Question

Three measuring rods are of lengths \(120 \, \text{cm}, 100 \, \text{cm}\) and \(150 \, \text{cm}\). Find the least length of a fence that can be measured an exact number of times using any of the rods. How many times each rod will be used to measure the length of the fence?

Hint:

For such problems, always use LCM to find the least common length and then divide to find number of uses.

Answer 1

Input:
Rod lengths: 120 cm, 100 cm, 150 cm

Objective: Determine the shortest fence length measurable by all rods.
This requires finding the Least Common Multiple (LCM) of 120, 100, and 150.

Method:
1. Prime Factorization:
- \(120 = 2^3 \times 3 \times 5\)
- \(100 = 2^2 \times 5^2\)
- \(150 = 2 \times 3 \times 5^2\)

2. LCM Calculation: Use the highest powers of each prime factor.
- Highest power of 2: \(2^3\)
- Highest power of 3: \(3^1\)
- Highest power of 5: \(5^2\)

\[\text{LCM} = 2^3 \times 3 \times 5^2 = 8 \times 3 \times 25 = 600 \text{ cm}\]

3. Rod Usage: Calculate how many times each rod is used to measure the fence length.
Number of uses = \(\frac{\text{LCM}}{\text{rod length}}\)
- 120 cm rod: \(\frac{600}{120} = 5\)
- 100 cm rod: \(\frac{600}{100} = 6\)
- 150 cm rod: \(\frac{600}{150} = 4\)

Results:
Shortest fence length = 600 cm
120 cm rod used 5 times
100 cm rod used 6 times
150 cm rod used 4 times