Problem 29 - Intermediate Algebra

Question

Oysters. According to the Guinness Book of World Records, the record for opening oysters is 100 in 140 seconds by Mike Racz in Invercargill, New Zealand, on July \(16,1990\). If it would take a novice \(8 \frac{1}{2}\) minutes to perform the same task, how long would it take them working together to open 100 oysters?

Step-by-Step Solution

Answer

Together, they open 100 oysters in about 110 seconds.

Step 1: Convert Novice Time to Seconds

First, convert the novice's time from minutes to seconds. Given: \(8 \frac{1}{2}\) minutes.This is equivalent to \(8.5\) minutes. Since 1 minute is 60 seconds, multiply:\[8.5 \times 60 = 510 \text{ seconds}\]So, the novice takes 510 seconds to open 100 oysters.

Step 2: Calculate Individual Opening Rates

Next, calculate the rate at which each person can open oysters.- Mike's rate is \(\frac{100}{140} \approx 0.714\) oysters per second.- The novice's rate is \(\frac{100}{510} \approx 0.196\) oysters per second.

Step 3: Combine the Opening Rates

Add the individual opening rates to find the combined rate when both are working together:\[0.714 + 0.196 = 0.91 \text{ oysters per second}\]Together, they can open 0.91 oysters per second.

Step 4: Calculate Combined Time to Open 100 Oysters

To find how long it takes them together to open 100 oysters, divide 100 by their combined opening rate:\[\frac{100}{0.91} \approx 109.89 \text{ seconds}\]Therefore, it would take them approximately 109.89 seconds.

Key Concepts

Rate Problems

Rate problems often involve figuring out how fast something can be completed over time. In essence, a rate tells you how much of something is done (or used) per unit of time. In the case of this oyster problem, we're interested in determining the rate at which oysters can be opened.

To understand a rate problem, you typically need to know two things: the total quantity involved and the time taken to complete it. Here, Mike opens 100 oysters in 140 seconds, giving us his opening rate as \( \frac{100}{140} \) oysters per second. Similarly, the novice takes 510 seconds for the same task, making their rate \( \frac{100}{510} \) oysters per second.

A clear understanding of how to calculate rates allows us to handle a variety of inefficiencies and energies required in tasks—something that is useful across different scenarios and subjects.

Collaborative Task

When more than one person is involved in completing a task, they often contribute differently to the overall effort. This is a collaborative task, and it's important to understand how each participant's contribution affects the overall timing.

To tackle collaborative tasks in problems, identify each individual's role and contribution:

• Determine each participant's rate of completing the task.
• Add up these rates to get a combined rate for the group.
• Use the combined rate to calculate how long the task will take.

In the oyster example, combining Mike's and the novice's rates gives us a total opening rate of 0.91 oysters per second. Understanding collaborative tasks can be insightful for teamwork, ensuring efficiencies and understanding everyone's contribution.

This approach helps in various real-life scenarios, such as project management or orchestration of tasks where teamwork is vital.

Time Calculation

Understanding how to efficiently calculate time is crucial in solving many math problems, especially those involving rate and collaborative tasks. Time, in many problems, is not always straightforward and might require conversion or additional calculations to make sense of it.

Take careful steps:

• Ensure time units are consistent throughout your calculations.
• Utilize conversion factors, like 1 minute equals 60 seconds, to shift between units when necessary.
• Reflect on whether your calculated time makes logical sense within the context of the problem.

In our exercise, we convert all times to seconds to keep calculations consistent. For instance, the novice's time of \(8 \frac{1}{2}\) minutes translates to 510 seconds. Consistency in units is key to accurate time calculations, avoiding errors and making sure each part of a problem aligns.

Unit Conversion

Conversion between units transforms amounts from one measurement system to another, ensuring that data is in the most appropriate unit for calculation. This is essential in solving problems accurately when different measurements are used.

Here's how to handle unit conversions effectively:

• Identify the units that need to be converted.
• Use the appropriate conversion factor, like multiplying by 60 when converting minutes to seconds.
• Keep track of the conversions so all quantities in a problem are consistent before proceeding with calculations.

In the oyster task, converting the novice's time from minutes to seconds allowed for a seamless calculation alongside Mike's already-in-seconds data. Mastering unit conversions promotes clearer interpretation of results, forming the groundwork for correctness in multi-step problem-solving.