 Hi and welcome to the session. Let's work out the following question. The question says if the bankers gain on a bill, the one ninth of the bankers discount the rate of interest being 10% per annum, find the unexpired period of the bill. Let's start with the solution to this question. Let T years be the unexpired period of the bill. Rupies S be the face value of the bill. R be the rate of interest per rupee per annum. Here, R is given to be 10% that is 10 by 100 that is equal to 1 by 10. Now, we calculate the bankers discount and bankers discount is equal to SRT that is S into 1 by 10 into T that is equal to S by 10. We also call this BT. Now, true discount that is TT is equal to SRT divided by 1 plus RT that is equal to S into T divided by 10 divided by 1 plus 1 upon 10 into T that is equal to ST upon 10 plus T therefore bankers gain is equal to BD minus TD that is bankers discount minus true discount that is equal to ST upon 10 minus ST upon 10 plus T but it is given that bankers gain is equal to 1 by 9 of bankers discount. Therefore, ST upon 10 minus ST upon 10 plus T is equal to 1 upon 9 into ST upon 10 or ST into 1 upon 10 minus 1 upon 10 plus T is equal to 1 upon 9 into ST upon 10 or 10 plus T minus 10 divided by 10 into 10 plus T is equal to 1 upon 90 because ST gets cancelled from both the sides. This further implies that T divided by 10 into 10 plus T is equal to 1 upon 90 this implies T is equal to 10 into 10 plus T divided by 90 this implies T is equal to 10 plus T divided by 9 this implies 90 minus T is equal to 10 this implies 8 T is equal to 10 and this implies that T is equal to 10 upon 8 which is further equal to 5 upon 4 S. So the unexpired period of the bill is 5 by 4 S. This is also our answer to this question. I hope that you understood the solution and enjoyed the session. Have a good day.