The marks obtained by Vijay and Amith are in the ratio 4 : 5, and the marks obtained by Amith and Abhishek are in the ratio 3 : 2. What is the ratio of the marks obtained by Vijay and Abhishek?

Introduction / Context:
This question requires combining two ratios that share a common term in order to deduce the ratio between two other persons. We are given the ratio of marks of Vijay and Amith and the ratio of marks of Amith and Abhishek, and we must find the ratio of Vijay to Abhishek. This is a standard technique in ratio and proportion problems and is very useful in many competitive examinations.

Given Data / Assumptions:

• Vijay : Amith = 4 : 5.
• Amith : Abhishek = 3 : 2.
• All marks are positive numbers.
• We are required to compute Vijay : Abhishek.

Concept / Approach:
The method is to express all marks in terms of a single variable by using the common person between the given ratios, which is Amith. From Vijay : Amith = 4 : 5, we can write Vijay and Amith in terms of one variable. From Amith : Abhishek = 3 : 2, we can write Amith and Abhishek in terms of another variable. Equating Amith's values from both expressions allows us to find a consistent set of values and then form the desired ratio Vijay : Abhishek.

Step-by-Step Solution:
Let Vijay's marks be 4x and Amith's marks be 5x. Also, from the second ratio, let Amith's marks be 3y and Abhishek's marks be 2y. Since Amith has the same marks in both descriptions, set 5x = 3y. This gives y = (5x) / 3. Now Abhishek's marks are 2y = 2 * (5x / 3) = 10x / 3. Vijay's marks are 4x. Therefore Vijay : Abhishek = 4x : (10x / 3). Multiply both terms by 3 to clear the denominator: 12x : 10x. Cancel x to get 12 : 10, which simplifies to 6 : 5.

Verification / Alternative check:
We may choose numerical values that satisfy both initial ratios. For example, let Amith have 15 marks. Then Vijay must have (4 / 5) * 15 = 12 marks. From Amith : Abhishek = 3 : 2 with Amith = 15, Abhishek has (2 / 3) * 15 = 10 marks. So Vijay : Abhishek = 12 : 10, which simplifies to 6 : 5. This matches the algebraic result.

Why Other Options Are Wrong:
The option 4 : 5 repeats the Vijay : Amith ratio, which is not requested. The ratio 3 : 2 is the Amith : Abhishek ratio, not Vijay : Abhishek. The ratio 1 : 3 is unrelated to the given proportions, and 5 : 6 is simply the inverse of the correct ratio. Only 6 : 5 correctly represents Vijay : Abhishek.

Common Pitfalls:
A common error is to multiply or divide the ratios incorrectly, for example directly combining 4 : 5 and 3 : 2 to get 12 : 10 without making sure which pair of names is being compared. Always use the common term (here Amith) to align the ratios properly before forming the new ratio.

Final Answer:
The ratio of the marks obtained by Vijay and Abhishek is 6 : 5.