The length, breadth and height of a room are 8 m 50 cm, 6 m 25 cm and 4 m 75 cm respectively. Find the length of the longest rod that can measure the dimensions of the room exactly.

To determine the length of the longest rod that can measure the dimensions of the room exactly, we need to find the greatest common divisor (GCD) of the room's dimensions. The dimensions given are 8 m 50 cm, 6 m 25 cm, and 4 m 75 cm. First, let's convert these measurements into a consistent unit, preferably centimeters, for easier calculation.

Converting Measurements

We can convert the dimensions as follows:

• 8 m 50 cm = (8 × 100) + 50 = 850 cm
• 6 m 25 cm = (6 × 100) + 25 = 625 cm
• 4 m 75 cm = (4 × 100) + 75 = 475 cm

Finding the GCD

Now that we have the dimensions in centimeters, we can find the GCD of 850, 625, and 475. The GCD is the largest number that divides all three numbers without leaving a remainder.

Step-by-Step Calculation

We can use the prime factorization method or the Euclidean algorithm to find the GCD. Here, we'll use the Euclidean algorithm:

1. First, find the GCD of the first two numbers, 850 and 625:
• 850 ÷ 625 = 1 remainder 225
• 625 ÷ 225 = 2 remainder 175
• 225 ÷ 175 = 1 remainder 50
• 175 ÷ 50 = 3 remainder 25
• 50 ÷ 25 = 2 remainder 0
2. So, GCD(850, 625) = 25.

Next, we find the GCD of this result (25) with the third dimension (475):

1. 475 ÷ 25 = 19 remainder 0

Since the remainder is 0, the GCD of 25 and 475 is also 25.

Final Result

The longest rod that can measure the dimensions of the room exactly is therefore 25 cm. This means that any length of rod that is a multiple of 25 cm can be used to measure the room's dimensions without any leftover space.