 Hi and welcome to the session. Today we will learn about operations on rational numbers. First of all let us see addition of rational numbers. So let us learn how to add two rational numbers with same denominators. Let us add 5 by 9 and minus 3 by 9. We will add these two rational numbers using the number 9. First of all let us draw the number line. So here we have the number line and on this number line the distance between any two consecutive points is equal to 1 by 9. Now we need to add minus 3 by 9 to 5 by 9 so we will start from 5 by 9. Now as we need to add minus 3 by 9 which is a negative rational number so that means we will move to the left of 5 by 9. Now the numerator of minus 3 by 9 is 3. That means we will make 3 jumps from 5 by 9 to the left of 5 by 9. So by making 3 jumps we reach to the point 2 by 9. That means 5 by 9 plus minus 3 by 9 is equal to 2 by 9. Now we have one more way of adding two rational numbers with same denominators. So let us add 5 by 9 and minus 3 by 9. Here we will keep the denominator same that is 9 and we will add the numerators of both the rational numbers. So we get 5 plus minus 3 in the numerator so this gives us 2 by 9. That means 5 by 9 plus minus 3 by 9 is equal to 2 by 9. Thus while adding rational numbers with same denominators we add the numerators keeping the denominator same. Now let's see how to add two rational numbers with different denominators. Let us add two rational numbers minus 2 by 3 and 3 by 4. Here the denominators of these two rational numbers are different. So for this first of all we will express these two rational numbers with same denominators. So let us take the LCM of 3 and 4. The LCM of 3 and 4 is equal to 12. Now we will express these two rational numbers with denominator as 12. So minus 2 upon 3 can be written as minus 2 into 4 upon 3 into 4 which is equal to minus 8 upon 12. And 3 upon 4 can be written as 3 into 3 upon 4 into 3 which is equal to 9 upon 12. So from this we can say that minus 2 upon 3 plus 3 upon 4 is same as minus 8 upon 12 plus 9 upon 12. So let us add these two rational numbers. Now the denominator of these two rational numbers is same and we know how to add two rational numbers with same denominators. We will keep the denominator as 12 and we will add the numerators. So in numerator we get minus 8 plus 9 so this gives us 1 upon 12. Thus the sum of minus 2 upon 3 and 3 upon 4 is equal to 1 upon 12. Now let's see what is the additive inverse of a rational number. Suppose we have a rational number p by q then p by q plus minus p by q will be equal to p plus minus p upon q which is equal to 0. So from this we can say that minus p by q is the additive inverse of rational number p by q and p by q is the additive inverse of rational number minus p by q. Let's take an example. Suppose we have a rational number minus 2 by 5 then additive inverse of minus 2 by 5 will be equal to 2 by 5. Now let's move on to subtraction of rational numbers. While subtracting rational numbers the additive inverse of the rational number that is being subtracted to the other rational number. Let's take an example for this. Let us subtract the rational number minus 5 by 7 from minus 2 by 5. So we have minus 2 by 5 minus minus 5 by 7. Now to subtract minus 5 by 7 from minus 2 by 5 we will add the additive inverse of minus 5 by 7 to minus 2 by 5. So this will be equal to minus 2 by 5 plus additive inverse of minus 5 by 7 which is equal to 5 by 7. So this is equal to minus 2 by 5 plus 5 by 7. Now to add these two rational numbers we will express the two rational numbers with same denominators. For this we will take the LCM of 5 and 7. Now LCM of 5 and 7 is 35 so the equivalent fraction of minus 2 by 5 will be equal to minus 14 by 35 plus and the equivalent fraction of 5 by 7 will be 25 upon 35. So we get minus 14 upon 35 plus 25 upon 35 which is equal to minus 14 plus 25 upon 35 that gives 11 upon 35. That means minus 2 by 5 minus minus 5 by 7 is equal to 11 upon 35. Next we have multiplication of rational numbers. First of all let us learn how to multiply a rational number by a positive integer. Suppose we want to multiply the rational number minus 2 by 7 by the positive integer 3. We will multiply this with the help of a number line. So let us draw the number line first. So this is the number line. Here the distance between two consecutive points is 1 by 7. As we want to multiply minus 2 by 7 which is a negative rational number. So we will start from 0 and we will move to the left of 0. Now we are multiplying minus 2 by 7 by 3 so that means we will make three jumps of 2 by 7 to the left of 0. Like this. So by making three jumps we reach at the point minus 6 by 7 that means minus 2 by 7 into 3 is equal to minus 6 by 7. Now we use one more way to multiply a rational number by a positive integer. Now we need to multiply minus 2 by 7 by 3 so this will be equal to minus 2 into 3 upon 7. Here we have multiplied the positive integer 3 with the numerator of the rational number. So we get minus 2 into 3 upon the denominator of the rational number that is 7. So this is equal to minus 6 upon 7. From this we conclude that by multiplying a rational number by a positive integer we multiply the numerator by that integer keeping the denominator unchanged. Now let's see how to multiply a rational number by a negative integer. Let's multiply minus 2 by 7 by minus 3. For this we will multiply the numerator minus 2 by the negative integer minus 3 and we will keep the denominator as 7 itself. So this will be equal to 6 by 7. Next we will see how to multiply two rational numbers. Suppose we want to multiply minus 2 by 7 and 3 by 5 then minus 2 by 7 into 3 by 5 will be equal to for this we will multiply the numerators of both the rational numbers and we will get minus 2 into 3 upon. Now we will multiply the denominators of both the rational numbers that is 7 into 5. So this is equal to minus 6 upon 35. Now let's see the reciprocal of a rational number. Suppose we have a rational number p by q then the reciprocal of the rational number p by q will be equal to q by p. For example suppose we have a rational number minus 2 by 3 then the reciprocal of minus 2 by 3 will be equal to 3 by minus 2 which is equal to minus 3 by 2. Now we have a property that the product of a rational number with its reciprocal is always 1. Let us verify this property here reciprocal of minus 2 by 3 is minus 3 by 2. So let us multiply minus 2 by 3 by its reciprocal that is minus 3 by 2. So this will be equal to minus 2 into minus 3 upon 3 into 2 which will be equal to 6 upon 6 that is 1. Now let's move on to division of rational numbers to divide one rational number by the other rational number we multiply the rational number by the reciprocal of the other. Here is an example for this we need to divide minus 7 upon 15 by 2 upon 3. So minus 7 upon 15 divided by 2 upon 3 will be equal to minus 7 upon 15 into the reciprocal of 2 upon 3 that is 3 upon 2. So this will be equal to minus 7 into 3 upon 15 into 2 which is equal to minus 21 upon 30 which is equal to minus 7 upon 10 in its standard form. Thus in this session we have learnt operations on rational numbers with this we finish this session hope you must have understood all the concepts goodbye take care and have a nice day.