A civil engineering team is installing three different types of sensor cables along a newl… — State PSC Exam

A civil engineering team is installing three different types of sensor cables along a newly constructed bridge. The standard supplier rolls for these cables come in lengths of $\frac{55}{2}$ meters, $\frac{77}{4}$ meters, and $\frac{121}{5}$ meters. To ensure modular replacement without any waste, the team must cut all the rolls into smaller, equal-length segments. What is the maximum possible length, in meters, of each segment?

Answer: 0.55 meters

Answer

0.55 meters

The problem asks for the maximum equal length that can divide all three roll lengths without any remainder, which mathematically corresponds to the Highest Common Factor (HCF). Using the standard formula for fractions: HCF = HCF(55, 77, 121) / LCM(2, 4, 5) = 11 / 20 = 0.55 meters.

Step-by-Step Solution

Determine the correct mathematical operation based on the problem context.

The requirement to cut multiple rolls into equal segments of maximum possible length with zero waste indicates the need to find the Highest Common Factor (HCF) of the three lengths.

HCF is used to find the largest common quantity that can perfectly divide multiple numbers.

State the standard formula for finding the HCF of fractions.

HCF = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

This is the mathematical rule for finding the HCF of rational numbers.

Calculate the HCF of the numerators.

The numerators are

55

77

, and

121

. Their prime factorizations are

5 \times 11

7 \times 11

, and

11 \times 11

. The HCF is

11

The largest common prime factor across all three numerators is 11.

Calculate the LCM of the denominators.

The denominators are

2

4

, and

5

. The lowest common multiple is

20

20 is the smallest positive integer divisible by 2, 4, and 5.

Compute the final decimal value.

\frac{11}{20} = 0.55

Converting the fraction to a decimal yields the exact required length.

Key Concept

Highest Common Factor (HCF) of Fractions

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