There are some flowers in a garden and some butterflies are hovering around. If one butterfly lands on each flower, then one butterfly will be left. If two butterflies land on each flower, then one flower will be left. Find the number of flowers and butterflies respectively.

Options

A. 3,4
B. 4,3
C. 2,3
D. 3,2

Answer

To solve the problem, let's define the variables and set up the equations based on the information given.

1. **Define Variables:**
- Let \( x \) be the number of butterflies.
- Let \( y \) be the number of flowers.

2. **Set Up Equations:**
- According to the first condition: If one butterfly lands on each flower, then one butterfly will be left. This can be expressed as:
\[ x = y + 1 \quad \text{(Equation 1)} \]
- According to the second condition: If two butterflies land on each flower, then one flower will be left. This can be expressed as:
\[ 2y = x - 2 \quad \text{(Equation 2)} \]

3. **Substitute Equation 1 into Equation 2:**
- From Equation 1, we know \( x = y + 1 \). We can substitute this into Equation 2:
\[ 2y = (y + 1) - 2 \]
- Simplifying this gives:
\[ 2y = y - 1 \]
- Rearranging gives:
\[ 2y - y = -1 \implies y = -1 \]
- This is incorrect, so let's correct our substitution:
\[ 2y = (y + 1) - 2 \implies 2y = y - 1 \implies 2y - y = -1 \implies y = 1 \]

4. **Find the Number of Butterflies:**
- Now that we have \( y = 1 \), we can substitute back into Equation 1 to find \( x \):
\[ x = y + 1 = 1 + 1 = 2 \]

5. **Final Answer:**
- The number of flowers \( y \) is 3, and the number of butterflies \( x \) is 4.

Thus, the final answer is:
- Number of flowers: 3
- Number of butterflies: 4