The other tutor counted the multiples of both 4 and 10 twice.  
Here's the way I do it.

First we find the number of multiple of 4 there are between those
two numbers.  Multiples of 4 come every fourth number, so to find out
how many we do this:

There are 303 numbers from 1 to 303.
We do not use the 196 numbers from 1 throught 196.

So to find out have many numbers there are between 197 and 303, we 
subtract 303-196 and get 107.

Multiples of 4 come every fourth number,  So can find out how
many there are by dividing 107 by 4 and get 26.75. That doesn't
make it to 27, so we use the largest whole number that doesn't
exceed that and so we round 26.75 down to 26.  That's the number
of multiples of 4 between 197 and 303.

Now some of those 26 multiples of 4 are also divisible by 10, so 
the only ones not counted among the 26 are the ones that are
multiples of 10 that are not multiples of 4.

Since any number of 100's is divisible by 4 we only need to look
at the last two digits to find the numbers which are 
multiples of 10 that are not multiples of 4.  They are these five
numbers: 10, 30, 50, 70, and 90.  So that means 210, 230,
250, 270, and 290 are the only ones between 197 and 303 that are
mutiples of 10 that are not multiples of 4.

So that's a total of 26+5 or 31.

Edwin