Problem 45 - Basic College Mathematics

Question

Amy offers \(\$ 6400\) for a used Ford Taurus advertised at \(\$ 8000\). The first offer from Jim, the car's owner, is to "split the difference" and sell the car for \((6400+8000) \div 2,\) or \(\$ 7200\). Amy's second offer is to split the difference between Jim's offer and her first offer. Jim's second offer is to split the difference between Amy's second offer and his first offer. If this pattern continues and Amy accepts Jim's third (and final) offer, how much will she pay for the car?

Step-by-Step Solution

Answer

Amy will pay $6950 for the car.

Step 1: Calculate Jim's First Offer

Jim suggests splitting the difference between Amy's initial offer (\( 6400 \) ) and the advertised price (\( 8000 \) ).\(\frac{6400 + 8000}{2} = 7200\). Thus, Jim's first offer is \(7200\).

Step 2: Calculate Amy's Second Offer

Amy splits the difference between her first offer (\( 6400 \) ) and Jim's first offer (\( 7200 \) ).\(\frac{6400 + 7200}{2} = 6800\). Thus, Amy's second offer is \(6800\).

Step 3: Calculate Jim's Second Offer

Jim splits the difference between Amy's second offer (\( 6800 \) ) and his first offer (\( 7200 \) ).\(\frac{6800 + 7200}{2} = 7000\). Thus, Jim's second offer is \(7000\).

Step 4: Calculate Amy's Third Offer

Amy splits the difference between her second offer (\( 6800 \) ) and Jim's second offer (\( 7000 \) ).\(\frac{6800 + 7000}{2} = 6900\). Thus, Amy's third offer is \( 6900 \).

Step 5: Calculate Jim's Final Offer

Jim splits the difference between Amy's third offer (\( 6900 \) ) and his second offer (\( 7000 \) ).\(\frac{6900 + 7000}{2} = 6950\). Thus, Jim's third (and final) offer is \( 6950 \).

Key Concepts

average

The average is a measure that sums up all values in a set and divides the sum by the number of values. To provide clarity with an example, if Amy and Jim are determining an average price for the car, they combine both offers and divide by two. For example, combining Amy's offer of \(6400 and Jim's offer of \)8000 results in an average of \( \frac{6400 + 8000}{2} = 7200 \). This measure helps them find a fair middle ground price.

arithmetic mean

The term 'arithmetic mean' is a fancy way to describe the average. It involves the same process of adding numbers together and dividing by the count of numbers. This concept is crucial in negotiations, as shown by Amy and Jim's interactions. Each step involves finding the arithmetic mean of their previous offers. In mathematical terms, the arithmetic mean of two numbers \( a \) and \( b \) is given by \( \frac{a + b}{2} \). This finds application in their iterative negotiation process.

iterative calculation

Iterative calculation refers to the repeated application of a formula or process. In the negotiation between Amy and Jim, we see several iterations where they continually take the average of their previous offers. Each person's subsequent offer is a result of iteratively splitting the difference between the latest offers. For instance, starting with \( 7200 \), the second offer iteration is \( \frac{6400 + 7200}{2} = 6800 \) and so on. These steps form a sequence that converges closer to a final agreed price.

problem-solving steps

Solving problems step by step makes complex tasks manageable. In our example, Amy and Jim's negotiation is broken down into clear steps. Jim's initial offer is calculated first. Then, Amy responds with her second offer. Jim then iterates with his second offer and finally, they end at Jim's third and last offer. Each step involves straightforward calculations to find the average. Following a structured approach ensures that each step logically follows from the previous one, making the problem easier to solve and understand.