Question

Kamla can complete a piece of work in 24 days. Nirmala can complete this work in 18 days. Kamla started the work alone and worked for X days. After this Nirmala alone completed the remaining work in Y days, in this way the work is completed in 20 days. If X and Y are both positive integers, then select the correct option.

• A. \(X^2 + 1\) is a multiple of 13.
• B. \((\frac{4X}{Y}) + 1\) is an even integer.
• C. 2X + Y = 22
• D. X – Y is an odd integer.

Hint:

In time and work problems, always convert time taken into work rate (work per unit time). The fundamental equation is always: (Rate \(\times\) Time) = Work Done. Summing up the work done by all parties should equal 1 if the job is completed.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the individual rates at which Kamla and Nirmala can complete a work. They work sequentially for X and Y days, respectively, to complete the work in a total of 20 days. We need to find the values of X and Y and then check which of the given options is correct.
Step 2: Key Formula or Approach:
1. Find the rate of work for each person. Rate = \(\frac{1}{\text{Time taken to complete work}}\).
2. The total work done is the sum of the work done by each person, which equals 1 (representing the whole work).
3. Set up a system of equations based on the work done and the total time taken.
Step 3: Detailed Explanation:
Kamla's time to complete the work = 24 days. Kamla's rate of work = \(\frac{1}{24}\) of the work per day.
Nirmala's time to complete the work = 18 days. Nirmala's rate of work = \(\frac{1}{18}\) of the work per day.
Kamla worked for X days, so work done by Kamla = \(X \times \frac{1}{24} = \frac{X}{24}\).
Nirmala worked for Y days, so work done by Nirmala = \(Y \times \frac{1}{18} = \frac{Y}{18}\).
The total work is completed, so the sum of their work is 1: (1) \(\frac{X}{24} + \frac{Y}{18} = 1\)
The total time taken is 20 days: (2) \(X + Y = 20\)
Now we solve these two equations. From equation (2), we can write \(Y = 20 - X\).
Substitute this into equation (1): \[ \frac{X}{24} + \frac{20 - X}{18} = 1 \] To solve for X, find the least common multiple (LCM) of 24 and 18, which is 72. Multiply the entire equation by 72: \[ 72 \left( \frac{X}{24} \right) + 72 \left( \frac{20 - X}{18} \right) = 72(1) \] \[ 3X + 4(20 - X) = 72 \] \[ 3X + 80 - 4X = 72 \] \[ 80 - X = 72 \] \[ X = 80 - 72 = 8 \] Now find Y using \(Y = 20 - X\): \[ Y = 20 - 8 = 12 \] So, X=8 and Y=12. Both are positive integers.
Now we check the options: (A) \(X^2 + 1\) is a multiple of 13. \(8^2 + 1 = 64 + 1 = 65\). Since \(65 = 13 \times 5\), 65 is a multiple of 13. This statement is CORRECT.
(B) \((\frac{4X}{Y}) + 1\) is an even integer. \((\frac{4 \times 8}{12}) + 1 = \frac{32}{12} + 1 = \frac{8}{3} + 1 = \frac{11}{3}\). This is not an integer. So, (B) is incorrect.
(C) 2X + Y = 22. \(2(8) + 12 = 16 + 12 = 28\). This is not 22. So, (C) is incorrect.
(D) X – Y is an odd integer. \(8 - 12 = -4\). This is an even integer. So, (D) is incorrect.
Step 4: Final Answer:
The only correct option is (A).