A sum of money is sufficient to pay the wages of worker K alone for 10 days or worker L alone for 15 days. If both K and L are paid from this same amount at the same time, for how many days will the money last?

Introduction / Context:
This question involves the concept of total available money being used to pay wages at fixed daily rates. The money can pay one worker for a certain number of days or another worker for a different number of days. We are asked to find how long that same amount of money will last if both workers are paid simultaneously from it.

Given Data / Assumptions:

• The money can pay worker K alone for 10 days.
• The same money can pay worker L alone for 15 days.
• Daily wages of each worker remain constant over time.
• Both K and L start receiving wages on the same day from the same fund.
• The entire amount is used only for their wages.

Concept / Approach:
Let the daily wage of K be k, and that of L be l. The total money available is the same in both equivalent descriptions. From this we can express the fund in two ways: 10k and 15l. Equating these expressions lets us find a relationship between k and l. Once we know the combined daily wage k + l, we can find the number of days the money will last by dividing the total fund by the total daily payout.

Step-by-Step Solution:
Let the total money be M.If it pays K alone for 10 days, M = 10k.If it pays L alone for 15 days, M = 15l.Equate the two expressions: 10k = 15l.So k / l = 15 / 10 = 3 / 2, hence l = (2 / 3)k.Combined daily wage when both work together = k + l = k + (2 / 3)k = (5 / 3)k.Total money M = 10k, so number of days with both = M / (k + l) = 10k / ((5 / 3)k) = 10 * (3 / 5) = 6 days.

Verification / Alternative check:
Assume a convenient value for K's daily wage, say k = Rs. 30. Then total money M = 10 * 30 = Rs. 300. From the earlier ratio, l = (2 / 3) * 30 = Rs. 20. If both are paid together, daily payout = 30 + 20 = Rs. 50. The number of days = 300 / 50 = 6 days, which matches our algebraic result. This confirms the correctness of the solution.

Why Other Options Are Wrong:
Nine days would require a much smaller combined daily wage and contradicts the fact that K alone uses up the fund in only 10 days. Twelve days and four days also do not satisfy the relationship between the wages derived from the given conditions. Only six days is consistent with both K and L's possible payment periods from the same fund.

Common Pitfalls:
A frequent mistake is to average the two given days as (10 + 15) / 2 = 12.5, which has no basis in the actual wage structure. Another error is to treat wages as equal without using the ratio implied by the total fund. Always translate such problems into algebra using a symbol for the daily wages and then equate the two expressions for the same money.

Final Answer:
The money will last for 6 days when both K and L are paid simultaneously from it.