Problem 51 - Elementary Algebra

Question

Answer the question by setting up and solving an appropriate equation. \(15 \%\) of what number is \(6.3\) ?

Step-by-Step Solution

Answer

The number is 42.

Step 1: Understand the Problem

We need to find a number such that 15% of it equals 6.3. This means we are looking for the whole when we know a part and the percentage it represents.

Step 2: Set up the Equation

We know that 15% of some number (let's call it \( x \)) is equal to 6.3. We can write this as an equation: \( 0.15 \times x = 6.3 \).

Step 3: Solve for the Unknown

To find \( x \), we need to divide both sides of the equation by 0.15: \[ x = \frac{6.3}{0.15} \]

Step 4: Perform the Calculation

Calculate \( \frac{6.3}{0.15} \) to find the value of \( x \). By doing the division, we get: \( x = 42 \).

Step 5: Review the Solution

Finally, confirm that 15% of 42 is indeed 6.3: \( 0.15 \times 42 = 6.3 \). The solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, helping us find unknown quantities based on given conditions. An equation is essentially a mathematical statement of equality between two expressions. In this case, we used an equation to determine what number, when multiplied by 15%, equals 6.3.

To solve an equation:

• Identify the unknown variable – here, it's the number we're trying to find, denoted by \( x \).
• Set up the equation by translating the word problem into mathematical terms, which results in \( 0.15 \times x = 6.3 \).
• Solve for the variable – divide both sides of the equation by 0.15 to isolate \( x \).

This process involves isolating the variable on one side by performing operations that simplify the equation. The goal is to make the equation true for the unknown value, revealing the number we seek.

Basic Algebra

Basic algebra is the branch of mathematics in which we use symbols, typically letters, to represent numbers in equations. These symbols allow us to solve problems and find unknown values through manipulation of the equations.

In this example:

• We use the letter \( x \) to represent the unknown number we are trying to find.
• We express the percentage as a decimal in the equation (15% converted to 0.15).
• Essential algebraic operations such as multiplication and division help us rearrange and simplify the equation to solve for \( x \).

Learning basic algebra equips you with the tools to break down problems into manageable parts, utilize formulas, and systematically solve for unknowns in different mathematical scenarios.

Mathematical Reasoning

Mathematical reasoning involves the logical thought process used to understand, interpret, and solve mathematical problems. It requires analyzing the situation, forming an equation, and using logical steps to arrive at a solution.

In our problem:

• First, we recognized the relationship between part, whole, and percentage.
• Next, mathematical reasoning guided the formation of the equation \( 0.15 \times x = 6.3 \).
• Finally, careful calculation and checking ensure that the solution makes sense and confirms the problem's conditions (\( 0.15 \times 42 = 6.3 \)).

Developing strong mathematical reasoning skills helps you see patterns, make connections, and apply appropriate strategies to solve complex problems, facilitating understanding beyond rote memorization of formulas.