Problem 8 - Prealgebra

Question

Jim sold 43 candles to raise money for a class trip. This is 15 less than the number Diana sold. Write and solve an equation to find the number of candles Diana sold.

Step-by-Step Solution

Answer

Diana sold 58 candles.

Step 1: Identify the known quantities

We know that Jim sold 43 candles. We also know that this is 15 less than the number of candles Diana sold.

Step 2: Define the variable

Let's define a variable to represent the unknown quantity. Let \( x \) be the number of candles that Diana sold.

Step 3: Set up the equation

Since Jim sold 15 fewer candles than Diana, we can set up the equation: \( x - 15 = 43 \).

Step 4: Solve the equation

To find \( x \), we need to isolate it on one side of the equation. Add 15 to both sides: \[ x - 15 + 15 = 43 + 15\]This simplifies to:\[ x = 58\]

Step 5: Verify the solution

If Diana sold 58 candles, then Jim sold 15 less, which is \( 58 - 15 = 43 \). This matches the number of candles Jim sold, confirming our solution is correct.

Key Concepts

Problem Solving

Problem solving in mathematics involves breaking down the problem step by step. This makes it easier to understand and find a solution. When Jim and Diana sold candles, we first needed to establish what we knew and what we didn't. By understanding that Jim sold 43 candles, which was 15 fewer than Diana, we could frame our approach.

• Identify known quantities: These are the facts given in the problem. Here, Jim sold 43 candles.
• Identify unknown quantities: These are what you need to find out. In this case, the number of candles Diana sold.
• Frame the problem as an equation: Use the relationship given in the problem to form a mathematical statement. This step sets the stage for solving the problem.

Achieving accuracy in problem solving depends on attentive reading and precise calculations. Once you have framed the equation, solving it becomes a systematic process of simplification.

Algebraic Expressions

An algebraic expression is a way to express information using symbols and numbers. In the problem about Jim and Diana, an algebraic expression allows us to represent unknown quantities with variables.
The key algebraic expression here is "15 less than the number of candles Diana sold," represented as:
- If Diana sold \( x \) candles, then Jim sold \( x - 15 \) candles.
By forming this expression, it helps bridge the gap between the words in the problem statement and the numerical solutions we seek.
Algebraic expressions:

• Use variables to represent unknown numbers.
• Consist of numbers, symbols, and operations like addition and subtraction.
• Enable solving equations through manipulation.

Algebraic expressions serve as the foundation of an equation. Understanding how to properly set them up is crucial for accurate solutions. They translate real-world problems into mathematical language which makes the problem solvable using mathematical techniques.

Variable Definition

Defining variables is an essential step in simplifying complex problems. It's a straightforward yet powerful tool to organize information and work towards a solution.
In this exercise, we defined the variable \( x \) to represent the number of candles Diana sold. By doing this:

• We put a clear label on the unknown quantity.
• It becomes easier to manipulate the equation based on this variable.
• It helps in verifying our results since \( x = 58 \) showed Diana sold 58 candles.

Variables act as placeholders for unknown values, allowing us to set up equations that model real-world situations. By isolating variables, you can find precise solutions, making variables fundamental in algebra and problem solving. Getting comfortable with defining and working with variables lets you tackle a broader range of mathematical challenges effectively.