A garrison of 3300 men had provisions for 32 days, when given at the rate of 850 gms. per head. At the end of 7 days, a reinforcement ar rives and it was found that the provisions will last for 17 days more, when given at the rate of 825 gms per head. What is the strength of the reinforcement?

Options

A. 1500
B. 1700
C. 1800
D. 2000

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Calculate the total provisions available initially.
The total provisions can be calculated using the formula:
\[ \text{Total provisions} = \text{Number of men} \times \text{Daily consumption per man} \times \text{Number of days} \]
Given:
- Number of men = 3300
- Daily consumption per man = 850 grams
- Number of days = 32

Calculating:
\[ \text{Total provisions} = 3300 \times 850 \times 32 \]
\[ = 3300 \times 27200 = 89760000 \text{ grams} \]

### Step 2: Calculate the provisions consumed in the first 7 days.
The provisions consumed in the first 7 days can be calculated as:
\[ \text{Provisions consumed} = \text{Number of men} \times \text{Daily consumption per man} \times \text{Number of days} \]
Calculating:
\[ \text{Provisions consumed} = 3300 \times 850 \times 7 \]
\[ = 3300 \times 5950 = 19635000 \text{ grams} \]

### Step 3: Calculate the remaining provisions after 7 days.
\[ \text{Remaining provisions} = \text{Total provisions} - \text{Provisions consumed} \]
Calculating:
\[ \text{Remaining provisions} = 89760000 - 19635000 = 70125000 \text{ grams} \]

### Step 4: Determine how many days the remaining provisions will last for the new total number of men.
After 7 days, it is given that the provisions will last for 17 more days at a rate of 825 grams per head. Let \( x \) be the total number of men after reinforcement.

The equation for the remaining provisions is:
\[ \text{Remaining provisions} = \text{Total men} \times \text{Daily consumption per man} \times \text{Number of days} \]
Substituting the known values:
\[ 70125000 = x \times 825 \times 17 \]

### Step 5: Solve for \( x \).
\[ 70125000 = x \times 14025 \]
\[ x = \frac{70125000}{14025} \approx 5000 \]

### Step 6: Calculate the strength of the reinforcement.
The strength of the reinforcement is given by:
\[ \text{Reinforcement} = x - \text{Initial number of men} \]
Calculating:
\[ \text{Reinforcement} = 5000 - 3300 = 1700 \]

Thus, the strength of the reinforcement is **1700 men**.