Question

A merchant sells three different sizes of canned tomatoes. A large can costs the same as 5 medium cans or 7 small cans. If a customer purchases an equal number of small and large cans of tomatoes for the same amount of money needed to buy 200 medium cans, how many small cans does she purchase? a. 35 b. 45 c. 72 d. 199 e. 208

Short answer

The correct answer is not listed among the options given. The customer purchases 245 small cans.

Step-by-step solution

Define the variables

Let's define the cost of a small can as \(s\), a medium can as \(m\) and a large can as \(l\). The given problem implies that \(l = 5m = 7s\). Therefore we can say that the price of a medium can is \(m = l / 5 = 7s / 5 \) and the price of a small can is \(s = l / 7\).

Set up the equation

According to the second statement the cost for an equal number of small and large cans is equal to the cost of 200 medium cans, we can write this as \(nl + ns = 200m\) where \(n\) is the number of large and small cans.

Simplify the equation

We can substitute the values of s, m in terms of l obtained from step 1 into the equation from step 2. We get \(nl + nl/7 = 200*7l/5 \rightarrow l*(8/7)n = 280l\).

Solve for n

Since l is a common factor, we can cancel it out and solve the equation to get the number of cans. This gives \(n = 280 * 7/8 = 245 cans\)

Key concepts

Understanding Algebra in Word Problems

When approaching algebra in word problems, it's important to break down the problem into manageable parts. Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. In this problem, defining variables is the first crucial step. We used three variables: \(s\) for small cans, \(m\) for medium cans, and \(l\) for large cans. This helps in representing unknown costs in a simplified manner.

Word problems often rely heavily on understanding relationships between these variables, as seen in the statement that a large can costs the same as 5 medium cans or 7 small cans. Such relationships form the basis of the algebraic equations used later in the problem.

Interpreting Word Problems with Accuracy

Word problems demand careful reading to extract mathematical equations from their descriptions. Here, the key is to understand what is equal or comparable. The problem states that the total cost of an equal number of small and large cans is like buying 200 medium cans. We express this relationship as \(nl + ns = 200m\), where \(n\) represents the number of small or large cans.

To solve the problem, clarity in interpreting sentences and translating them into mathematical expressions is vital. Word problems often include hidden equations that you need to tease out with these interpretations. This problem-solving skill is honed by practice with many such problems.

Creative Problem-Solving Strategies

Solving problems creatively involves exploring different ways to apply what we know, to find the solution. In our exercise, we substituted expressions derived from the relationships given (like \(l = 5m = 7s\)) into our main equation \(nl + ns = 200m\).

Creative problem-solving might include manipulating equations like substituting variables, cancelling out similar terms, or finding common denominators. Here, substituting made the problem more approachable by removing complexity step by step. This type of reasoning is at the heart of many GRE math problems, encouraging flexibility in analytical thought processes.

Mastering Step-by-Step Solutions

Developing a systematic approach to solving math problems is essential. The detailed steps laid out in this solution guide a learner through logical reasoning. We began by defining variables, moved to setting up the equation, simplified it, and finally solved for \(n\), the number of cans.

Each step in this chain builds on the last. By breaking down the solution, it becomes much easier to follow and understand, especially for complex calculations. Practice in following such step-by-step processes increases accuracy and confidence when faced with similar GRE math problems. Making sure to thoroughly understand each part before moving on is key to mastering GRE algebra problems.