 Hello friends welcome again to another session on number systems so far we were doing conversion of decimal numbers of terminating decimal numbers and pure form of non-terminating recurring decimal numbers in this session we are going to do conversion of decimal numbers which are mixed type non-terminating recurring right so what do I mean so in the earlier sessions we saw in the earlier sessions in the earlier sessions we saw that there are two types of decimal representation one is terminating terminating decimal representation and the other one is non-terminating non-terminating and repeating or recurring whichever way you want call it right recurring decimal representation so terminating example was let us say 0.25 another was 0.325 and things like that non-terminating would be 0.333 and it continues or 0.313131 like that right now in non-terminating recurring also we had two two types one is pure form pure form where it was all the decimals after the decimal point was repeating so for example this or 0.37 bar correct so all the digits will repeat now there is a mixed form also mixed form is you know what is mixed form so few decimals or few digits after the decimal will will not be repeating others will be repeating for example 0.223 bar that is this is nothing but 0.223223 so on and so forth right now in the previous two sessions we covered this fully how to convert terminating decimal numbers into p by q form we also followed covered this form your form how to convert in this session we are going to deal with mixed mixed decimal non-terminating recurring decimals so let us let us see how so let me take an example first so let me let me say I have to convert 0.231 bar into in the form of what p upon q our rational number we know that this is a rational number so it must be represented in the form of p by q so now look at the process carefully now if you if you see in the previous case when we were dealing with the pure form there let us say the number was simply 0.31 bar so in that case what did we used to do we used to count the number of repeating digits so in this case there are two isn't it n is equal to and hence we used to multiply first of all we will be calling this as x and then we will multiply the given number by one followed by how many zeros two zeros so one zero zero x in the left hand side and then on the right hand side it was again if you multiply 0.31 which is nothing but 0.313131 like that by 100 you will get 31.31 bar isn't it now what did we used to do you we used to subtract these two equation this is one and this is two so we used to get what 100 x minus x in the left hand side will be equal to 31.31 bar minus 0.31 bar and we used to get 31 right and hence 99 x was equal to 31 so hence you can find out x is equal to 31 upon 99 so easy to find out now in this case you can't really multiply this by 100 so here again if you see n is equal to two how many digits are getting repeated two right but if you multiply this let us say this is x but if you multiply this by 100 what will you get 100 x will be equal to 23.131313 so on and so forth isn't it now if you try to subtract let us say if you want to subtract 100 x 100 x and minus x and this is 0.2313131 like that it will be very difficult to subtract these two parts isn't it so it will not work it will not work so what do we do in this case first of all first of all shift the decimal to the position which position exactly before the repeating where the repeating starts right so two is not getting repeated so I will try to shift this decimal just next to two and before three right how can I do that so you can simply see you can multiply this by 10 right so hence if you multiply that by 10 you will get 10x is equal to 2.31 bar isn't it now this is my first step now I will repeat the process which I have learned in the previous session that is this one now again now count n n is equal to how much 2 how many digits are getting repeated 2 right now you multiply this now you call this equation as equation number 1 now multiply equation number 1 by 1 followed by 2 zeros right so 100 into 10x is equal to 100 into 2.31 bar so hence you'll get 1000 x is equal to 231.31 bar this is my second equation now you operate 2 minus 1 that means 1000 x is the left hand side minus 10x is equal to 231.31 bar minus 0.31 bar okay so what will you get you will get 990 x is equal to 231 isn't it 990 is I'm sorry no not 2.31 this will be this will not be 0.31 it will be 2.31 bar right so this is simply 2.31 bar so if you subtract it is nothing but 231 minus 2 or I'll write the full steps so hence you will get 231.31 bar minus 2.31 bar is nothing but so this 0.31 bar will cancel because of this you know so this both the 0.31 bar after the decimal will just get eliminated so it will be left with 231 minus 2 which is equal to 229 okay so hence x is 229 upon 900 sorry not 900 990 990 understood understood how to find it out and if you can simplify this further you should simplify this fraction now let us take another example to check whether we learned it or not so let us say now x is equal to let us say you have to you have to convert 3.32123 bar this is the decimal number right this is non-terminating but repeating right you have to now convert this into which form p by q how to do first of all what is the step take the decimal point just next to the repeating digit so one is the first repeating digit so I'll have to shift this decimal here what to do how to do so you can shift that by two places by doing what multiplying by 100 so you'll get 332.123 bar now by this step I have ensured that after the decimal all the digits must repeat so now you see after the decimal all the digits are repeating now what to do next now you follow the same process so what is n here n is 3 because three digits are repeating so hence I will multiply equation number one by one followed by three zeros so it is nothing but hundred x into thousand is equal to 332.123 bar into thousand so LHS will become 1 0 0 0 0 x is equal to 332 123.123 bar now this is my equation number two now do what you know 2 minus 1 that means 1 0 0 0 0 0 0 x minus 100 x is equal to 332 123.123 bar minus what is that this is 332.123 bar now if you see we have we have equated or let's say you know we have got the same repetition after the decimal so that gets eliminated so it is simply nothing but 9 9 9 0 0 okay triple 9 99,900 right x is equal to what is it so it will be nothing but 332 123 123 be careful with the calculations because you may lead to it may lead to some calculation error so this is the most simplified form so 3 minus 2 is 1 12 minus 3 is 9 this becomes 10 10 minus 3 is 7 so 1 3 3 right so just check whether we have got the right calculation 1 plus 2 is 3 so how to check check 1 plus 2 is 3 right now 9 plus 3 is 12 carry 1 1 plus 7 8 8 plus 3 is 11 that is 1 here carry 1 again so 1 plus 1 is 2 3 and 3 so this is correct calculation so what is x now x is nothing but 3 3 1 7 9 1 upon 9 9 9 double 0 okay double 0 so now if you see the numerator and denominator both are multiples of 3 right so if you and for that matter yes it is multiple of 3 only not 9 so it is 11 0 I am now dividing the numerator by 3 so it is 11 and 0 and then 5 and then 9 and 7 okay and the denominator will be 3 3 3 double 0 right let us check if the numerator again is divisible by 3 no it's not no it's not so so let's check the calculation once again how to check multiply the numerator what you got by 3 to see whether you got you you get this number or not so 3 times 7 is 21 carry 2 3927 plus 2 is 9 carry 2 again 3 5s are 15 plus 2 17 carry 1 3 0 or 0 plus 1 is 1 and 3 1s are 3 and 3 1s are 3 so looks like the calculation is correct okay so you must also have a habit of checking the calculation so this is my representation of what this was x and what was x x was 3.3 2 1 2 3 bar so if you do this calculation you will get 3.3 2 1 2 3 bar okay so please check the problem solving sessions for more such problems and you know once you follow the steps you repeat it for some time you try some 56 problems you will be able to get a command over the process I hope you understood the process of converting a mixed non-terminating repeating decimal into p by q form thank you.