How to make rate work problems easier

Example 1: If 10 workers complete a work in 5 days working 8 hours a day, how much work will be done by 6 workers in 10 days working 2 hours a day? Here is what it looks like:   10 workers……………..5 days …………….. 8 hours ……………1 work 6 workers………………10 days …………… 2 hours …………… ? work We need to find the amount of work done, so we start with the work done in the first case and then multiply it by the respective ratios: Work done = 1 * (6/10) * (10/5) * (2/8) = 3/10 We multiply by 6/10 because number of men decreases from 10 to 6. The work done will reduce, so we multiply by 6/10 (the fraction less than 1). We also multiply by 10/5 because number of days increases from 5 to 10. Because of this, the work done will increase, so we multiply by 10/5 (the fraction more than 1). We also multiply by 2/8 because number of hours decreases from 8 to 2. Because of this, the work done will decrease, hence, we multiply by 2/8 (the fraction less than 1). So the process is super simple – start with what you need to find out, say x, and multiply it by the ratio of each thing that changes from A to B. Whether you multiply by A/B or B/A depends on whether with this change increases or reduces x. If x increases, you will multiply by the fraction that is greater than 1, however if x decreases, you will multiply by the fraction that is less than 1. On this same concept, let’s look at the 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? Statement 1: Mules work more slowly than horses. Statement 2: 48 mules can haul the same load of lumber in 16 minutes. Let’s see what data we have in the question stem: 16 horses …….. 24 mins ………. 1 work 12 horses …….. 14 mins ………. ? work Work done = 1*(14/24)*(12/16) = (7/16)th of the work We multiply by 14/24 because if the time taken to do the work decreases, the work done will also decrease. 14/24 is less than 1 so it will decrease the work done. We also multiply by 12/16 because if the number of horses decreases, the work done will also decrease. 12/16 is less than 1 so it will decrease the work done. All in all, we now know that 12 horses complete 7/16th of the work in 14 mins. So there is still 1 – 7/16 = 9/16 of the work left to do. Now let’s review the two statements. Statement 1: Mules work more slowly than horses. This statement doesn’t give us any figures, so how can we analyse it mathematically? What we can do is find the range in which the time taken by all the horses and mules together will lie according to this statement. Case 1: When mules work at a rate that is infinitesimally smaller than the rate of horses. In this case, 12 mules are equivalent to 12 horses. So we have a total of 12 + 12 = 24 horses working together to complete (9/16)th of the work. 16 horses …….. 24 mins ………. 1 work 24 horses ……… ? mins ………. 9/16 work Time taken = 24*(16/24)*(9/16) = 9 mins Since the mules are slower than the horses, the time taken to complete the work will be more than 9 minutes. How much more than 9 minutes, we do not know. Now look at the flip side: Case 2: When the mules work at a rate close to 0. If the mules work slower, time taken will be more till the point when mules work so slowly that they do almost no work. 16 horses …….. 24 mins ………. 1 work 12 horses ……… ? mins ………. 9/16 work Time taken = 24*(16/12)*(9/16) = 18 minutes Therefore, depending on how fast/slow the mules are, the time taken to do the rest of the work could be anywhere from 9 minutes to 18 minutes. Therefore the time taken could be either less or more than 15 minutes – this statement alone is not sufficient. Statement 2: 48 mules can haul the same load of lumber in 16 minutes. We now know exactly how fast the mules are, so this must be sufficient to say whether the time taken to do the rest of the work was less or more than 15 minutes – we don’t need to actually find the time taken here – therefore, the answer is B, Statement 2 alone is sufficient. However, if you would like to find out for practice, just find the equivalence between the horses and the mules first. To haul the load in 16 minutes, we need 48 mules To haul the load in 24 minutes, we need 48 * (16/24) = 32 mules So 32 mules are equivalent to 16 horses (because 16 horses haul the load in 24 minutes). This means that 2 mules are equivalent to 1 horse, and 12 mules are, therefore, equivalent to 6 horses. So now, in effect we have a total of 12 + 6 = 18 horses, and the situation now becomes this: 16 horses …….. 24 mins ………. 1 work 18 horses ……… ? mins ………. 9/16 work Time taken = 24*(16/18)*(9/16) = 12 minute – less than a quarter-hour to finish the work.