# Ratio and Proportion Question

Am doing a math text book, and I came across this one:

120 men can lay a road which is 80km long in 20days. 100 men can lay a road of 90km in how many days?

I tried finding per man = 1/30 km per day; and solving, but it didn't work.

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Imagine $120$ people working on the $80km$ road. We can assume that each person works on a small part of it, such that after 20 days everybody has completed his/her section and the road is complete. Thus, we have the $80 km$ road divided between $120$ persons, and this means that each person lays $\frac{80}{120} km$ of road in 20 days. This also means that the length of road laid down by each person per day is $\frac{1}{20}\cdot \frac{80}{120} km = \frac{1}{30} km$.

Now, we have $100$ persons working on a $90km$ road. Let's first ask ourselves how many kilometers of road the team of $100$ people can lay down each day: it is simply what each can do, times the number of people, that is $100 \cdot \frac{1}{30} km = \frac{10}{3} km$.

Knowing that the team can lay $\frac{10}{3} km$ of road per day means that, during $n$ days, the team can lay $\frac{10}{3} km \cdot n$ of road. Since we want to find $n$ when the team lays $90km$ of road, we state the equality:

$$\frac{10}{3} km \cdot n = 90km$$ Divide by $\frac{10}{3} km$ on both sides and you find $n$, that is the number of days it takes to build the road.

Note: I took the long path here because it is very important to understand properly what we are doing. You can make a drawing of the situation to help you, or try to image in your head a geometric representation of the situation.

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Thank you very much. This was a longer one, but It helped. – Ramana Jan 16 '14 at 5:31
Sorry, I cannot vote up for both answers, because I have only 1 reputation. – Ramana Jan 16 '14 at 5:32
No problem, I'm happy to know it helped! – Olivier Jan 16 '14 at 5:33

I think the calculation is OK. 1 man in 1 day can lay 1/30 km. Then 100 men in 1 day can lay 100/30 km. 100 men can lay 90 km for 90/(100/30)=(90 x 30)/100=27 days.

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Thankyou very much! It helped a lot! Again, thankyou very much. – Ramana Jan 16 '14 at 4:46