplying both sides by 2 gives \(2 * \frac{k}{2} = 2 * 8\), which simplifies to \(k = 16\).

Key Concepts

Mental Math

Combining quick thinking with basic mathematical principles, mental math is an essential skill to speed up calculations without the need for a pen and paper or a calculator. It's all about doing math 'in your head'. For example, solving the equation \(\frac{k}{2} = 8\) can be handled through mental math by asking yourself, 'what number do I multiply by 2 to get 16?' Quickly, you'll realize that the answer is 8, so \(k\) must be 16. To be proficient at mental math, practice regularly with exercises that test your ability to perform calculations quickly and accurately.

Here are a few mental math strategies:

• Break down the problem into smaller, more manageable parts.
• Recognize patterns that simplify the process, like knowing that multiplying by 2 is the same as doubling a number.
• Use benchmark numbers to easily add, subtract, multiply or divide.
• Practice with times tables, squares, and cubes to speed up multiplication and division.

Memorization of the multiplication tables and familiarizing yourself with shortcuts can make a significant difference in solving equations rapidly.

Inverse Operations

In the realm of mathematics, inverse operations are pairs of operations that undo each other. Common examples include addition and subtraction, or multiplication and division. Understanding and applying inverse operations are critical in solving linear equations such as \(\frac{k}{2} = 8\).

To isolate \(k\), you perform the inverse of division by 2, which is multiplication by 2. This approach not only simplifies the equation to \(k = 16\), but also underlines the importance of using inverse operations for rearranging equations to make one variable the subject. An easy way to think of it is as a 'balancing act'—whatever you do to one side of the equation, you must also do to the other to maintain equilibrium.

• To undo addition, subtract the same number from both sides.
• To undo subtraction, add the same number to both sides.
• To reverse multiplication, divide both sides by the same number.
• To reverse division, multiply both sides by the same number.

Keeping these concepts in mind, inverse operations are a fundamental technique to achieve the solution for many algebra problems.

Algebraic Manipulation

Algebraic manipulation involves the strategic rearrangement and simplification of equations to find an unknown variable. It's a blend of different operations, properties of numbers, and understanding of how equations work. In the context of our problem, \(\frac{k}{2} = 8\), algebraic manipulation is showcased by the various steps taken to isolate the variable \(k\).

This process may include but is not limited to, using inverse operations as described earlier, combining like terms, or applying the distributive property. For someone new to algebra, it important to:

• Familiarize oneself with basic operations and their properties.
• Practice rewriting expressions in equivalent forms.
• Understand that whatever you do to one side of an equation, you must also do to the other side.

By mastering algebraic manipulation, solving for unknown variables in equations becomes a structured and logical process rather than a guessing game. It enables the solving of increasingly complex algebraic expressions and builds a strong foundation for advancing in mathematics.