The average working hours per month of the staff aged over 50 years in a factory were 160 and that of the staff aged under 50 years were 210. The mean working hour per month of all the staff was 200. The ratio of the number of the staff aged over 50 to that the staff aged under 50 is:
1 ∶ 4
Let the number of staff aged over 50 be \(a\) and under 50 be \(b\). By the weighted-average condition:
\[\frac{160a+210b}{a+b}=200\]
\[160a+210b=200a+200b \implies 10b=40a \implies \frac{a}{b}=\frac{1}{4}\]
Alligation check: deviations from 200 are \(|160-200|=40\) and \(|210-200|=10\), so \(a:b=10:40=1:4\).
Hence the required ratio is 1 : 4.
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