One garden had 5 times as many raspberry bushes as a second garden. After 22 bushes are transplanted from the first garden to the second garden, the number of raspberry bushes in the gardens are the same. How many bushes did each garden have originally?
The Correct Answer and Explanation is:
Correct Answer:
- The first garden originally had 55 raspberry bushes.
- The second garden originally had 11 raspberry bushes.
Explanation (300+ words):
Let’s break this problem down step by step using algebra.
We are told:
One garden had 5 times as many raspberry bushes as a second garden.
Let’s call the number of raspberry bushes in the second garden x.
Then the number of raspberry bushes in the first garden is 5x, since it had 5 times as many.
So originally:
- First garden: 5x bushes
- Second garden: x bushes
Then, 22 bushes are transplanted from the first garden to the second. This changes the number of bushes in both gardens:
- First garden now has: 5x − 22 bushes
- Second garden now has: x + 22 bushes
According to the problem, after this transfer, both gardens have the same number of bushes.
So we set up the equation: 5x−22=x+225x – 22 = x + 22
Solving the equation:
5x−22=x+225x – 22 = x + 22
Subtract x from both sides: 4x−22=224x – 22 = 22
Add 22 to both sides: 4x=444x = 44
Divide both sides by 4: x=11x = 11
So, the second garden originally had x = 11 bushes.
The first garden had: 5x=5×11=55 bushes5x = 5 × 11 = 55 \text{ bushes}
Verification:
Let’s check if the final numbers are equal after transplanting 22 bushes:
- First garden: 55 − 22 = 33 bushes
- Second garden: 11 + 22 = 33 bushes
✔️ Yes, the numbers match. Both gardens end up with 33 bushes, as stated in the problem.
Conclusion:
Originally:
- First garden: 55 bushes
- Second garden: 11 bushes
This algebraic approach ensures accuracy and logical consistency with the scenario described. Would you like a visual diagram to illustrate the transfer?