 Hi and welcome to the session, today we will learn about multiplication of fractions. So first of all let us see multiplication of a fraction by a whole number. To multiply a whole number with a proper or improper fraction, multiply the whole number by the numerator of the fraction keeping the denominator same. Now we already know that a proper fraction represents a part of a whole and in this numerator is less than the denominator. And improper fraction is a combination of a whole and a proper fraction and in this denominator is less than the numerator. Let's take an example for this. Let's multiply the whole number 2 with a proper fraction 3 by 5. So for this we will multiply the whole number 2 with the numerator of the fraction that is 3. So we will get 2 into 3 upon. Now we will keep the denominator same so we get 5. So this is equal to 6 by 5. Now to multiply a next fraction to a whole number convert the next fraction to an improper fraction and then multiply. Let us multiply 3 with 2 1 by 5. Now first of all we will convert 2 1 by 5 that is the next fraction into improper fraction. So this will be equal to 3 into 11 by 5. Now we will multiply this and we will get 3 into 11 upon 5 which is equal to 33 upon 5. Next we have fraction as an operator of now of represents multiplication of 15 means 1 by 5 into 15. Let's take an example. In a class 60 students by 3 of total students like to study math find out how many students like to study maths. So number of the students will be equal to 1 by 3 of total students that is 60. Now of means multiplication so this will be equal to 1 by 3 into 60 which is equal to 60 upon 3. So this will be equal to 20 in its waste form that means 20 students like to study maths. See the multiplication of a fraction by a fraction. Fractions are multiplied by multiplying their numerators and denominators separately to write the product as product of numerators upon product of denominators. For example let's multiply 3 by 8 with 6 by 4. So this will be equal to product of numerators that is 3 into 6 upon product of denominators that is 8 into 4. So this will be equal to 18 upon 32. Now this will be equal to 9 upon 16 in its lowest form. So 3 by 8 into 6 by 4 is equal to 9 by 16. Let's move on to our next topic value of the products. First of all we have when two proper fractions are multiplied the product is less than each of the fractions. Let's take an example. Suppose we have two proper fractions 2 by 7 and 5 by 7. So let's multiply them. So 2 by 7 into 5 by 7 is equal to 10 by 49. Now let's compare 10 by 49 and 2 by 7. Now we can write 2 by 7 as 14 by 49 as an equivalent fraction. So we need to compare 10 by 49 and 14 by 49. This is very clear that 10 by 49 is less than 14 by 49. This implies 10 by 49 is less than 2 by 7. Let's compare 10 by 49 and 5 by 7. 7 can be written as 35 by 49 as an equivalent fraction. So we need to compare 10 by 49 and 35 by 49. From this we can say that 10 by 49 is less than 35 by 49. Or this implies 10 by 49 is less than 5 by 7. Thus the product of two proper fractions that is 10 by 49 is less than each of the fractions 2 by 7 and 5 by 7. Next we have the product greater than each of the fractions. Let's take an example. Suppose we have two improper fractions 3 by 2 and 5 by 4. So let us multiply 3 by 2 with 5 by 4. So we will get 15 by 8. Now clearly 15 by 8 is greater than 3 by 2. 15 by 8 is greater than 5 by. Thus the product of two improper fractions that is 15 by 8 is greater than each of the fractions 3 by 2 and 5 by 4. Lastly we have product of a proper fraction is less than the improper fraction and greater than the proper fraction. Let's take an example for this. Suppose we have a proper fraction 2 by 5 and an improper fraction 7 by 3. Then 2 by 5 into 7 by 3 is equal to 14 by 15. And clearly 14 by 15 is less than 7 by 3 and 14 by 15 is greater than 2 by 5. So that means product of a proper and improper fraction is less than the improper fraction and greater than the proper fraction. Now let us try to solve this question. In this we are given that 2 by 5 into unknown fraction is equal to 14 by 13. We know that to multiply two fractions we multiply the numerators and denominators separately. So that means 2 into this number will be equal to 14 and we know that 2 into 7 is equal to 14. So we must have 7 over here. And now 5 into this number will be equal to 30 and we know that 5 into 6 is 30 that means here we must have 6. This is how we can solve such type of questions. With this we finish this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.