Enigma of Alligation - Part 3

The unanswered question of the last post should have been an easy one.

Since we are finding the weighted average of Speeds, the only way that A₁ × w₁ or Speed × w can make any sense is when w’s, the weights, are Time.

Thus, the found ratio, 2 : 1, is that of Time travelled at the two speeds of 30 kmph and 60 kmph.

Since the total time travelled is 6 hours, time travelled at 30 kmph will be 4 hrs and the required distance will be 120 km.

Well, to move on, this post was not required and you would have already learnt the ‘behind the scene’ theory of Alligation. This post is just to have some fun with Alligation. You would be surprised by the number of ways in which Alligation can be performed in the above question. So let’s repeat the question and see the numerous ways ……

I cover a total of 240 km in 6 hours, partly at 30 kmph and rest at 60 kmph. What distance was covered at 30 kmph?

Method 1: ‘Alligating’ with Speeds

We have already seen this ……

Individual two speeds are 30 kmph and 60 kmph and average speed is 40 kmph. Thus, 

And we started this post by stating that this is the ratio of time and NOT of distances covered at the two speeds.

Method 2: ‘Alligating’ with reciprocal of Speeds

Let’s say we need to tweak around such that the ratio found is that of distances covered at the two speeds.

Does finding the weighted average of reciprocals of speed make any sense?

 

Now we will have to think, w being a variable for what, will ‘’ make any sense. Only when w will be distance, will the expression become a unit of time. Thus, the ratio found refers to the ratio of the distances travelled at the two speeds.

Method 3: Eliminating fractions in the above.

If the fractions unsettle someone, one way to get rid of it is as follows:

Covering entire 240 kms at speed of 30 kmph will take 8 hours.

Covering entire 240 kms at speed of 60 kmph will take 4 hours.

But covering them partly at 30 kmph and partly at 60 kmph requires 6 hours as per the data given in the questions.

‘Alligating’, these times,

Now, this appears very very simple, as compared to the earlier method. And we directly get the ratio of the distances covered at the two speeds. (You think why is it the ratio of the distance and not of time).

One could also solve them by ‘alligating’ a couple of ‘manipulated variables’, but hope you get the idea.

The Icing on the cake – The oral approach

And believe what, all the above are quite an inefficient way to solve this question. We are not referring to the equation way to solve (if these are inefficient, the equation way is suicidal)

The oral way to solve the question is ……

Assume the entire time, 6 hours, is travelled at 30 kmph. One would have covered 180 kms. Then, in this case, how did one cover an extra 60 kms?

Because part of the time was travelled, not at 30 kmph, but at 60 kmph, i.e. an additional speed of 30 kmph. At an additional speed of 30 kmph, an additional distance of 60kmph is covered. Hence time travelled at 60 kmph will be 2 hours. The rest of the time, 4 hours, is travelled at 30 kmph i.e. distance covered in this stretch will be 120 km.

Homework:

Solve each of the following in two ways, one by using Alligation (just to master the process) and secondly by the oral method explained as ‘icing on the cake’

1. A total of Rs. 180 is divided among 100 students such that each girl student receives Rs. 2 and each boy student receives Rs. 1.5. Find the number of girl students.

2. Amit takes a loan of Rs. 10,000, partly at 6% simple interest and rest at 8% simple interest. If he pays a total interest of Rs. 1500 in 2 years, find the amount taken on loan at 6%.

3. In CAT, each correct answer fetches 4 marks and each wrong answer fetches a penalty of 1 mark. If a student attempts 48 questions and scores 132 marks, find the number of correct and wrong answers.

4. In a zoo, there are only pigeons and rabbits. If there are a total of 200 heads and 560 legs, find the number of pigeons and rabbits.

Remember, equations are to be shunned like plague; while doing Alligation, try as many ways as possible; and finally to solve in the exam, master the oral approach. And in no case should you use a pencil, irrespective of the method being tried.

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