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Componendo

If m a+n b: b...

Question

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

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Solution

(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.

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