 Hello and welcome to the session. In this session we shall discuss how to find ratio of fractions and unit rate of different quantities. Suppose one is making a chocolate cake and the ingredients that are used to make one pound of chocolate cake are 2 cups of flour, 1 chocolate bar, 3 tablespoons of sugar and 4 cups of milk. Now we know ratio is a comparison of two quantities using the region. If two numbers are x and y, then ratio of x to y is written as x by y or x is to y. For one pound of chocolate cake, one needs 2 eggs and 3 spoons of sugar. So here, the ratio of 2 sugar and 1 cake is 2 is to 3. The order in ratio is absolutely essential. That is, x is to y is different from y is to x. As in the above example, ratio of x to sugar in one's cake is 2 is to 3 and ratio of sugar to egg is 3 is to 2. We also know that like a fraction, a ratio should also be reduced. Reducing a ratio is called simplifying. Now in the above example, if one wants to make 2 pounds of chocolate cake, he needs 4 eggs and 6 spoons of sugar. Then the ratio of x to sugar becomes 4 is to 6. That is, 4 by 6 which becomes 2 by 3 or we can also write it as 2 is to 3. Now we will find the ratio of fractions. It means when both numerator and denominator are given in fractions. That is, if we have a fraction of the type a upon b whole upon c upon d, here both numerator and denominator are fractions. And we can write it as pound b divided by c upon d. Now we invert the second fraction to change in multiplication. So we have a upon b into d upon c which is equal to a d upon b c. So a upon b whole upon c upon d is equal to a d upon b c. Let us take an example. Suppose n gets 1 part of 7 parts of a tomato pizza and 2 parts of 5 parts of the onion pizza. Now the fractions of parts of pizza that n gets is 1 by 7 and 2 by 5. As we know that ratio is a comparison of two quantities using division. So the ratio of parts of tomato pizza and onion pizza that n gets is 1 by 7 upon 2 by 5. Now here both numerator and denominator are fractions. So we write it as 1 by 7 divided by 2 by 5 which can be written as 1 by 7 multiplied by 5 by 2. And we get 5 by 14. There are some points that should be kept in mind to solve such problems. First is every integer can be written in fraction form. It means every integer has 1 in its denominator. For example 3 can be written as 3 upon 1. Second we have if there is a mixed fraction 2 and 1 by 3. We can simplify it as 1 numerator 2 multiplied by 3 that is 6 and add numerator 1 that is 7. And the denominator remains same as 3. So we have the fraction 7 upon 3. Let us take an example. Simplify 3 and 1 upon 2 is to 3 and 1 upon 3. And we write it as 2 and 1 upon 2 upon 3 and 1 upon 3. Here we have mixed fraction. So we convert it into simple fraction using the above result. And we get 2 multiplied by 2 that is 4 plus 1 that is 5 in the numerator. And the denominator 2 remains the same whole upon 3 multiplied by 3 that is 9 plus 1 that is equal to 10 in the numerator. And we have the denominator 3. And this is equal to 5 upon 2 into 3 upon 10 that is 15 by 20 which can be written as 3 by 4 or 3 is to 0. So we say 3 is to 0 is the required ratio. Now we are going to study weight. A ratio that compares 2 quantities of different units is called weight. For example, 120 miles upon 3 hours. Here miles and hours are different units. Now we will see what a unit weight is when a weight is simplified so that it has denominator of 1 unit is called unit weight. For example, 120 miles by 3 hours. Here in denominator we have 3. Now to find the unit weight we need 1 in the denominator so we divide both numerator and denominator by 3. And we get 120 divided by 3 whole upon 3 divided by 3 which is equal to 40 upon 1 and is written as 40 miles per hour. Now let's take another example. If Sam spends 20 of dollars in 4 weeks how much money does he spend there in 1 week? So here we have to find the unit weight of 20 of dollars in 4 weeks that is 20 of dollars by 4 weeks. And here we have 4 in the denominator and to find the unit weight we need 1 in the denominator so we divide both numerator and denominator by 4. And we get 28 divided by 4 upon 4 divided by 4 which is equal to 7 by 1 that is 7 dollars per week. Now let us learn how to find unit weight of fractions. For this consider an example. If a person works 1 by 2 miles in each 1 by 4 hours find the unit weight for unit weight is equal to 1 by 2 miles upon 1 by 4 hours here to make the denominator as 1. We divide both numerator and denominator by 1 by 4 that is 1 by 2 divided by 1 by 4 whole upon 1 by 4 divided by 1 by 4. And therefore we get 1 by 2 into 4 by 1 upon 1. Here we have inverted the second fraction to convert into multiplication and therefore we get 4 by 2 upon 1 that is 4 upon 2 which is equal to 2 so we have 2 miles per hour. So the unit weight is 2 miles per hour. This completes our session. Hope you enjoyed this session.