 Hello and welcome to the session. In this session we will discuss k-method for proportions. If a series of fractions are equal, that is say we have p upon q is equal to r upon s is equal to x upon y is equal to so on. Then each of these fractions or you can say each ratio is equal to the sum of the numerators that is p plus r plus x plus and so on divided by the sum of the denominators that is q plus s plus y plus and so on. That is this is equal to the sum of the antecedents of the given ratios upon the sum of consequence of the given ratios. So let us suppose that we are given upon q is equal to r upon s is equal to x upon y. Then we have to prove that each of the given ratios is equal to p plus r plus x upon q plus s plus y. After the proof we take let the given ratios p upon q is equal to r upon s is equal to x upon y is equal to k. Then this means that p would be equal to q into k, r would be equal to s into k and x would be equal to y into k. Now consider p plus r plus x upon q plus s plus y. Now this would be equal to in place of p we put qk so qk plus sk that is in place of r we put sk plus x which is yk. This whole upon q plus s plus y. Now taking k commons on the numerator we have k into q plus s plus y the whole and this whole upon q plus s plus y. q plus s plus y cancels in the numerator and denominator and we are left with k. And we had assumed each ratio to be equal to k and also we get that p plus r plus x upon q plus s plus y equal to k. So we can now say that p upon q is equal to r upon s is equal to x upon y is equal to p plus r plus x upon q plus s plus y. So we have proved that each ratio is equal to sum of the antecedents upon the sum of their consequence. We can extend this proof to prove this result for a series of fractions equal to each other. Let us consider one example to illustrate this result that is of k method. Suppose we are given that p is to q is equal to r is to s and we are supposed to prove that p plus q upon r plus s is equal to fourth root of p is to the power 4 plus q is to the power 4 upon fourth root of r to the power 4 plus s to the power 4. We suppose that the ratios be equal to k that is p upon q is equal to r upon s is equal to k. From here we have p is equal to q k r is equal to s k. Now we will consider the LHS which is plus q upon r plus s. Now in place of p we put q k and in place of r we put s k. So now taking q common in the numerator q into k plus 1 d whole upon taking s common in the denominator we have s into k plus 1 d whole. Now these cancel with each other and we are left with q upon s. So we have the LHS as q upon s. In the same way let us now consider the LHS which is fourth root of p to the power 4 plus q to the power 4 upon fourth root of r to the power 4 plus s to the power 4. Now in place of p we put q k and in place of r we put s k. So we have fourth root of q to the power 4 into k to the power 4 plus q to the power 4 upon fourth root of s to the power 4 into k to the power 4 plus s to the power 4. Now we can take fourth root of q to the power 4 common from the numerator. So we have q into fourth root of k to the power 4 plus 1 d whole upon taking fourth root of s to the power 4 common in the numerator we have s into fourth root of k to the power 4 plus 1. Now these cancel and so we have q upon s which is same as the LHS. Thus we have the LHS is equal to the LHS and hence we have proved the required result. Now we will discuss application of k method to continue proportions. Let us consider an example for this. If we have p to r s r n continued proportion then we need to prove that p to the power 4 is true. q to the power 4 is equal to p minus q whole to the power 4 is true q minus r whole to the power 4. Now as p to r r n continued proportion then p upon q is equal to q upon r is equal to r upon s. And we suppose let each of these ratios be equal to k. So from here we have p is equal to qk, q is equal to rk and r is equal to sk. Now in place of r here we will put sk so we get q is equal to sk square and in place of q we put sk square here so we get p equal to skq. So we have p equal to skq, q equal to sk square and r equal to sk. Now we need to prove this so first of all we will consider the LHS which is p to the power 4 is true q to the power 4 that is p to the power 4 upon q to the power 4. Now in place of p we will put skq so this is equal to s to the power 4 into k to the power 12. This upon q to the power 4 that is sk square to the power 4 which is s to the power 4 into k to the power 8. So this is equal to k to the power 4. Now in the same way we will consider the RHS which is p minus q whole to the power 4 is 2 minus r whole to the power 4. That is this is equal to p minus q whole to the power 4 upon q minus r whole to the power 4. Now in place of p we will put skq and in place of q we will put sk square. So we have skq minus sk square this whole to the power 4 upon in place of q we have sk square minus in place of r we have sk this whole to the power 4. Now taking sk square whole to the power 4 common in the numerator we have this into k minus 1 whole to the power 4. This upon taking sk whole to the power 4 common in the denominator we have this into k minus 1 this whole to the power 4. Now these cancel with each other and this is equal to s to the power 4 into k to the power 8 upon s to the power 4 into k to the power 4. Now s to the power 4 s to the power 4 cancels and here we are left with k to the power 4. So we have RHS is k to the power 4 and also the LHS is k to the power 4. That means we now have LHS is equal to RHS. Hence we have proved the given result. This is how k method is applied in case of continuous proportions. So this complete session hope you understood the concept of k method in proportions.