Example: Calculate 31% of 724

Solution: Method 1: The conventional method to calculate 31% of 724 is: 

\(\frac{{31}}{{100}} \times 724=244.44\) 

Method 2: In this method we split the percentage. This method gives better and faster results though it looks longer at the first glance.

31% of 724 = 30% of 724 + 1% of 724

= 217.2 + 7.24 = 224.44

Splitting of given percentage into two or more suitable parts will make calculation easier and faster. Suppose we have to calculate 34% of 99, it is clear that 34% of 99 is same as 99% of 34 and 99% of any number is 100% of that number – 1% of the same number, thus

99% of 34 = 100% of 34 – 1% of 34 = 34 – 0.34 = 33.66

Example: What is 22% of 431?

Solution: Method 1: 22% of 431

=\(\frac{{22}}{{100}} \times 431\,\, = 94.82\)

Method 2: 22% of 431 is 20% of 431 + 2% of 431

= 2 – 10% of 431 + 2 – 1% of 431 

= 2 – 43.1 + 2 – 4.31 = 86.2 + 8.62 = 94.82

Example: 2314 is what percentage of 7189?

Solution: As discussed in example 2, 2314 is approximately one - third of the number 7189, thus if we change 2314 by a number x, the denominator must be changed by 3x. Let’s subtract 189 from 7189 so that it becomes 7000 (a round number), then in order to maintain the same ratio we must subtract 189/3 = 63 from numerator.

Required fraction = \(\frac{{2314 - 63}}{{7189 - 189}} = \frac{{2251}}{{7000}} = 0.3217\)

Thus the required percentage = 32.17%. If we directly divide 2314 by 7189 by long division, we get 32.18%.

Example: 1341 is what %  of 5131? 

Solution: By simple division, \(\frac{{1341}}{{5131}} \times 100\) 

= 26.13%. But this will be a time consuming calculation. Let us try the method discussed above:

1341 is roughly one fourth of the number 5131, now when we subtract 131 from 5131 to make it a round number, we must subtract 131/4 from the numerator, hence the required fraction becomes \(\frac{{1341 - \frac{{131}}{4}}}{{5131 - 131}} = \frac{{1341 - 32.7}}{{5000}} \approx \frac{{1309}}{{5000}}\)= 0.2618

Alternate method:  Sometimes calculation can be made easier with the help of the numbers given in the choices. We have to make 1341 with the help of above data. In this method we can use choices at the best level, like just having a view it is clear 134 is between 20 – 30%. Now have a look at the choices if there is only one option, we can mark it without calculating further. If there are 2 or 3 options, then we can go closer.

2 – 10% = 2 – 513.1 = 1026.2

Now it is 1341 – 1026.2 = 314.8

6 – 1% = 6 – 51.13 = 306.78  

Means answer has to between 26 – 27%

For closer view we require (314.8 – 306.78) = 8.08

8.08 \( \approx \) 5.13 + .5131

= 0.1% + 0.03% = 0.13%

So the answer is very close to 26.13%

Example: 90 is how much % more than 70?

Solution: Percentage change = \(\left[ {\cfrac{{{\rm{difference}}}}{{{\rm{base}}}}} \right] \times 100\) 

As we know difference is 20, we should carefully identify the base number, generally base is the reference number from which comparison is done. 

Here base is 70 and the difference is 20. So the solution is \(\frac{{20}}{{70}} \times 100 = 28.56\)