a man has some sheep and some hands if the number of feet exceed the number of head by 164 and the number of sheep and number of hens by 12 find the number of sheep

Let's solve this problem step by step.

### Step 1: Define Variables
- Let the number of sheep be \( S \).
- Let the number of hens be \( H \).

### Step 2: Set Up Equations
1. **Number of Feet vs. Number of Heads:**
   - Each sheep has 4 feet and 1 head.
   - Each hen has 2 feet and 1 head.
   - The total number of feet is \( 4S + 2H \).
   - The total number of heads is \( S + H \).
   - According to the problem, the number of feet exceeds the number of heads by 164:
     \[
     4S + 2H = S + H + 164
     \]

2. **Number of Sheep and Hens:**
   - The number of sheep exceeds the number of hens by 12:
     \[
     S = H + 12
     \]

### Step 3: Solve the Equations
1. **First Equation (Feet vs. Heads):**
   \[
   4S + 2H = S + H + 164
   \]
   Subtract \( S + H \) from both sides:
   \[
   3S + H = 164 \quad \text{(Equation 1)}
   \]

2. **Substitute \( S = H + 12 \) into Equation 1:**
   \[
   3(H + 12) + H = 164
   \]
   Expand and simplify:
   \[
   3H + 36 + H = 164
   \]
   Combine like terms:
   \[
   4H + 36 = 164
   \]
   Subtract 36 from both sides:
   \[
   4H = 128
   \]
   Divide by 4:
   \[
   H = 32
   \]

3. **Find the Number of Sheep \( S \):**
   \[
   S = H + 12 = 32 + 12 = 44
   \]

### Final Answer
The number of sheep is \( \boxed{44} \).

Would you like to see more details or have any questions about the solution? 

Here are five related questions:
1. How would the solution change if the difference in the number of feet and heads was 200 instead of 164?
2. What if the number of sheep exceeded the number of hens by 20 instead of 12?
3. Can you formulate a general rule for similar problems involving animals with different numbers of feet and heads?
4. How would you solve the problem if the number of feet exceeded the number of heads by a different value?
5. What mathematical concepts are used to solve this type of problem?

**Tip:** When solving word problems, translating the problem into equations using variables is key to finding the solution efficiently.

A man has some sheep and some hens. If the number of feet exceeds the number of heads by 164 and the number of sheep exceeds the number of hens by 12, find the number of sheep.

This problem involves solving a system of linear equations to find the number of sheep and hens based on given conditions. Learn how to use algebra to solve for unknowns in this word problem, suitable for students in Grades 7-9.

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