 Hello and welcome to the session. I am Asha and I am going to help you with the following question that says, 150 workers were engaged to finish a job in a certain number of days. Four workers dropped out on the second day, four more were dropped out on the third day and so on. It took eight more days to finish the work, find the number of days in which the work was completed. Let us now start with the solution and let 150 workers complete the work in n days. Therefore, one worker shall complete the work in 150 into n days and this further implies that one worker completes one upon 150 into n work in one day and therefore 150 workers complete 150 upon 150 into n work one day. Now on the second day, four workers were dropped therefore on second day work done by 146 workers is equal to 146 upon 150 into n and on the third day four more workers were dropped therefore on third day work done by 142 workers will be equal to 142 upon 150 in. Now in this manner four workers are dropped daily therefore it will take eight more days to finish the work and thus we can say that number of days to finish the work when four workers are dropped daily is equal to n plus eight and the work done on the first days 150 upon 150 into n work done on the second day is 146 upon 115 into n work done on the third days 142 upon 115 into n plus so on and the number of days are n plus 8 so the number of terms are also n plus 8 and this is equal to 1. Now taking 1 upon 150 into n common we have 150 plus 146 plus 142 plus so on up to n plus 8 terms so equal to 1 or we have 150 plus 146 plus 142 plus so on up to n plus 8 term is equal to 150 into n. Now let us find the sum of this AP sequence now where the first term is 150 and the common difference is minus 4 so the sum of these n plus 8 terms which are n plus 8 in number will be equal to n plus 8 upon 2 into 2 times the first term plus the number of terms minus 1 into the common difference which is minus 4 this is equal to 150 into n or we further up n plus 8 into 300 plus n plus 7 into minus 4 is equal to 300 n or this is further equal to n plus 8 into 300 minus 4 in minus 28 is equal to 300 n now opening the brackets we have 300 in minus 4 n square minus 28 in plus 2400 on multiplying 8 with minus 4 in we have minus 32 in minus 224 this is equal to 300 in now in cancelling we have 4 n square plus 60 in minus 2176 is equal to 0 now taking 4 common if n square plus 15n minus 544 is equal to 0 which further implies that n square plus 15n minus 544 is equal to 0 now by splitting the middle term this can further written as n square plus 32n minus 17n minus 544 is equal to 0 now taking n common from the first two terms and minus 17 from the last two terms this can further written as n into n plus 32 minus 17 into n plus 32 is equal to 0 or we have n plus 32 into n minus 17 is equal to 0 now we know that if the product of two numbers is equal to 0 then at least one of them is 0 so this implies either n is equal to minus 32 or n is equal to 17 now minus 32 is rejected since the number of days cannot be negative and hence n is equal to 17 and therefore after dropping 4 workers each day the number of days in which the work was completed is equal to n plus 8 that is 17 plus 8 which is equal to 25 hence our answer is 25 days so this completes the session take care and bye for now