Finding the Combined Ratio A : B : C : D

This problem requires us to determine the combined ratio A : B : C : D. We are given three separate ratios that link these variables: A : B, B : C, and C : D. To find the combined ratio, we need to make sure the terms common to multiple ratios have the same numerical value.

Given Ratios

The initial ratios provided are:

• A : B = 2 : 3
• B : C = 3 : 4
• C : D = 3 : 5

Method for Combining Ratios

The process involves linking the ratios through shared variables. We start by combining the first two ratios and then incorporate the third ratio.

Step 1: Combining A : B and B : C

First, let's look at the ratios involving A, B, and C:

• A : B = 2 : 3
• B : C = 3 : 4

The variable B is common to both ratios. In the first ratio, B corresponds to 3. In the second ratio, B also corresponds to 3. Since the value for B is the same in both ratios, we can directly combine them.

Therefore, the combined ratio A : B : C is:

Step 2: Combining A : B : C with C : D

Next, we combine the ratio found in Step 1 with the third given ratio:

• A : B : C = 2 : 3 : 4
• C : D = 3 : 5

The variable C is common here. In the ratio A : B : C, C corresponds to 4. In the ratio C : D, C corresponds to 3.

The values for C (4 and 3) are different. To combine these ratios, we must make the value for C the same in both. We do this by finding the Least Common Multiple (LCM) of the two values, 4 and 3.

The LCM of 4 and 3 is 12.

Now, we adjust each ratio to make the value for C equal to 12:

• For the ratio A : B : C = 2 : 3 : 4, we multiply each part by 3 (because 12 / 4 = 3): A : B : C = (2 \times 3) : (3 \times 3) : (4 \times 3) = 6 : 9 : 12
• For the ratio C : D = 3 : 5, we multiply each part by 4 (because 12 / 3 = 4): C : D = (3 \times 4) : (5 \times 4) = 12 : 20

With the value for C now matching (12) in both adjusted ratios, we can combine them to find the final ratio A : B : C : D:

Final Answer Verification

The resulting combined ratio is 6 : 9 : 12 : 20. Comparing this with the given options, we find that it matches the fourth option.