 Hi and how are you all today? I am Priyanka and we will be doing this question together. It says express the following in the form p by q where p and q are integers and q is not equal to 0. So basically we need to convert these decimal that is non-terminating repeating decimals in the form of p by q that is fraction. So let us proceed on with our solution of 1 by 1. The first part which is given to us is 0.6 bar. Now in this type of question let us take x as 0.6 bar. Now on multiplying both the sides by 10 we have 10x is equal to 6.666 which goes on. Let this be the first equation and let this be the second equation. Now if we carefully observe we can write 10x is equal to 6 plus this can be written as 0.6 bar whereas we can also have 10x is equal to 6 plus this can be written as x. Right? So further we can have 9x is equal to 6 as x will get subtracted from 10x and we are left with x is equal to 6 by 9 which on simplifying gives us the answer of x as 2 by 3. So x is equal to 2 by 3 which we took as 0.6 bar. So we converted 0.6 bar into a fraction that is in the form of p by q. Right? So this completes the first part of the question and the answer is 2 by 3. Proceeding on further to the second part we have the expression as 0.47 bar. So again let us take x equal to 0.47 bar which means 0.4777 which goes on. Now on multiplying both the sides by 10 we have 10x is equal to 4.7 bar. So we can write which means that the number which we have is 4.777 which goes on. Now we need to multiply x is equal to 0.47 bar with 100 also to get 100x is equal to 47.7 bar which means now we have 47.777 which goes on. Now if we carefully analyze and let this be the first equation and second equation. Now on subtracting from second we have that is the equation 100x minus 10x in the left hand side and 47.7 bar minus 4.7 bar on the right hand side. Further this will be 90x is equal to 43 which makes the answer of x is equal to 43 by 90. So the answer to the second part comes to us as 43 by 90. Proceeding on further to the last and final part. Now in the third part we are given 0.001 bar. Now we have three numbers so let this be x. And now we need to multiply both the sides by 1000 so that we are able to move our decimal to this position. So now we have 1.001 bar. Now this be the first equation and this be the second equation. Now on subtracting first equation from second equation we have 999x equals 1 which gives us the value of x as 1 divided by 999 and this is the answer of the last and final part. And this completes the question also. So I hope you enjoyed the session and bye for now.