Three numbers in the ratio 2 : 5 : 7 are taken as the first, second, and third terms, respectively. If their fourth proportional is 140, find the middle number.
40
Let the three numbers be \(2k, 5k, 7k\) as the first, second and third terms.
The fourth proportional \(x\) satisfies first : second = third : fourth, that is \(2k : 5k = 7k : x\).
So \(x = \frac{5k \times 7k}{2k} = \frac{35k}{2}\).
Given \(\frac{35k}{2} = 140\), we get \(35k = 280\), so \(k = 8\).
The middle number is the second term \(5k = 5 \times 8 = 40\).
Hence, the middle number is 40.
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