Question

Arpita and Nikita, working together, can complete an assigned job in 12 days. If Arpita works initially to complete 40% of the job, and the remaining job is completed by Nikita alone, then it takes 24 days to complete the job. The possible number of days that Nikita requires to complete the entire job, working alone, is ________

Hint:

In work-rate problems, always convert the information into rates (e.g., jobs per day). When solving systems of equations, be prepared for quadratic equations which may yield multiple valid solutions. Contextual clues, like different names for workers, can sometimes help in choosing the intended answer if a unique solution is expected.

Answer 1

Correct Answer: 20

Step 1: Understanding the Concept:
This is a work and time problem. We can solve it by defining the rates of work for each person and setting up equations based on the information given.
Step 2: Key Formula or Approach:
1. Let Arpita's rate be A jobs/day and Nikita's rate be N jobs/day. 2. The total work is 1 job. 3. From the first statement: \( A + N = \frac{1}{12} \). 4. From the second statement: Time taken by Arpita + Time taken by Nikita = 24 days. This translates to: \( \frac{0.4}{A} + \frac{0.6}{N} = 24 \). 5. Solve this system of two equations for N. The time Nikita takes alone is \(1/N\).
Step 3: Detailed Explanation:
We have the system of equations: 1) \( A + N = \frac{1}{12} \) 2) \( \frac{0.4}{A} + \frac{0.6}{N} = 24 \) From equation (1), we can express A as \( A = \frac{1}{12} - N \). Substitute this into equation (2): \[ \frac{0.4}{\frac{1}{12} - N} + \frac{0.6}{N} = 24 \] \[ \frac{0.4}{\frac{1-12N}{12}} + \frac{0.6}{N} = 24 \] \[ \frac{4.8}{1 - 12N} + \frac{0.6}{N} = 24 \] Multiply the entire equation by \( N(1 - 12N) \) to clear the denominators: \[ 4.8N + 0.6(1 - 12N) = 24N(1 - 12N) \] \[ 4.8N + 0.6 - 7.2N = 24N - 288N^2 \] \[ -2.4N + 0.6 = 24N - 288N^2 \] Rearrange into a standard quadratic form \(ax^2+bx+c=0\): \[ 288N^2 - 26.4N + 0.6 = 0 \] Multiply by 10 to remove the decimal: \[ 2880N^2 - 264N + 6 = 0 \] Divide by 6 to simplify: \[ 480N^2 - 44N + 1 = 0 \] We solve this quadratic equation for N using the quadratic formula \( N = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \): \[ N = \frac{44 \pm \sqrt{(-44)^2 - 4(480)(1)}}{2(480)} = \frac{44 \pm \sqrt{1936 - 1920}}{960} = \frac{44 \pm \sqrt{16}}{960} = \frac{44 \pm 4}{960} \] This gives two possible values for N:
Case (i): \( N = \frac{44+4}{960} = \frac{48}{960} = \frac{1}{20} \).
Case (ii): \( N = \frac{44-4}{960} = \frac{40}{960} = \frac{1}{24} \).
If \( N = 1/20 \), Nikita takes \( 1/N = 20 \) days. Arpita's rate is \( A = 1/12 - 1/20 = (5-3)/60 = 2/60 = 1/30 \), so Arpita takes 30 days. This is a valid solution.
If \( N = 1/24 \), Nikita takes \( 1/N = 24 \) days. Arpita's rate is \( A = 1/12 - 1/24 = (2-1)/24 = 1/24 \), so Arpita also takes 24 days. In this case, their rates are equal.
Step 4: Final Answer:
Assuming Arpita and Nikita have different work rates, Nikita's rate N is 1/20. The time required for Nikita to complete the job alone is \( 1/N = 20 \) days.