Question

if (ma+nb):b::(mc+nd):d, prove that a,b,c,d are in proportion

Answer 1

(ma+nb)/b = (mc+nd)/d
d(ma+nb)=b(mc+nd)
dma+dnb=bmc+bnd
dma-bmc=bnd-dnb
m(da-bc)=n(bd-db)
m(da-bc)=n(0)
m(da-bc)=0
da-bc=0
da=bc
a/b = c/d
Thus, a,b,c and d are in proportion.

Answer 2

Hence,a,b,c,d are in proportion.

Step-by-step explanation:

Given,

(ma+nb):b:(mc+nd):d

We have to prove that,

a,b,c,d are in proportion

Now,

$\aerial {\frac{ma+nb}b = \frac{mc+nd}{d}}$

⇒ $d(ma+nb)=b(mc+nd)$

⇒ dma+dnb=bmc+bnd

⇒ dma-bmc=bnd-dnb

⇒ m(da-bc)=n(bd-db)

⇒ m(da-bc)=n(0)

⇒ m(da-bc)=0

⇒ da-bc=0

⇒ da=bc

So,$\bold {\frac{a}{b}= \frac{c}{d}}$

Hence,a,b,c,d are in proportion.