One - fourth of a herd of deer have gone to the forest. One - third of the total number of deer are grazing in a field and the remaining 15 are drinking water on the bank of a river . Find the total number of deer in the herd.

Answer

To solve the problem step by step, let's denote the total number of deer in the herd as \( x \).

### Step 1: Define the variables
Let the total number of deer in the herd be \( x \).

### Step 2: Express the number of deer in different locations
- One-fourth of the herd has gone to the forest:
\[ \text{Deer in the forest} = \frac{x}{4} \]
- One-third of the total number of deer are grazing in a field:
\[ \text{Deer in the field} = \frac{x}{3} \]
- The remaining deer are drinking water on the bank of a river, which is given as 15:
\[ \text{Deer drinking water} = 15 \]

### Step 3: Set up the equation
The total number of deer can be expressed as the sum of deer in the forest, deer in the field, and deer drinking water:
\[ \frac{x}{4} + \frac{x}{3} + 15 = x \]

### Step 4: Find a common denominator
To solve the equation, we need to find a common denominator for the fractions. The least common multiple of 4 and 3 is 12. We will rewrite the equation:
\[ \frac{3x}{12} + \frac{4x}{12} + 15 = x \]

### Step 5: Combine the fractions
Now combine the fractions on the left side:
\[ \frac{3x + 4x}{12} + 15 = x \]
This simplifies to:
\[ \frac{7x}{12} + 15 = x \]

### Step 6: Isolate \( x \)
To isolate \( x \), we will subtract \( \frac{7x}{12} \) from both sides:
\[ 15 = x - \frac{7x}{12} \]
This can be rewritten as:
\[ 15 = \frac{12x - 7x}{12} \]
Which simplifies to:
\[ 15 = \frac{5x}{12} \]

### Step 7: Solve for \( x \)
To solve for \( x \), multiply both sides by 12:
\[ 15 \times 12 = 5x \]
\[ 180 = 5x \]
Now divide both sides by 5:
\[ x = \frac{180}{5} = 36 \]

### Conclusion
The total number of deer in the herd is \( \boxed{36} \).

---