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Question

Difficulty: MediumHCF and LCM

A metal fabrication company has three long copper rods measuring 545\frac{54}{5} meters, 8110\frac{81}{10} meters, and 10825\frac{108}{25} meters. The company needs to cut all three rods into smaller segments of equal length such that no copper material is left over. To minimize the total number of segments, what is the maximum possible length of each individual segment?

  1. A
    16225\frac{162}{25} meters
  2. B
    3245\frac{324}{5} meters
  3. 2750\frac{27}{50} metersAnswer
  4. D
    275\frac{27}{5} meters

Answer

The maximum possible length of each segment is 27/50 meters.
The maximum possible length for the equal segments is found by calculating the HCF of the given fractions. Using the mathematical rule for fractions, HCF = HCF(numerators) / LCM(denominators). The HCF of 54, 81, and 108 is 27. The LCM of 5, 10, and 25 is 50. Therefore, the maximum length is 27/50 meters.

Step-by-Step Solution

1
Identify the mathematical operation required to find the maximum possible equal length.
The problem requires finding the Highest Common Factor (HCF) of the three fractional lengths.
The rods must be cut into equal pieces without wastage, and the length of each piece must be maximized to minimize the number of segments.
2
Recall the formula for finding the HCF of fractions.
HCF of fractions = (HCF of numerators) / (LCM of denominators).
This formula is necessary to accurately compute the greatest common divisor of non-integer values.
3
Calculate the HCF of the numerators: 54, 81, and 108.
The factors give 54 = 2 × 27, 81 = 3 × 27, and 108 = 4 × 27. The HCF is 27.
We need the greatest integer that divides all the numerators evenly.
4
Calculate the LCM of the denominators: 5, 10, and 25.
The multiples of 25 are 25, 50, 75, etc. Since 50 is divisible by both 5 and 10, the LCM is 50.
We need the smallest common multiple for the denominators to complete the fraction formula.
5
Construct the final fraction.
The maximum length is 27/50 meters.
Dividing the computed numerator HCF by the denominator LCM gives the correct value.

Key Concept

HCF of Fractions
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Topics

HCF and LCM

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