A profit of Rs. 960 is shared between A and B in the ratio 1/3 : 1/2. What is the difference in rupees between the amounts of profit received by A and B?

Introduction / Context:
This problem involves dividing a total profit between two people, A and B, according to a ratio expressed in fractional form. The question then asks for the difference between their individual shares. It tests understanding of ratios, handling of fractions, and the ability to translate a profit sharing agreement into actual rupee amounts.

Given Data / Assumptions:

• Total profit to be shared between A and B is Rs. 960.• The ratio of their shares is 1/3 : 1/2.• The ratio refers to the relative quantities of profit received by A and B.• We must find the difference between B's share and A's share in rupees.

Concept / Approach:
To work with a ratio like 1/3 : 1/2, it is convenient to clear denominators by multiplying both terms by the least common multiple of 3 and 2, which is 6. This converts the ratio into whole numbers. Once we have a standard ratio in whole numbers, we can represent the shares as multiples of a common variable. Then we use the condition that their sum is equal to Rs. 960 to determine the value of that variable and hence each individual share. The final step is to subtract A's share from B's share to obtain the required difference.

Step-by-Step Solution:
Step 1: The ratio is 1/3 : 1/2.Step 2: Multiply both terms by 6 to remove fractions: (1/3)*6 = 2 and (1/2)*6 = 3, so the ratio becomes 2 : 3.Step 3: Let A's share be 2k and B's share be 3k, where k is a common multiplier.Step 4: The total profit is 960, so 2k + 3k = 960.Step 5: Combine like terms: 5k = 960.Step 6: Solve for k: k = 960 / 5 = 192.Step 7: Compute A's share: 2k = 2 * 192 = 384.Step 8: Compute B's share: 3k = 3 * 192 = 576.Step 9: Difference between their shares = 576 − 384 = 192 rupees.

Verification / Alternative check:
We can check by confirming that the ratio of the actual shares equals 1/3 : 1/2. First, form the ratio 384 : 576. Divide both numbers by 192: 384 / 192 = 2 and 576 / 192 = 3, giving the simplified ratio 2 : 3. Now match this with the original fractional ratio. Since 2 : 3 corresponds to (1/3)*6 : (1/2)*6, it is consistent. Also, the total 384 + 576 = 960 equals the given profit. Therefore, the calculation is correct and the difference of Rs. 192 is verified.

Why Other Options Are Wrong:
• Rs. 120: This would come from using the wrong total or from misinterpreting the ratio.• Rs. 160: This might come from mistakenly taking 1/5 of the total as the difference, but that is not how the distribution works here.• Rs. 240: This value would not preserve both the correct ratio and the correct total profit when back calculated.

Common Pitfalls:
Some learners forget to convert the fractional ratio into a whole number ratio and try to work directly with fractions in a confusing way. Others may mistakenly add the fractions 1/3 and 1/2 incorrectly or treat them as percentages. Another error is to miscalculate the value of k by incorrectly dividing the total by something other than the sum of the ratio parts. Care must also be taken not to confuse the difference between the ratios (such as 3 − 2) with the difference between the actual rupee amounts. Each step must be done systematically to avoid these mistakes.

Final Answer:
The difference between the profit shares of A and B is Rs. 192.