 Hello and welcome to the session. In this session, we will discuss about ratio. Ratio is a relationship between two quantities, same kind. The ratio of say P and Q, these are the two quantities of same kind. This is expressed as P is to Q or we can also write it as P upon Q. So, we can say ratio of one quantity to another of same kind is a fraction of which the numerator and denominator are respectively the measures of these quantities and they are expressed in the same unit. These quantities P and Q are the terms of the ratio P is to Q. The ratio the antecedent like this P is the antecedent and the second term of the ratio is called the consequent. So, in this case this Q which is the second term of the ratio P is to Q is the consequent. To compare two quantities they must be expressed in the same unit. The ratio no unit since a ratio expresses the number of times one quantity contains another and so every ratio is an abstract quantity and thus we can say that a ratio has no unit. The value of a ratio say a is to b which could be written as a upon b is unaltered if both the antecedent and the consequent are multiplied or divided by the same number. So, a upon b could be expressed as m a upon m b that is we multiply the antecedent and the consequent by m. So, they would be equivalent also when we divide the antecedent by m and the consequent also by m then also we obtain a upon b or you can say a is to b. Next we discuss ratio of equality or inequality. Now, this ratio P is to Q could be a ratio of greater inequality. The ratio P is to Q or you can say P upon Q is greater than unity or we can also say that the antecedent of the ratio that is P is greater than the consequent of the ratio which is Q. So, if we are given a ratio say P is to Q then we would say that this ratio is a greater inequality if this is greater than one or the antecedent of the ratio is greater than the consequent. Then in the same way this ratio P is to Q would be a ratio of less inequality if the given ratio that is P is to Q or P upon Q is less than one or we can also say that the antecedent which is P is less than the consequent which is Q. In the same way the given ratio would be a ratio of equality if the ratio P is to Q or you can say P upon Q is equal to one the antecedent P is equal to the consequent Q. Let us now discuss one theorem which says that a ratio of less inequality is increased and a ratio of greater inequality is decreased by adding the same number to both its terms. Let us now prove this theorem just for we assume let P upon Q be any ratio now adding to both the terms of the ratio P upon Q we get the new ratio as P plus X upon Q plus X. We know that the ratio is of less inequality if its antecedent is less than the consequent and it is of greater inequality if its antecedent is greater than its consequent. So this means we can say that the ratio P upon Q is of less inequality if P is less than Q greater inequality P is greater than Q. Now consider upon Q plus X minus P upon Q. Now taking LCM we get this is equal to Q plus XT whole into Q in the denominator and in the numerator we have P plus XT whole into Q minus P into Q plus XT whole. So this is equal to plus QX minus PQ minus PX and this whole upon Q into Q plus XT whole. This PQ minus PQ cancels and we have X into Q minus P the whole this one upon Q into Q plus XT whole. This is P plus X upon Q plus X minus P upon Q. That is we have upon Q plus X that is the new ratio minus the given ratio which is P upon Q is equal to X into Q minus P the whole upon Q into Q plus XT whole. Now if would be greater than P then this means that Q minus P is greater than 0 that is this term would be positive and hence the whole term would be positive. So if Q is greater than P then P plus X upon Q plus X minus P upon Q would be positive or we can say that P plus X upon Q plus X is greater than P upon Q. This is the case when we have Q greater than P and in case we have Q less than P that is Q minus P is less than 0 that is this term would be negative and hence the whole of this term would be negative. So if Q minus P is less than 0 then in that case upon Q plus X minus P upon Q would be less than 0 or we can say that P plus X upon Q plus X would be less than P upon Q. We have two cases that is P plus X upon Q plus X would be greater than P upon Q if Q is greater than P and we know that when Q is greater than P or you can say P is less than Q then the ratio P upon Q is of less inequality and according to this theorem we have that the ratio of less than equality is increased when we add same number to both the terms of the given ratio and from this we can conclude that the ratio of less than equality is increased. Also we have that P plus X upon Q plus X is less than P upon Q if Q is less than P. Now Q less than P or say P greater than Q means that the ratio P upon Q is of greater inequality and according to the theorem we have that a ratio of greater inequality is decreased when we add the same number to both the terms of the given ratio and so this means that the ratio of greater inequality is decreased when we add same number to both the terms of the given ratio. So hence proof the theorem let's now discuss another theorem according to which we have that a ratio less inequality is decreased a ratio of greater inequality is increased by subtracting from both its terms any number which is less than each of these terms. Now start with the proof of this theorem first of all we take let P upon Q be any given ratio now we take X to be less than the term P and also it is less than the term Q. Now subtracting from both the terms of ratio P is to Q we have P minus X upon Q minus X as the new ratio. Now consider upon Q minus X minus P upon Q now taking LCM we have Q minus XT whole into Q now in the numerator we have P minus XT whole into Q minus P into Q minus XT whole. So further we have PQ minus QX minus PQ plus PX this whole upon Q minus XT whole into Q now PQ cancels with minus PQ and here we have X into P minus Q the whole upon Q into Q minus XT whole. So we now have minus X minus P upon Q is equal to X into P minus Q the whole upon Q into Q minus XT whole. Now if we have P greater than Q so this means that P minus Q is greater than 0 that is this term would be positive this term is already positive this term is also positive. Now as we have less than Q so this means minus X will also be positive so if we have P greater than Q then upon Q minus X minus P upon Q would be greater than 0 that is this term would be positive or you can say that minus X upon Q minus X would be greater than P upon Q given that P is greater than Q so we get this if P is greater than Q or Q is greater than X and also P is greater than X. Now if in this term we have is less than Q so this means that P minus Q is less than 0 that is this term P minus Q would be negative X is positive Q is positive and Q minus X is also positive as X is less than Q. So this means when P minus Q is less than 0 so this term that is we now have P minus X upon Q minus X minus P upon Q is less than 0 or you can say that P minus X upon Q minus X is less than P upon Q if we are given that P is less than Q Q is greater than X and also P is greater than X. Hence we get that P minus X upon is greater than P upon Q if P is greater than Q Q is greater than X or you can say X is less than Q and also P is greater than X. We know that as in the ratio P is to Q and T is equivalent that is P is greater than the consequent therefore this is the ratio of greater inequality and from this theorem we have that the ratio of greater inequality is increased by subtracting from both of its terms any number which is less than each of these terms. So this results shows that the ratio of greater inequality is increased when a number X is subtracted from each of the terms of the given ratio P upon Q. Also we have P minus X upon Q minus X is less than P upon Q if P is less than Q Q is greater than X and P is greater than X. We already know that P less than Q is a ratio of lesser inequality and from the theorem we have that the ratio of less inequality is decreased by subtracting from both of its terms any number which is less than each of the given terms. So this shows a ratio of less inequality is decreased by subtracting from both the terms of the given ratio that is from P and Q a number X which is less than P and less than Q. So hence we have proved the theorem so this completes the session hope you understood the concept of ratios.