There are 120 legs and 50 heads of some ducks and horses grazing in a field. Find the number of horses.

Options

A. a.20
B. b.10
C. c.30
D. d.40

Answer

To solve the problem of finding the number of horses given the total number of legs and heads of ducks and horses, we can follow these steps:

### Step 1: Define Variables
Let:
- \( d \) = number of ducks
- \( h \) = number of horses

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of heads (ducks + horses) is 50:
\[ d + h = 50 \quad \text{(Equation 1)} \]
2. The total number of legs (ducks have 2 legs, horses have 4 legs) is 120:
\[ 2d + 4h = 120 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing everything by 2:
\[ d + 2h = 60 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( d + h = 50 \) (Equation 1)
2. \( d + 2h = 60 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[ (d + 2h) - (d + h) = 60 - 50 \]
This simplifies to:
\[ h = 10 \]

### Step 5: Find the Number of Ducks
Now that we have the number of horses, we can find the number of ducks using Equation 1:
\[ d + 10 = 50 \]
Thus,
\[ d = 50 - 10 = 40 \]

### Final Answer
The number of horses is \( h = 10 \).

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