If (ma + nb) : b : : (mc + nd) : d, prove that a, b, c, d are in proportion.
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Given, (ma + nb) : b : : (mc + nd) : d.
⇒(ma+nb)b=(mc+nd)d⇒d(ma+nb)=b(mc+nd)⇒mad+nbd=bmc+bnd⇒mad=bmc\Rightarrow \dfrac{(ma + nb)}{b} = \dfrac{(mc + nd)}{d} \\[0.5em] \Rightarrow d(ma + nb) = b(mc + nd) \\[0.5em] \Rightarrow mad + nbd = bmc + bnd \\[0.5em] \Rightarrow mad = bmc \\[0.5em]⇒b(ma+nb)=d(mc+nd)⇒d(ma+nb)=b(mc+nd)⇒mad+nbd=bmc+bnd⇒mad=bmc
On dividing equation by m,
⇒ad=bc⇒ab=cd⇒a:b::c:d.\Rightarrow ad = bc \\[0.5em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[0.5em] \Rightarrow a : b : : c : d.⇒ad=bc⇒ba=dc⇒a:b::c:d.
Hence, proved that a, b, c, d are in proportion.
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