Question
Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?
Answers
GPT-4o mini
To find the shortest height at which the stacks of boxes (12 inches tall and 18 inches tall) will be the same height, we need to find the least common multiple (LCM) of the two heights, 12 and 18.
1. **Find the prime factorizations:**
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
2. **Determine the LCM:**
- For each prime factor, take the highest power that appears in the factorizations.
- For \( 2 \): the highest power is \( 2^2 \).
- For \( 3 \): the highest power is \( 3^2 \).
Therefore, the LCM is given by:
\[
\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36
\]
3. **Conclusion:**
The shortest height at which the two stacks will be the same height is **36 inches**.
To double-check, we see how many boxes we need for each stack:
- For the 12-inch boxes:
\[
\frac{36}{12} = 3 \text{ boxes}
\]
- For the 18-inch boxes:
\[
\frac{36}{18} = 2 \text{ boxes}
\]
Both stacks at 36 inches confirm our result.
Thus, the answer is \( \boxed{36} \) inches.
1. **Find the prime factorizations:**
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
2. **Determine the LCM:**
- For each prime factor, take the highest power that appears in the factorizations.
- For \( 2 \): the highest power is \( 2^2 \).
- For \( 3 \): the highest power is \( 3^2 \).
Therefore, the LCM is given by:
\[
\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36
\]
3. **Conclusion:**
The shortest height at which the two stacks will be the same height is **36 inches**.
To double-check, we see how many boxes we need for each stack:
- For the 12-inch boxes:
\[
\frac{36}{12} = 3 \text{ boxes}
\]
- For the 18-inch boxes:
\[
\frac{36}{18} = 2 \text{ boxes}
\]
Both stacks at 36 inches confirm our result.
Thus, the answer is \( \boxed{36} \) inches.