 Hello and welcome to the session. In this session we will discuss ratios. When we need to compare two quantities, we use ratios. We know that the ratio for two different comparisons may be the same and when we need to compare two quantities, the units must be the same. Suppose that we need to find the ratio of three meters to twenty centimeters. Now as you can see that the units of both these quantities are different, one is in meters and the other in centimeters. First we need to convert both the units to the same unit. That is we have three meters is equal to three into hundred centimeters that is three hundred centimeters. Thus we have the required ratio that is three meters is to twenty centimeters is given as three hundred is to twenty which is equal to fifteen is to one. So the ratio of three meters to twenty centimeters is fifteen is to one. Next is equivalent ratios. Different ratios can be compared with each other to know whether they are equivalent or not. For this what we do is we first convert the given ratios in the form of fractions. Then these fractions are converted to like fractions and if we get that the like fractions are equal, then the given ratios are equivalent ratios. Consider the ratios five is to twenty and three is to twelve. Let's see if these two ratios are equivalent or not. First we have the ratio five is to twenty. We convert this into fractions so we have five upon twenty. Then the ratio three is to twelve is written as three upon twelve. Now we need to convert them to like fractions. So for that we multiply both the numerator and denominator of this fraction by three. So we get fifteen upon sixty and here we multiply the numerator and denominator by five. So as to get fifteen upon sixty. Now as you can see that the like fractions that we have obtained are equal. So we say that the ratios five is to twenty and three is to twelve are equivalent ratios. Now let's consider the ratios one is to six and one is to four. Let's see if they are equivalent or not. Now one is to six is written as one upon six. One is to four is written as one upon four. Now we convert them to like fractions for that we multiply both the numerator and denominator of this fraction by four. So we get four upon twenty four and here we multiply both the numerator and denominator by six. So we get six upon twenty four. Now as you can see that these like fractions are not equal. So we have that four upon twenty four is less than six upon twenty four or you can say that one upon six is less than one upon four and thus the two ratios one is to six and one is to four are not equivalent. Then we have if the two ratios are equivalent then the four quantities are said to be in proportion. Like as you can see in this example the ratio five is to twenty and three is to twelve are equivalent. So we say that five twenty three and twelve are in proportion. Next we discuss keeping things in proportion and getting solutions. Suppose that we have some apples with us and we have that the cost of two kg of apples is rupees twenty then we need to find the cost of five kg of apples. This can be done in two ways. First we assume that the cost of five kg of apples is rupees x. Now we consider the proportions that is we have two is to twenty is equal to five is to x that is two upon twenty is equal to five upon x. So from here we get x is equal to twenty into five upon two. So this gives us fifty that is we get that the cost of five kg apples is rupees fifty. Now the other method of doing this is by using unitary method. In the unitary method we first find the value of one unit and then the value of the required number of units like as we are given that the cost of two kg apples is rupees twenty. So first we find out the cost of one kg apples which would be equal to rupees twenty upon two that is equal to rupees ten. Now we find out the cost of five kg of apples which would be equal to rupees ten into five and that is equal to rupees fifty. So the cost of five kg of apples is rupees fifty. So this completes the session. Hope you have understood the concept of ratios.