In a cricket match, Virat, Dhoni and Rohit together score 285 runs. The ratio of Virat’s runs to Rohit’s runs is 3 : 2 and the ratio of Rohit’s runs to Dhoni’s runs is also 3 : 2. How many runs did Virat score in that match?

Introduction / Context:
This question tests understanding of chained ratios, where several people or quantities are compared in linked proportions. Here, the runs scored by Virat, Dhoni and Rohit are related through two given ratios, and we must convert these into a single combined relationship to find the exact runs scored by Virat.

Given Data / Assumptions:
• Total runs scored by Virat, Dhoni and Rohit together = 285.
• Virat : Rohit = 3 : 2.
• Rohit : Dhoni = 3 : 2.
• All runs are whole numbers and the ratios are exact.

Concept / Approach:
When we have chained ratios, the key idea is to express all quantities in terms of a common variable using both ratios. We first write Virat and Rohit in one ratio and Rohit and Dhoni in another. By matching the common player Rohit between the two ratios, we can express Virat, Rohit and Dhoni all in terms of a single parameter. Then we use the total sum to solve for the parameter and finally obtain Virat’s individual score.

Step-by-Step Solution:
Step 1: From Virat : Rohit = 3 : 2, let Virat = 3x and Rohit = 2x.Step 2: From Rohit : Dhoni = 3 : 2, let Rohit = 3y and Dhoni = 2y.Step 3: Since Rohit is the same player in both expressions, set 2x = 3y. This gives x = 3y / 2.Step 4: Substitute x into Virat’s runs: Virat = 3x = 3 * (3y / 2) = 9y / 2.Step 5: Rohit = 3y and Dhoni = 2y, so total runs = (9y / 2) + 3y + 2y = (9y / 2) + 5y = (9y + 10y) / 2 = 19y / 2.Step 6: Set 19y / 2 = 285. So 19y = 570, giving y = 30.Step 7: Virat’s runs = 9y / 2 = 9 * 30 / 2 = 270 / 2 = 135.

Verification / Alternative check:
Using y = 30, Rohit’s runs are 3y = 90, and Dhoni’s runs are 2y = 60. Check the totals: Virat 135 + Rohit 90 + Dhoni 60 = 285, which matches the given total. Also verify ratios: Virat : Rohit = 135 : 90 = 3 : 2, and Rohit : Dhoni = 90 : 60 = 3 : 2. All conditions are satisfied, so the value for Virat’s runs is consistent and correct.

Why Other Options Are Wrong:
Option 154 would give remaining runs of 285 - 154 = 131 for Rohit and Dhoni combined, which cannot be split to maintain both given ratios simultaneously.
Option 112 and option 106 similarly fail when you try to assign runs to Rohit and Dhoni while keeping both ratios and the total sum correct. Only 135 leads to integer values that exactly satisfy all constraints.

Common Pitfalls:
A frequent mistake is to add ratios directly instead of converting them into a combined relation. Another error is to assume Virat : Dhoni is directly 3 : 2, which is not given. It is essential to use Rohit as the common link and set the different expressions for Rohit equal to each other so that a single variable description is obtained.

Final Answer:
The number of runs scored by Virat in the match is 135.