Solving A Classic Fruit Puzzle A Step-by-Step Mathematical Approach

In this intriguing mathematical problem, we delve into the world of a fruiterer who initially possesses a collection of 200 apples and pears. As the fruiterer engages in sales, the number of fruits changes, leading to a specific relationship between the remaining apples and pears. Our mission is to determine the initial quantity of apples the fruiterer had. This problem is a classic example of a word problem that requires careful analysis and the application of algebraic principles to arrive at the solution. The keywords here are apples, pears, and the relationship between the quantities before and after the sales.

To tackle this problem effectively, we need to translate the given information into mathematical equations. Let's denote the initial number of apples as 'A' and the initial number of pears as 'P'. From the problem statement, we know that the total number of fruits initially is 200. This can be expressed as our first equation:

A + P = 200

Next, we consider the sales made by the fruiterer. He sells 79 apples and 9 pears. This means the remaining number of apples is A - 79, and the remaining number of pears is P - 9. The problem states that after these sales, the fruiterer has three times as many apples as pears. This crucial piece of information leads us to our second equation:

A - 79 = 3(P - 9)

These two equations form a system of linear equations that we can solve to find the values of A and P. The setup phase is critical because it lays the foundation for the subsequent steps. A clear and accurate representation of the problem in mathematical terms is essential for a successful solution. This involves identifying the unknowns, defining variables, and translating the given information into equations that capture the relationships between the variables. In this case, we've identified the initial number of apples and pears as our unknowns and have established two equations based on the total number of fruits and the ratio of apples to pears after the sales.

Now that we have our system of equations, the next step is to solve for the unknowns. There are several methods to solve a system of linear equations, such as substitution, elimination, or matrix methods. In this case, the substitution method appears to be the most straightforward approach. We can solve the first equation for one variable and substitute that expression into the second equation. Let's solve the first equation for P:

P = 200 - A

Now we substitute this expression for P into the second equation:

A - 79 = 3((200 - A) - 9)

This equation now contains only one variable, A, which we can solve for. Let's simplify and solve for A:

A - 79 = 3(191 - A) A - 79 = 573 - 3A 4A = 652 A = 163

So, we have found that the initial number of apples, A, is 163. To find the initial number of pears, P, we can substitute this value back into the equation P = 200 - A:

P = 200 - 163 P = 37

Therefore, the fruiterer initially had 163 apples and 37 pears. This part of the solution process highlights the importance of algebraic manipulation and problem-solving skills. The ability to solve a system of equations is a fundamental concept in mathematics and has wide applications in various fields. Here, we used the substitution method, which involves expressing one variable in terms of another and then substituting that expression into the other equation. This method allows us to reduce the system of equations to a single equation with one variable, which can then be solved using basic algebraic operations. The key is to perform the algebraic manipulations accurately and systematically to avoid errors and arrive at the correct solution.

It's always a good practice to verify our solution to ensure its accuracy. We can do this by plugging the values we found for A and P back into the original equations and checking if they hold true. Let's start with the first equation:

A + P = 200 163 + 37 = 200 200 = 200

The first equation holds true. Now let's check the second equation:

A - 79 = 3(P - 9) 163 - 79 = 3(37 - 9) 84 = 3(28) 84 = 84

The second equation also holds true. This confirms that our solution is correct. The verification step is crucial because it provides a check for any potential errors in our calculations or reasoning. By plugging the values back into the original equations, we can ensure that our solution satisfies the given conditions of the problem. This step not only increases our confidence in the solution but also helps us identify and correct any mistakes that may have been made. In this case, the verification process confirms that the fruiterer initially had 163 apples and 37 pears, which satisfies both the total number of fruits and the ratio of apples to pears after the sales.

Therefore, the fruiterer had 163 apples at first. This final answer is the culmination of our step-by-step analysis and solution process. It represents the answer to the original question posed in the problem statement. The ability to clearly and concisely state the answer is an important aspect of problem-solving, as it demonstrates a complete understanding of the problem and its solution. In this case, we have clearly stated the answer, which is the initial number of apples the fruiterer had. This answer is the result of our careful analysis, equation setup, solution process, and verification, ensuring that it is accurate and reliable. The keywords here are the initial number of apples which is the core of the problem.

Word Problem, Algebra, System of Equations, Fruiterer, Apples, Pears, Mathematical Solution, Problem Solving, Equation Setup, Verification