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This is correct, but how did this person get 25/20?

Question : It took 20,000 people 20 years to build the Taj Mahal. Each worker works at the same rate. How long would it have taken for 25,000 workers to build the Taj Mahal?

Working out : If the total work is fixed, the time is inversely proportional to the number of workers.

Specifically, it takes 20,000 people build Taj Mahal in 20 years. Let's use this relation to find how long would it have taken for 25,000 workers to build the Taj Mahal.

Let's multiply the number of people by 25/20 to make it 25,000. Since the quantities are inversely proportional we must divide the time by 25/20 to get 16.

So, Taj Mahal would be built in 16 years if there were 25,000 people.

We are told that it took 20 years for 20,000 workers to build the Taj Mahal, and we want to find out how long it would take for 25,000 workers to build it.

To adjust the number of workers from 20,000 to 25,000, we need to multiply the number of workers by a factor. This factor is derived by comparing the two scenarios: 25,000 workers is 25,000/20,000 times 20,000 workers. In other words, the new number of workers is the original number of works multiplied by the quotient of new workers and old workers. Since 25,000/20,000 reduces to 25/20, this is the factor by which we multiply the original number of workers to get the new number of workers.

Since time is inversely proportional to the number of workers, the time it would take to build the Taj Mahal with 25,000 workers can be found by multiplying the original time by the inverse of the factor, which is 20/25.

Therefore, to find out how long it would have taken 25,000 workers to build the Taj Mahal, we multiply the original number of workers by the factor 25/20 to make it 25,000 workers, and multiply the original time taken by the inverse of that factor, which is 20/25:

[tex]\rm 20,000\times \dfrac{25}{20} \longrightarrow 20 \times \dfrac{20}{25}\;years\\\\\\\\\rm 25,000 \longrightarrow 16\;years[/tex]

[tex]\dotfill[/tex]

Additional Notes

The typical way to solve an inverse proportion question is as follows.

We are given that the time it takes the number of workers to build the Taj Mahal is inversely proportional to the number of workers:

[tex]\rm Time=\dfrac{k}{Number\; of\; workers}[/tex]

where k is the constant of proportionality.

Given that it took 20,000 people 20 years to build the Taj Mahal, we can substitute this into the equation and solve for k:

[tex]20=\dfrac{k}{20000}\\\\\\k=20\cdot20000\\\\\\k=400000[/tex]

Therefore, the equation that relates the time (in years) to the number of workers is:

[tex]\rm Time=\dfrac{400000}{Number\; of\; workers}[/tex]

To find out how long it would have taken 25,000 workers to build the Taj Mahal, we simply substitute 25,000 workers into the equation and solve for time:

[tex]\rm Time=\dfrac{400000}{25000}\\\\\\Time=16\;years[/tex]