What number should be added to each of the numbers 55, 100, 65 and 116, so that the resulting numbers are in continued proportion?
20
Let the number added be \(x\). For continued proportion \(\dfrac{55+x}{100+x} = \dfrac{65+x}{116+x}\). Cross-multiplying:
\[(55+x)(116+x) = (100+x)(65+x)\]
\[6380 + 171x + x^2 = 6500 + 165x + x^2\]
\[6x = 120 \Rightarrow x = 20\]
Check: 75, 120, 85, 136 give \(\tfrac{75}{120} = \tfrac{85}{136} = \tfrac{5}{8}\). Hence 20.
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