Question

If 𝑎 : 𝑏 : : 𝑐 : 𝑑 then prove that: 𝑎 : (𝑎 − 𝑏) : : 𝑐 : (𝑐 − 𝑑).

Answer 1

Given,

a : b : : c : d

⇒ `a/b=c/d`

If the original ratios are equal, their reciprocals are also equal (this is known as Invertendo):

⇒ `b/a=d/c`

Applying dividendo, we get:

⇒ `(b − a)/a = (d − c​)/c`

⇒ `−(a − b​)/a = −(c − d​)/c`

⇒ `(a − b​)/a  = (c − d)/c​ `

Now, take the reciprocal

∴ a − ba​ = c − dc​

Hence, proved that a : (a − b) :: c : (c − d).