Enigma of Alligation – Part 2

Understand that Alligation is the same as weighted average, with just a rearrangement of the terms ......

. This is weighted average.

On cross-multiplying and then re-arranging,

A_avg × w₁ + A_avg × w₂ = A₁ × w₁ + A₂ × w₂

w₁ × (A_avg – A₁) = w₂ × (A₂ – A_avg)

. This is Alligation and is usually worked and depicted as the following visual

The above is given to explain to you that whenever in doubt about what does the ratio found using Alligation refer to, you should think of the weights in the weighted average formula.

Considering the question about apples and mangoes asked in the part 1, using allegation we get ……

This is the easier part. Now is the tough part. What does the ratio 1 : 1 refer to?

The question asks the ratio of the number of apples and mangoes. Would this be 1 : 1?

Whenever faced with such a doubt, think about the weighted average formula. We are finding the weighted average of the profit percentage. So ask yourself when would , A₁ × w₁ i.e. 100% × w₁ make some sense. Would 100% of (number of apples) make any sense? Profit percentage is a percentage of what?

Since profit percentage is a percentage of the cost, to make some sense, 100% has to be of the cost of apples. Thus, the ‘weights’ in this case would refer to the costs of apples and the costs of mangoes.

Now to complete the answer …… costs of apples and mangoes are equal. And since each mango costs twice that of an apple, the number of apples and mangoes will be 2 : 1.

Thus, the right answer has been arrived in an oral manner, without the need to put pencil to paper to write any equation.

Just to cement what we have learnt, let’s take up another example ……

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?

Without resorting to writing equations, two individual speeds are 30 kmph and 60 kmph. And the net (average) speed is i.e. 40 kmph.

Thus, using Alligation, the ratio of ‘weights’ of the two speeds is 2 : 1.

What does this ratio refer to? Since the question is asking distance travelled at 30 kmph, do we have to divide the total distance, 240, in this ratio of 2 : 1?

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