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Question
Find two numbers such that the mean proportional between them is 28 and the third proportional to them is 224.
Solution
Let the two numbers are a and b.
∵ 28 is the mean proportional
∵ a : 28 : : 28 : b
∴ ab = (28)2 = 784
⇒ a = `(784)/b` ...(i)
∵ 224 is the third proportional
∴ a : b : : b : 224
⇒ b2 = 224a ...(ii)
Substituting the value of a in (ii)
b2 = `24 xx (784)/b`
⇒ b3 = 224 x 784
⇒ b2 = 175616 = (56)3
∴ b = 56
Now substituting the value of b in (i)
a = `(784)/(56)` = 14
Hence numbers are 14, 56.
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