Consider a problem, there are three types of products X ,Y ,Z costing 532,504 and 486 bucks respectivel.A person buys A, B and C number of X, Y and Z respectively such that A+B+C is 37 and the average cost per product of the final mixture is 506.We have to find 'A'.
In the solution, they have found out the ratios, A:B=1:13 and A:C=10:13(I can understand till this part).Then, they have added the 'A' parts together(1+10) and arrived at A:B:C=11:13:13 .I can't understand the method by which they have combined the ratios(usually L.C.M. of the no. of parts of the common component will be used).Please explain.
I'm assuming you know the basics of modular arithmetic. If you have trouble following any of this, please comment and I'll explain further. We have $$\begin{align}A+B+C&=37\tag{1}\\ {532A+504B+486C\over37}&=506\end{align}$$ Substituting the first equation in the second gives $$\begin{align} 532A+504B+486(37-A-B)&=506\cdot37\\ (532-486)A+(504-486)B&=(506-486)\cdot36\\ 46A+18B&=20\cdot37\\ 23A+9B&=370\tag{2} \end{align}$$
We have only one equation for two unknowns, but we haven't yet used the fact that $A$ and $B$ must be nonnegative integers.
Reducing equation $(2)$ modulo $23$ gives $$9B\equiv2\pmod{23}$$ and we find by trial that $$B\equiv13\pmod{23}$$ Since we know frm $(1)$ that $0\leq B\leq37$, the only choices for $B$ are $13$ and $36$. Substituting $B=36$ in $(2)$ gives $A=2$ and then $(1)$ gives $C=-1$, which is impossible, so we try $B=13$. Then $(2)$ gives $A=11$ and $(1)$ gives $C=13$.