If k : l = 4 : 3 and l : m = 5 : 3, what is the combined ratio k : l : m?

Introduction / Context:
This problem is a straightforward ratio chaining question. We are given relationships between k and l, and between l and m. The goal is to connect these pieces of information and determine a single three term ratio involving k, l and m. Problems like this highlight how ratios can be combined when one quantity is common in both ratios.

Given Data / Assumptions:

• k : l = 4 : 3.
• l : m = 5 : 3.
• We must find the triple ratio k : l : m.
• All quantities are positive and proportional as stated.

Concept / Approach:
The key is to make the intermediate term l consistent in both ratios so that they can be combined. We express k and l in terms of one variable from the first ratio, and l and m in terms of another variable from the second ratio. Then we equate the two expressions for l, find the relationship between the variables, and compute k, l and m in comparable units. Finally, we write k : l : m and simplify the numbers if needed.

Step-by-Step Solution:
From k : l = 4 : 3, let k = 4x and l = 3x. From l : m = 5 : 3, let l = 5y and m = 3y. Equate l from both descriptions: 3x = 5y. So y = 3x / 5. Now m = 3y = 3 * (3x / 5) = 9x / 5. To eliminate denominators, multiply all terms by 5. k becomes 4x * 5 = 20x. l becomes 3x * 5 = 15x. m becomes (9x / 5) * 5 = 9x. Thus k : l : m = 20x : 15x : 9x = 20 : 15 : 9.

Verification / Alternative check:
We can check by reconstructing the original pairwise ratios from 20 : 15 : 9. For k : l, we obtain 20 : 15, which reduces to 4 : 3 as required. For l : m, we obtain 15 : 9, which reduces to 5 : 3. Since both original ratios are satisfied, 20 : 15 : 9 is consistent and correct.

Why Other Options Are Wrong:
Ratios such as 18 : 24 : 11 or 9 : 15 : 1 do not reduce to the given pairwise ratios when tested. The ratio 21 : 7 : 3 yields k : l = 3 : 1, which does not match 4 : 3. Similarly, 12 : 9 : 5 is inconsistent with l : m = 5 : 3. Only 20 : 15 : 9 satisfies both given conditions simultaneously.

Common Pitfalls:
A common mistake is to simply place the numbers side by side as 4 : 3 : 5 : 3 without properly aligning the common term. Others may try to average or mix numbers incorrectly. Always treat the shared variable carefully and ensure that the value of l is the same in both representations before combining ratios.

Final Answer:
The combined ratio of k : l : m is 20 : 15 : 9.