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## Arithmetic - Use of Variation

Today, we would like to discuss a Veritas Prep work question. The intent is to show you how simple your calculations can get when you use the methods we discussed in our Ratios module. I couldn’t say it enough – develop a love for ratios. You will save a huge amount of time in lots of questions.

Question: 16 horses can haul a load of lumber in 24 minutes. 12 horses started hauling a load and after 14 minutes, 12 mules joined the horses. Will it take less than a quarter-hour for all of them together to finish hauling the load?

(1) Mules work more slowly than horses.
(2) 48 mules can haul the same load of lumber in 16 minutes.

Solution: First do this question on your own and see the calculations involved. Thereafter, check out the solution given below to know how we can solve the question using our joint variation method.

We are given that 16 horses can complete the work in 24 mins. Let’s find out how much work is done by 12 horses in 14 mins (before the mules join in)

16 horses ……… 24 mins ………. 1 work
12 horses ……… 14 mins ………. ?? work

Work done = 1*(14/24)*(12/16) = 7/16 work

So in 14 mins, the 12 horses can complete 7/16 of the work i.e. they do 1/16 of the work every 2 mins.

How much work is leftover for the mules and horses to do together? 1 – 7/16 = 9/16

Leftover work = 9/16

This makes us think that 12 horses alone will take 9*2 = 18 mins to finish the work. When 12 mules join in, depending on the rate of work of mules, time taken to complete this work could be less than or more than 15 mins.

Statement 1: Mules work more slowly than horses.

This statement doesn’t give us enough information. It just tells us that mules work slower than horses. Say if they work very slowly so that, effectively, they are not adding much to the work done, the work will get done in approximately 18 mins. If they work faster, time taken will keep decreasing. If they work as fast as the horses, the rate at which the work will be done will double (because we already have 12 horses and we will add 12 mules which will be equivalent to 12 horses) and time taken will become half i.e. it will be 9 mins. So the time taken will vary in the range 9 mins to 18 mins depending on the rate of work of mules. This statement alone is not sufficient.

Statement 2: 48 mules can haul the same load of lumber in 16 minutes.

We now know the rate of work of mules. The point is that now we can easily calculate the exact time taken by 12 horses and 12 mules to complete 9/16 of the work. Once we calculate the exact time, we will be able to say whether the time taken will be less than or more than 15 mins. Hence this statement alone is sufficient to answer the question. We don’t really need to find out exactly how much is taken by the 24 animals together since it is a DS question. Ideally, we should mark the answer as (B) and move on.

Nevertheless, let’s do the calculations if only to practice application of work concepts.

Let’s try to find the equivalence of mules and horses (the way we did with cars in the previous post)

We know that 16 horses can haul a load of lumber in 24 minutes. Let’s find out the number of mules that are needed to complete the work in 24 mins.

48 mules …….16 mins.

?? mules ……..24 mins

No. of mules required = 48*16/24 = 32 mules

So, 32 mules do the same work in the same time as done by 16 horses. Or we can say that 2 mules are equivalent to 1 horse. Hence, 12 mules are equivalent to 6 horses. When 12 mules join the 12 horses, equivalently we get 12+6 = 18 horses.

16 horses ……… 24 mins ………. 1 work

18 horses ……….. ?? mins ……….. 9/16 work

Time taken by 18 horses (i.e. 12 horses and 12 mules) = 24*(9/16)*(16/18) = 12 mins

Yes, the horses and mules together will take less than a quarter-hour to finish hauling the load.