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?
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.