Art of Problem Solving: Joint Proportion

Summary

In this engaging and relatable video, the speaker tackles a real-world math problem about construction delays caused by reduced workers and increased road size. Using concepts like joint and inverse proportions, the speaker explains how to calculate the new time required to finish the roadwork after the number of workers decreases from 8 to 5 and the road length increases. Through multiple approaches, including worker daily output and proportional reasoning, the speaker arrives at the conclusion that it will take 64 days to complete the job, all while adding humor and practical math insights for the audience.

Takeaways

• 😀 The speaker encounters issues shooting videos due to external construction noise and comes in early to avoid disturbances.
• 😀 The speaker initially consults the construction foreman, Lino, to estimate the time required for a road project.
• 😀 Lino estimates that with 8 workers, it will take about 30 days to complete 750 yards of road, but the number of workers reduces, complicating the timeline.
• 😀 The speaker explains the math of road construction using joint proportions, showing how time, workers, and road length are interrelated.
• 😀 The road length and time are directly proportional: more road means more time with the same number of workers, while keeping time constant requires more workers.
• 😀 A joint proportion equation is used to model the relationship between road length, time, and the number of workers, where these factors are multiplied by a constant.
• 😀 The speaker calculates the constant using initial data, then applies it to the new situation with fewer workers (5 workers for 1,000 yards of road).
• 😀 The result shows that with 5 workers, it will take 64 days to finish 1,000 yards of road, using both a joint proportion and worker-by-worker calculations.
• 😀 The speaker also considers how much work one worker can do in a day, calculating the daily output for 8 workers and scaling it to 5 workers.
• 😀 In another approach, the speaker uses the inverse proportionality of workers and time, explaining how reducing workers leads to more time required to complete the project.
• 😀 Ultimately, all methods of calculation converge on the result that the project will take 64 days to complete with the reduced workforce and increased road length.

Q & A

Why did the speaker come in early to shoot videos?

The speaker came in early to shoot videos to avoid the noise from construction, ensuring the recording would not be disturbed by jackhammers.

What did the foreman, Lino, say about the construction timeline?

Lino mentioned that with 8 workers, it would take about 30 days to complete 750 yards of road, as there were only 8 workers assigned to the project.

What change did the speaker notice regarding the number of workers?

The speaker noticed that on the second day, only 5 workers were present, instead of the expected 8, which led to a longer project timeline.

How did the speaker approach Lino about the changes in the project?

The speaker asked Lino about the issue, and Lino explained that the workers were reassigned to other sections of the road, which extended the timeline.

What is the mathematical relationship between the amount of road, time, and workers?

The amount of road, time, and workers are directly proportional in this scenario. If the road length increases, more workers are needed to maintain the same amount of time, and vice versa.

What is a joint proportion in math, as described in the script?

A joint proportion is a mathematical relationship where the amount of road is directly proportional to both time and the number of workers, expressed as road = constant * time * workers.

How did the speaker calculate the constant in the proportion?

The speaker used the initial data (750 yards of road, 30 days, and 8 workers) to calculate the constant by dividing the road by the time and the number of workers, which resulted in a constant value.

What was the calculated time for 5 workers to complete 1,000 yards of road?

Using the joint proportion, the speaker calculated that with 5 workers and 1,000 yards of road, it would take 64 days to complete the project.

How did the speaker calculate how much work one worker could do in a day?

The speaker calculated that if 8 workers could complete 25 yards of road in one day, then each worker could complete 25/8 yards per day.

What was the alternative approach the speaker used to solve the problem?

The speaker calculated how much work a single worker could do per day and then used that to determine how many days it would take for 5 workers to complete 1,000 yards of road, which also resulted in 64 days.