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.
Similar Questions
- An amount is divided between two friends in the ratio 4: 7. If the second part is ₹6 more than the first, then the difference in amount between two friends is:
- If x : y = 8:3, then \((x - y) : (x + y) = ?\)
- If (4x + 5) : (3x + 11) = 13:17, then (5x + 4) : (4x - 1) =?
- If \(4/9 : x = 2/3 : 16/81\), then \(x = ?\)
- If (4x + 5) : (3x + 11) = 13:17, then (5x + 4) : (4x - 1) = ?
If 11.6: x :: x: 2.9, and x > 0, then find the value of x.
Two plumbers, X and Y, are paid a total of ₹770 per day by their employer. If X is paid 140% of the sum paid to Y, then how much is Y paid (in ₹, rounded off to two decimal places) per day?
A workshop mixes coolant concentrate and water in a 3:2 ratio for a machine cooling system. If the total mixture required is 25 liters, how many liters of coolant concentrate are required?
Given the ratio 3:4::6:8, which operation verifies the proportion?
If a : b = 3 : 2 and b : c = 4 : 5, then a : b : c = ?
Important Questions from Ratio and proportion
- An amount is divided between two friends in the ratio 4: 7. If the second part is ₹6 more than the first, then the difference in amount between two friends is:
- If x : y = 8:3, then \((x - y) : (x + y) = ?\)
- If the third proportional of 9 and 60 be x, then what is the value of x?
- If 36% of first number is equal to four-sevenths of second number, what is the ratio of first number to the second number?
- A, B and C invest a sum in the ratio of 61 : 43 : 90, respectively. If they earned a total profit of ₹2,910 at the end of the year, then what is the difference between share of B and C?
