If a + b + c = 534, 2a : b = 15 : 4 and b : c = 6 : 5, then what is the value of a?

Appeared in: CGHS SSC - 2018

Explanation

The first step is to simplify the ratio 2a : b = 15 : 4 to a : b = 15 : 8.
Next, combine the two ratios (a : b = 15 : 8 and b : c = 6 : 5) by finding the Least Common Multiple (LCM) of the common term 'b' (LCM of 8 and 6 is 24).
This results in a unified ratio of a : b : c = 45 : 24 : 20.
Let a = 45x, b = 24x, and c = 20x. The sum is 45x + 24x + 20x = 89x.
Given that the total sum is 534, we solve for the common multiplier: 89x = 534, which gives x = 6.
Finally, calculate the value of 'a' by substituting x: a = 45 * 6 = 270.

Option B: This value represents 'c', not 'a'. After finding the common multiplier x = 6, the value of c is calculated as c = 20x = 20 * 6 = 120.
Option C: This value (180) would result if the common multiplier 'x' was incorrectly calculated as 4 (180 / 45 = 4). If x were 4, the total sum would be 89 * 4 = 356, not 534.
Option D: This value (80) would result from a calculation error, possibly by misinterpreting the ratios or incorrectly calculating the multiplier 'x'. For example, if one incorrectly used c=20x and x=4, the result would be 80.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving for a variable using combined ratios as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical reasoning and problem-solving skills.
While not a direct clinical scenario, numeracy is a critical competency for nurses, essential for tasks like medication dosage calculations, fluid balance monitoring (intake/output), and interpreting lab results.
What if? If the first ratio was a : 2b = 15 : 4, the simplified ratio would become a : b = 15 : 2. The combined ratio would change, leading to a different final answer.

How to Approach the Question

First, analyze the given information: two separate ratios (2a:b and b:c) and the total sum of the variables (a+b+c).
The primary goal is to find the value of 'a'. To do this, you must first find a single ratio that connects a, b, and c.
Simplify the first ratio (2a : b = 15 : 4) to get the relationship between 'a' and 'b' (a : b = 15 : 8).
Identify the common term in both ratios, which is 'b'.
Standardize the ratios by finding the Least Common Multiple (LCM) of the values corresponding to 'b' (8 and 6). The LCM is 24.
Adjust both ratios to make the 'b' term equal to 24, which gives the combined ratio a : b : c = 45 : 24 : 20.

Concept Tested & Keywords

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Attempt every question from this paper in a timed mock, then review the full solution for each one.