If A : B = 1 : 4 and B : C = 2 : 3, then what is the combined ratio A : B : C for the three quantities?

Introduction / Context:
This question is a straightforward ratio chaining problem. We are given two separate ratios involving three variables A, B and C. To find the combined three term ratio A : B : C, we need to make the common term B consistent in both given ratios and then combine the information. This type of question checks understanding of how to merge partial ratio information into a single comprehensive ratio.

Given Data / Assumptions:
- A : B = 1 : 4.
- B : C = 2 : 3.
- B is the common term in both ratios.
- We are required to find the three term ratio A : B : C.

Concept / Approach:
When two ratios share a common term, we first rewrite them so that the value of the common term is represented by the same number in both ratios. Then, we can directly read off the corresponding values of A and C. This method leverages the idea that ratios may be multiplied by the same factor on both sides without changing their meaning. The key is to align the second term of the first ratio with the first term of the second ratio since both represent B.

Step-by-Step Solution:
Step 1: Write the first ratio: A : B = 1 : 4. Step 2: Write the second ratio: B : C = 2 : 3. Step 3: The common term B is 4 units in the first ratio and 2 units in the second ratio. Step 4: To make B equal in both ratios, find a common multiple of 4 and 2 which is 4. Step 5: Keep A : B = 1 : 4 as it is. Multiply B : C = 2 : 3 by 2 to get 4 : 6. Step 6: Now B is 4 in both, so A : B : C = 1 : 4 : 6.

Verification / Alternative check:
You can check by assuming an actual value for B. Suppose B corresponds to 4 units. From A : B = 1 : 4, A would be 1 unit. From B : C = 2 : 3, if B is 4 then 2 units map to 4, so 1 unit is 2, and 3 units is 6. Therefore, C is 6. This directly produces the ratio A : B : C = 1 : 4 : 6, confirming the earlier ratio alignment method is correct.

Why Other Options Are Wrong:
2 : 8 : 7 would imply C is 7 when B is 8, but that does not match the scaled ratio B : C = 2 : 3. Similarly, 2 : 8 : 5 and 1 : 4 : 5 misrepresent the relationship between B and C. Only 1 : 4 : 6 maintains both original conditions correctly when broken back into A : B and B : C.

Common Pitfalls:
Students sometimes attempt to directly join the ratios as 1 : 4 : 3 or 1 : 2 : 3, which is incorrect because it ignores the need to equalize the common term B. Another frequent error is multiplying the wrong side of one ratio or failing to scale both parts of the ratio by the same factor. Careful alignment of the common variable is critical in such problems.

Final Answer:
The combined ratio of the three quantities is 1 : 4 : 6.