A quick way to compute weighted average of large numbers

Hi

Weighted average is an easy concept that gets test quite often in the quantitative section of the TANCET test.

Let us take a look at a question that requires you to compute the weighted average and then evaluate an alternate shorcut approach to computing the same.
Question
If the average marks scored by 30 students of school A in Class XII board exam is 1072 and that by 20 students of school B is 1084, then what is the average marks scored by the students of the two schools taken together?
1. 1078.6
2. 1076.8
3. 1075.9
4. 1080.2
5. 1077.

Correct Answer
Choice (2). 1076.8

Explanation
The conventional way to solve this question is to find the weighted average of the two marks as show below.
(30*1072 + 20*1084)/(30 + 20)
= (32160 + 21680) / 50
= 53840 / 50
= 1076.8.

As you realize, conceptually there was nothing difficult with solving this question. The only pain point was the calculation. And it might not be possible for most of us to solve this calculation in under 1.2 minutes.

Now let us look at an alternate, shortcut approach to this question.
Step 1: Subtract 1072 from both 1072 and 1084.
Step 2: Compute the weighted average of the resulting numbers. i.e. ,0 and 12
The weighted average of 30 students scoring 0 and 20 students scoring 12
= (30*0 + 20*12) / (30 + 20) = 240 / 50 = 4.8
Step 3: Add 1072 to the weighted average obtained in step 2. i.e., 1072 + 4.8 = 1076.8

Therefore, whenever you have to compute the weighted average of two large numbers, subtract the smaller of the two numbers from both the numbers and then compute the weighted average and then add back the subtracted number to the result.