An industrial cutting machine is programmed to divide a rectangular metallic sheet measuri… — State PSC Exam

An industrial cutting machine is programmed to divide a rectangular metallic sheet measuring $\frac{24}{5}$ meters in length and $\frac{16}{3}$ meters in width into identical square plates. If the square plates must be of the maximum possible size without leaving any wasted material, what is the area of each resulting square plate?

$\frac{64}{225} \text{ m}^2$ Answer
B
2304 \text{ m}^2
C
64 \text{ m}^2
D
$\frac{256}{25} \text{ m}^2$

Answer

The area of each resulting square plate is \frac{64}{225} \text{ m}^2.

To cut the rectangle into identical squares of maximum size without waste, the side length of the square must be the Highest Common Factor (HCF) of the rectangle's dimensions. Using the fraction formula, the HCF of

\frac{24}{5}

and

\frac{16}{3}

\frac{\text{HCF}(24, 16)}{\text{LCM}(5, 3)} = \frac{8}{15}

meters. The area of the square is the side length squared, which is

(\frac{8}{15})^2 = \frac{64}{225} \text{ m}^2

Step-by-Step Solution

Determine the required mathematical operation for finding the maximum square size.

The side length of the square must be the Highest Common Factor (HCF) of the length (24/5) and width (16/3).

To divide a rectangle into identical squares without waste, the square's side must perfectly divide both dimensions. The 'maximum size' implies finding the greatest common divisor.

Apply the rule for finding the HCF of fractions.

HCF(a/b, c/d) = HCF(a, c) / LCM(b, d). Thus, HCF(24/5, 16/3) = HCF(24, 16) / LCM(5, 3).

The standard formula for the HCF of fractions requires taking the HCF of the numerators divided by the LCM of the denominators.

Calculate the specific HCF and LCM values.

HCF(24, 16) = 8 and LCM(5, 3) = 15. The side length of the square is \frac{8}{15} meters.

8 is the largest integer dividing both 24 and 16. 15 is the smallest integer divisible by both 5 and 3.

Calculate the area of the square.

Area = (\frac{8}{15})^2 = \frac{64}{225} \text{ m}^2.

The question specifically asks for the area of the plate, which is the square of its side length.

Key Concept

HCF of Fractions and Geometric Application

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