Question

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

Answer 1

Hint: First, we will convert the given dimensions into centimeters. Then we find the highest common factor, that is, H.C.F. of the given dimensions to find the longest rod which can measure the three dimensions of the room exactly.

Complete step-by-step answer:

Given that the length, breadth and height of a room are 8m 25cm, 6 m 75 cm and 4 m 50 cm, respectively.

We know that \[1{\text{ m}} = 100{\text{ cm}}\].

First, we will convert the length of the room, that is, 8 m 25 cm in centimetres.

\[
  8 \times 100 + 25 = 800 + 25 \\
   = 825{\text{ cm}} \\
\]

Now we will convert the breadth of the room, that is, 6 m 75 cm in centimetres.

\[
  6 \times 100 + 75 = 600 + 75 \\
   = 675{\text{ cm}} \\
\]

We will now convert the length of the room, that is, 4 m 50 cm in centimetres.

\[
  4 \times 100 + 50 = 400 + 50 \\
   = 450{\text{ cm}} \\
\]

Prime factorizing the numbers 825 cm, 675 cm and 450 cm, we get

\[825 = 3 \times 5 \times 5 \times 11\]
\[675 = 3 \times 3 \times 3 \times 5 \times 5\]
\[450 = 2 \times 3 \times 3 \times 5 \times 5\]

We will now find the common factors from the above prime factorization of 825, 765, and 450.

\[3 \times 5 \times 5\]

Multiplying the above numbers to find the H.C.F., we get

\[{\text{H.C.F}}. = 75\]

Thus, we now have that the highest common factor, i.e., H.C.F. of 825, 675 and 450 is 75.

Hence, the length of the longest tap is 75 cm, which can measure the three dimensions of the room exactly.

Note: In solving these types of questions, you should be familiar with the steps to find the highest common factor of the three numbers. The question is very simple to solve but students should be able to find the main point that the HCF of the length, breadth and height will be the length of the longest rod, which will measure the dimensions exactly. Then, calculate the H.C.F. without making any calculation mistakes.