If potatoes cost 50 cents per \(\mathrm{lb}\) and you use \(1 / 8\) of all the potatoes in the cupboard to make a meal, what was the value of all the potatoes used if \(7 \mathrm{lb}\) of potatoes remain in the cupboard? A. \(\$ 6\) B. \(\$ 3\) C. \(\$ 1\) D. 50 cents E. Cannot be determined

Before we dive into solving any problem, it's important to clearly understand the information given and what you are being asked to find. In this GMAT problem-solving exercise, we are given the cost of potatoes per pound, the fraction of the total potatoes used for a meal, and the remaining weight of the potatoes in the cupboard. This exercise demonstrates the importance of mathematical reasoning in breaking down a problem and systematically solving it.

Start by identifying what you know: the cost per pound (50 cents), the fraction used (1/8), and the remaining weight (7 pounds). Next, figure out the relationship between these quantities.

* What is the total weight of the potatoes initially?
* How much of this total was used?

By approaching the problem step by step and using logical reasoning, you can set up an equation that accurately represents the given information. Solving this equation will then allow you to find the value of the potatoes used in the meal.

Algebra plays a crucial role in solving this problem. Let's break down the algebraic steps involved. We start by letting the total weight of the potatoes be represented by \( x \) pounds. Since 7 pounds remaining implies one-eighth of the total potatoes were used, we can set up the following equation:


\[ x - \frac{x}{8} = 7 \]

This equation shows the relationship between the total weight, the fraction used, and the remaining weight. Next, combine like terms to simplify:

\[ \frac{8x - x}{8} = 7 \]

Simplify it further to:
\[ \frac{7x}{8} = 7 \]

By multiplying both sides by 8 to clear the fraction, you get:

\[ 7x = 56 \]

Finally, isolate \( x \) by dividing both sides by 7:
\[ x = 8 \]

Therefore, the total initial weight of the potatoes is 8 pounds. The use of algebra allows us to find unknown quantities and solve the problem efficiently.

Understanding and solving word problems can be challenging, but following a structured approach can help. This exercise showcases how to translate a real-life situation into a mathematical problem and solve it effectively.

Here’s a quick guide for tackling word problems:

* **Identify what you need to find:** In this problem, we need to find the value of the potatoes used.
* **Extract relevant information:** The cost per pound, the fraction used, and the remaining weight are crucial details.
* **Formulate equations:** Use algebra to represent the relationships described in the problem.
* **Solve step by step:** Break down the problem, solve the equation, and then use this solution to find the final answer.

By using these steps, you can convert a real-world problem into mathematical terms and solve it methodically. Remember to double-check your calculations to ensure accuracy.