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

Given:
Lengths of rods = 120 cm, 100 cm, and 150 cm

Step 1: Find the least length of the fence that can be measured exactly by all rods
This means find the Least Common Multiple (LCM) of 120, 100, and 150.

Step 2: Find prime factorization of each rod length
- \(120 = 2^3 \times 3 \times 5\)
- \(100 = 2^2 \times 5^2\)
- \(150 = 2 \times 3 \times 5^2\)

Step 3: Find LCM using highest powers of each prime
- Highest power of 2 = \(2^3\)
- Highest power of 3 = \(3^1\)
- Highest power of 5 = \(5^2\)

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

Step 4: Find how many times each rod will be used to measure the fence length
Number of times rod is used = \(\frac{\text{LCM}}{\text{rod length}}\)
- For 120 cm rod: \(\frac{600}{120} = 5\)
- For 100 cm rod: \(\frac{600}{100} = 6\)
- For 150 cm rod: \(\frac{600}{150} = 4\)

Final Answer:
Least length of fence = 600 cm
Rod of length 120 cm is used 5 times
Rod of length 100 cm is used 6 times
Rod of length 150 cm is used 4 times