In a hospital, the ratio between the number of doctors and the number of nurses was 1 : 3. When 12 new doctors joined the hospital and 24 nurses left the hospital, the ratio became 3 : 4. After the resignation of k more nurses from the hospital, the ratio becomes 1 : 1. Find the value of k.
12
Let the initial number of doctors be \(x\) and the initial number of nurses be \(3x\).
After 12 new doctors joined and 24 nurses left, the number of doctors became \(x + 12\) and the number of nurses became \(3x - 24\). The new ratio is given as 3 : 4.
Therefore, we can set up the equation:
\(\frac{x + 12}{3x - 24} = \frac{3}{4}\)
Cross-multiplying, we get:
\(4(x + 12) = 3(3x - 24)\)
\(4x + 48 = 9x - 72\)
\(5x = 120\)
\(x = 24\)
So, initially, there were 24 doctors and 72 nurses (\(3 \times 24\)).
After the changes, there were \(24 + 12 = 36\) doctors and \(72 - 24 = 48\) nurses.
Let \(k\) be the number of nurses who resigned. The new number of nurses is \(48 - k\). The ratio of doctors to nurses becomes 1 : 1.
Therefore:
\(\frac{36}{48 - k} = \frac{1}{1}\)
\(36 = 48 - k\)
\(k = 48 - 36\)
\(k = 12\)
Therefore, 12 more nurses resigned to make the ratio of doctors to nurses 1:1.
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