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For the question, "The average of $30$ numbers is $40$ and that of other $40$ numbers is $30$. Find average of all numbers.", the options given are:

  • $35$
  • $34$
  • $34\frac{1}{2}$
  • $34\frac{2}{7}$

I solved it like, $\frac{30\cdot40+40\cdot30}{70}=34\frac{2}{7}$

But I am trying to learn averages without using the formula.

If the count of both sets of numbers had been $30$, then answer would have been $35$.

Now weightage is more towards $30$ average, so answer would be less than $35$.

But that eliminates only one option.

How can I eliminate other options?

barak manos
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aarbee
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  • Hope I'm not being rude, but this just seems totally misguided. How are you going to really learn averages without a formula? Life is not (usually) a multiple choice test. – David Sep 11 '14 at 05:04
  • Hope I am not being rude, but life is not based on formulas. We must be able to see through simple things without the formulas. It is strange that it is coming from me, who needs help in this basic question, but I am hopeful that I am on the right track of learning and that things would be smoother with time. – aarbee Sep 11 '14 at 06:04
  • Please note that I already knew how to solve this question with formula. I am only trying to build on that. Because in real life, many people are able to solve it mentally. – aarbee Sep 11 '14 at 06:06
  • So, I still get the impression that you are looking for a formula, if perhaps a short cut rather than the standard one. I can't think of anything that helps in your example, but here is a different example to illustrate what I mean: to find the average of $112,107,111,102$ you could deduct $100$, find the average of four small numbers, then add $100$ back again. I really do not believe that anyone could find the average of these numbers without some kind of formula. Certain things in life are, and must be, based on formulas. PS you weren't being rude, please do not worry about it. – David Sep 11 '14 at 06:21
  • Exactly this is what I was looking for. Assumed mean. In my case, if I take it to be 35, I am left with only 10 numbers, whose values are 30 each. So, I need to subtract (5*10)/70 from 35 (5 being difference of 30 from 35). I got the answer. Thanks. Ps- your ps was quite warm. It made my day. Thanks again. – aarbee Sep 11 '14 at 11:47
  • Maybe I should have used different wording for post. Instead of formula, I might have asked a method other than the tradional method. Or perhaps something else, which is quicker. – aarbee Sep 11 '14 at 11:51

1 Answers1

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The key formula to remember is $$\mathrm{Average} \times \mathrm{Number} = \mathrm{Total},$$ or "ANT" for short. "Number" refers to the number of numbers added up. So, if the average of 30 numbers is 40, then $$40 \times 30 = \mathrm{Total} = 1200.$$ In other words, the sum of those 30 numbers is $1200$. Similarly, the total of the other 40 numbers is also $30 \times 40 = 1200$. So the total of all $30 + 40 = 70$ numbers is $1200 + 1200 = 2400$. So if we add up $70$ numbers and got a total of $2400$, what is their average?

If you don't want to use this formula, then you can very easily be led astray, so I don't recommend trying to take shortcuts. You can perhaps eliminate some of the answer choices as a checking mechanism, but in the end, you will want to solve the problem formally, using mathematics.

heropup
  • 135,869
  • Thanks for the answer. But sorry to disappoint you, I am not able to read this answer completely. Perhaps there is some formatting that my phone doesn't support. I'll go to a cafe in evening and would read it then. Thanks. – aarbee Sep 11 '14 at 06:09
  • -1. You only mentioned what I had already mentioned in the post. – aarbee Sep 11 '14 at 11:49