Question

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

Answer 1

Correct Answer: 35

Let's consider the number of patients admitted in hospital A and hospital B to be \( x \) and \( x + 21 \) respectively.

We are given that:

• The sum of recovery days for patients in hospital A is 200,
• The sum of recovery days for patients in hospital B is 152.
• The average recovery days for patients admitted in hospital A is 3 more than the average in hospital B.

Step 1: Setting up the Equation

The average recovery days for patients in hospital A is:

\[ \frac{200}{x} \]

The average recovery days for patients in hospital B is:

\[ \frac{152}{x + 21} \]

According to the problem, the average recovery days for hospital A is 3 more than the average recovery days for hospital B. Hence, we have the equation:

\[ \frac{200}{x} = \frac{152}{x + 21} + 3 \]

Step 2: Solving the Equation

First, subtract \( \frac{152}{x + 21} \) from both sides:

\[ \frac{200}{x} - \frac{152}{x + 21} = 3 \]

Now, cross-multiply to eliminate the fractions. Multiply both sides by \( x(x + 21) \):

\[ 200(x + 21) - 152x = 3x(x + 21) \]

Expanding both sides:

\[ 200x + 4200 - 152x = 3x^2 + 63x \]

Simplify the equation:

\[ 48x + 4200 = 3x^2 + 63x \]

Rearrange the equation to form a standard quadratic equation:

\[ 3x^2 + 15x - 4200 = 0 \]

Now solve this quadratic equation using the quadratic formula. The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \( 3x^2 + 15x - 4200 = 0 \), \( a = 3 \), \( b = 15 \), and \( c = -4200 \). Substituting these values into the quadratic formula:

\[ x = \frac{-15 \pm \sqrt{15^2 - 4(3)(-4200)}}{2(3)} \]

Simplify the discriminant:

\[ x = \frac{-15 \pm \sqrt{225 + 50400}}{6} \]

\[ x = \frac{-15 \pm \sqrt{50625}}{6}\]

Now calculate the square root of 50625:

\[ x = \frac{-15 \pm 225}{6} \]

Now solve for \(x\):

• Using the plus sign: \[ x = \frac{-15 + 225}{6} = \frac{210}{6} = 35 \]
• Using the minus sign: \[ x = \frac{-15 - 225}{6} = \frac{-240}{6} = -40 \quad \text{(which is not a valid solution since the number of patients cannot be negative)} \]

Step 3: Conclusion

Therefore, the number of patients admitted to hospital A is \( \boxed{35} \).