A garrison of 2000 soldiers has food provisions that will last for 54 days. After 15 days, a reinforcement of soldiers arrives, and it is then calculated that the remaining provisions will last only for 20 more days. How many soldiers are there in the reinforcement that joined the original garrison?

Introduction / Context:
This is a standard garrison provision problem that uses the idea that total food supply is fixed and daily consumption depends on the number of men. When reinforcements arrive, the daily consumption increases and the total duration for which the provisions last decreases. The task is to find how many additional men joined the original group.

Given Data / Assumptions:

• Initial number of soldiers = 2000.
• Provisions initially last for 54 days at that strength.
• After 15 days, reinforcement arrives.
• From that point, the provisions last for only 20 more days.
• Food consumption per soldier per day is constant and the total supply does not change except by consumption.

Concept / Approach:
The total amount of food can be measured in men days, which is the product of the number of men and the number of days the food lasts. Initially, the total men days of food are calculated from the original garrison. After 15 days some of this food has been consumed. The remaining food must support the larger garrison for 20 more days. Using these facts, we can set up equations to find the size of the reinforcement.

Step-by-Step Solution:
Let F be the total food measured in men days. Initially, with 2000 men and 54 days of food, F = 2000 * 54. After 15 days, food consumed = 2000 * 15 men days. Remaining food = F - 2000 * 15 = 2000 * (54 - 15) = 2000 * 39 men days. Let N be the total number of soldiers after the reinforcement joins. The remaining food lasts 20 days for N men, so 2000 * 39 = N * 20. Compute N: N = (2000 * 39) / 20 = 200 * 39 / 2 = 100 * 39 = 3900 men. Reinforcement size = N - 2000 = 3900 - 2000 = 1900 men.

Verification / Alternative check:
Check that the food calculation is consistent: 2000 men for 15 days consume 30000 men days. Then 3900 men for 20 days consume 78000 men days. The total is 108000 men days, which matches the initial F = 2000 * 54 = 108000. This confirms the arithmetic and the logic.

Why Other Options Are Wrong:
2100 as reinforcement would give 4100 total men, leading to inconsistent consumption compared with the remaining food. 1700 as reinforcement would give 3700 total men, which would require more days than 20 for the remaining food. 2000 as reinforcement suggests the number of men doubles, which would make the provisions last less than 20 days. 1500 reinforcement also does not satisfy the men days balance for the given durations.

Common Pitfalls:
Learners sometimes equate the days directly or try to use ratios of days without translating everything into men days. Another mistake is forgetting to subtract the food consumed during the first 15 days before the reinforcement arrives. Always compute total food in men days, subtract the used portion, and then equate the remaining food to the product of the new strength and the remaining days.

Final Answer:
The reinforcement consists of 1900 additional men.