 Hello and welcome to the session. In this session, we shall learn how to identify the constant of proportionality that is unit rate in table, equation and variable descriptions. Let us consider the following table. Now let us see the ratio that is y is to x for each column. Now in the first column we have y by x is equal to 2 by 3 as y is equal to 2 and x is equal to 3. In the second column we have y by x is equal to 4 by 6 that is 2 by 3 and in the third column y by x is equal to 6 by 9 as the value of y is equal to 6 and x is equal to 9 and we get 2 by 3 again. It means the ratio y by x is same in each column. So we say that all the ratios are proportional equal to 2 by 3. This 2 by 9 3 is a constant term so it is called constant of proportionality. The constant of proportionality we mean that the ratio between 2 quantities is always same for example if y and x are 2 quantities if y by x is equal to k where k is constant then k is called the constant of proportionality and this relationship can also be written as y is equal to k into x and if such relationship exists then we say that the 2 quantities are directly proportional. We should also note that constant of proportionality often represents the rate. Let us take an example. The following table shows how the number of who has ever taken in high school depends upon the number of weeks she has attended school and we have to find how many tests that Shania took in a week and we know that the constant of variation which often represents a weight will tell us the answer. To find the constant of proportionality we divide total number of tests by number of weeks and we get and we get the constant that is y upon x as we denote the total number of tests by y and number of weeks by x so in column 1 we get y upon x that is 4 upon 2 and in column 2 we get 12 upon 6 and in both the cases we can see that we get the constant of proportionality as 2. So the constant of proportionality thus we can say Shania takes 2 tests per week. Now we can learn how to identify the constant of proportionality using equation. We know that the equation when two quantities are proportional is given by y is equal to k into x where k is constant of proportionality. Let us take an example. Write equation of proportionality for given statement and the statement is David adds $3 to the savings account every week. Now let us find the table for the given statement. In first week David has $3 in his savings account. In second week he adds $3 more so total of $6 and so on and we get for the first week David has $3 in his account. In the second week he has $6. In the third week he has $9. In the fourth week he has $12 and so on. Now let us see the ratio that is dollar per week and we get 3 upon 1 which is equal to 6 upon 2 which is equal to 9 upon 3 which is equal to 12 upon 4 and we see that each ratio is equal to 3. Now we see if dollar is d week is w and the value of k is equal to 3 then we get the equation d upon w is equal to 3 which implies that d is equal to 3 into w where 3 is the constant of proportionality. We can also use diagrams for finding unit rate. For example it takes 20 minutes to make 5 sandwiches. How many sandwiches can be made in 40 minutes find its unit rate. With the help of this diagram we can find how many sandwiches can be made in 1 minute. We know that the total time taken is 20 minutes. The bar is divided into 5 equal parts which shows that 5 sandwiches are made in 20 minutes so to make 1 sandwich will be equal to 20 by 5 that is 4 minutes. So now our diagram shows time taken for making 1 sandwich that is 4 minutes. Now we need to find that how many sandwiches can be made in 30 minutes. From this diagram we can see that 5 blocks are made in bar which we get by total time taken divided by time to make 1 sandwich. So in 30 minutes we can make 40 divided by 4 that is 10 sandwiches. Now we shall learn how to identify the constant of proportionality by verbal description. Let us take an example. Describe the proportional relationship between the two quantities verbally and the statement is test writes her bike 12 miles per hour where we are relating the number of hours test writes her bike to the distance she has travelled. We are given in 1 hour she travels 12 miles that is in second hour she would again travel 12 miles it means in 2 hours she travels 24 miles that is 12 miles plus 12 miles which gives us 24 miles. Similarly in 3 hours she travels 36 miles in 4 hours she travels 38 miles and we see that each ratio that is 12 upon 1 is equal to 24 upon 2 is equal to 36 upon 3 is equal to 48 upon 4 which is same and is equal to 12. So there is a proportional relationship between the two quantities so we have the equation m is equal to 12 into h where m is miles is the number of hours and thus we can say the unit rate is 12. This completes our lesson. Hope you enjoyed this lesson.