A metal rod is divided into three parts, A, B and C. The lengths of A and B are in the mratio 9 : 20, and the lengths of C and B are in the ratio 5 : 11. If the difference between the lengths of A and C is 10 cm, find the total length (in cm) of the metal rod.

Correct Answer: The correct option is 4190 cm (Option 3).

Step-by-Step Explanation:

Step 1: Express the given ratios with a common term.

We are given the ratios of the lengths of three parts of a metal rod, A, B, and C:

Ratio of A to B = $A:B=9:20$

Ratio of C to B = $C:B=5:11$, which means the ratio of B to C is $B:C=11:5$.

To combine these into a single ratio $A:B:C$, we need to make the term for B equal in both ratios.

The values for B in the two ratios are 20 and 11. The Least Common Multiple (LCM) of 20 and 11 is $20\×11=220$.

Multiply the first ratio $A:B$ by 11:

$A:B=(9\×11):(20\×11)=99:220$

Multiply the second ratio $B:C$ by 20:

$B:C=(11\×20):(5\×20)=220:100$

Thus, the combined ratio of the lengths of A, B, and C is:

$A:B:C=99:220:100$

Step 2: Set up equations using a common variable.

Let the lengths of parts A, B, and C be represented as:

$A=99x$

$B=220x$

$C=100x$

where $x$ is a positive multiplier constant.

Step 3: Use the given difference to find $x$.

We are given that the difference between the lengths of C and A is 10 cm:

$C-A=10$

$100x-99x=10$

$1x=10$

$x=10$

Step 4: Calculate the total length of the metal rod.

The total length of the rod is the sum of the lengths of all three parts:

$\text{Total Length}=A+B+C$

$\text{Total Length}=99x+220x+100x=419x$

Substitute $x=10$ into the total length equation:

$\text{Total Length}=419\×10=4190\text{cm}$

Therefore, the total length of the metal rod is 4190 cm.