When two ratios are equal, then the four quantities compositing them are said to be proportiona. If `a/b=c/d,` then it is written `a:b=c:dor a:b::c:d.` Also if `a/b=c/d=lamdaimpliesa=blamdaand c=dlamdaimpliesa/b=c/d =(a+-c)/(b+-d)=(""^(n)sqrt((a)^(n)+-(c)^(n)))/(""^(n)sqrt((b)^(n)+-(d)^(n)))=lamda(lamdagt0)` important property of proportion :
1. If `a:b=c:d,then`
`(a+b)/(b)=(c+d)/(d)("Componendo")i.e.(a)/(b)=(c)/(d)implies(a)/(b)+1=(c)/(d)+1`
2. If `a:b=c"d, then`
`(a-b)/(b)=(c-d)/(d)("Dividenod")i.e.(a)/(b)=(c)/(d)implies(a)/(b)-1=(c)/(d)-1`
3. If `a:b:=c:d, then`
`(a+b)/(a-b)=(c+d)/(c-d)("Componendo and dividendo")`
`If ""(a+3d)/(a+9d)=(a+d)/(a+5d)=k,` then k is equal to `(a,d gt0)`

Options

A. `1//2`
B. 2
C. 6
D. `1//4`