An artisan is preparing metal rods for a custom fence. Two existing rods, which measure \f… — State PSC Exam

An artisan is preparing metal rods for a custom fence. Two existing rods, which measure $\frac{15}{4}$ meters and $\frac{25}{8}$ meters in length, must be cut into identical smaller pieces of the maximum possible length without wasting any material. What should be the length of each piece?

A
$\frac{75}{4}$ meters
B
$\frac{5}{4}$ meters
$\frac{5}{8}$ metersAnswer
D
$\frac{75}{8}$ meters

Answer

\frac{5}{8}

meters

The maximum possible length of each piece is found by calculating the HCF of the two fractional lengths. The correct formula is to divide the HCF of the numerators (15 and 25) by the LCM of the denominators (4 and 8). The HCF of 15 and 25 is 5, and the LCM of 4 and 8 is 8, resulting in exactly

\frac{5}{8}

meters.

Step-by-Step Solution

Identify the mathematical operation required for the scenario.

Finding the maximum possible identical length from two given lengths requires calculating their Highest Common Factor (HCF).

The pieces must be of equal length and as large as possible without leaving any remainder.

State the formula for finding the HCF of fractions.

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

This is the standard rule for determining the greatest common divisor of rational numbers.

Calculate the HCF of the numerators (15 and 25).

The factors of 15 are 1, 3, 5, 15. The factors of 25 are 1, 5, 25. The highest common factor is 5.

The numerator of our final answer must be the HCF of the original numerators.

Calculate the LCM of the denominators (4 and 8).

The multiples of 8 (8, 16, 24...) are already divisible by 4. Thus, the least common multiple is 8.

The denominator of our final answer must be the LCM of the original denominators.

Construct the final fraction.

\frac{5}{8}

meters.

Combining the results from the previous steps yields the correct maximum length.

Key Concept

Highest Common Factor (HCF) of fractions

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Key Concept

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