In a group of boys and dogs, the number of heads is 7 and number of legs is 20. How many boys and dogs are there re- spectively?

Options

A. 2 boys, 5 dogs
B. 3 boys, 5 dogs
C. 4 boys, 3 dogs
D. 5 boys, 2 dogs

Answer

To solve the problem of finding the number of boys and dogs in a group where the total number of heads is 7 and the total number of legs is 20, we can follow these steps:

### Step 1: Define Variables
Let:
- \( b \) = number of boys
- \( d \) = number of dogs

### Step 2: Set Up Equations
From the problem, we know:
1. Each boy and dog has one head.
- Therefore, the total number of heads gives us the equation:
\[ b + d = 7 \quad \text{(Equation 1)} \]

2. Each boy has 2 legs and each dog has 4 legs.
- Therefore, the total number of legs gives us the equation:
\[ 2b + 4d = 20 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing all terms by 2:
\[ b + 2d = 10 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( b + d = 7 \) (Equation 1)
2. \( b + 2d = 10 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[ (b + 2d) - (b + d) = 10 - 7 \]
This simplifies to:
\[ d = 3 \]

### Step 5: Substitute Back to Find \( b \)
Now that we have \( d = 3 \), we can substitute this value back into Equation 1 to find \( b \):
\[ b + 3 = 7 \]
Subtracting 3 from both sides gives:
\[ b = 4 \]

### Conclusion
Thus, the number of boys is \( 4 \) and the number of dogs is \( 3 \).

### Final Answer
- Number of boys: 4
- Number of dogs: 3

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