Using √3 ≈ 1.732, calculate the value of the expression √3 / (√3 + 3) correct to three decimal places.

Introduction / Context:
This question tests simple numerical approximation skills and basic fraction manipulation. You are asked to evaluate a surd expression numerically using an approximate value for sqrt(3) and then round the result to three decimal places. This is a common type of question in quantitative aptitude tests, especially when calculators are not allowed.

Given Data / Assumptions:

• Use the approximation √3 ≈ 1.732.
• The expression is √3 / (√3 + 3).
• We need the value correct to three decimal places.
• Standard rules of decimal division and rounding apply.

Concept / Approach:
First substitute the approximate value 1.732 for √3 in both numerator and denominator. Then compute the numerator and denominator as decimals and perform the division. Finally, round the result to three decimal places. Because both numerator and denominator contain the same approximation for √3, the final value is stable and does not require any rationalisation of the denominator.

Step-by-Step Solution:
Substitute √3 ≈ 1.732 into the expression.The numerator becomes 1.732.The denominator becomes 1.732 + 3 = 4.732.Now compute the fraction 1.732 / 4.732.Performing the division (for example by long division or an approximate method) gives a value close to 0.366.Rounded to three decimal places, the result is 0.366.

Verification / Alternative check:
You can check this division by multiplication. Take 0.366 and multiply by 4.732. Compute 4.732 * 0.3 = 1.4196. Then 4.732 * 0.06 = 0.28392, and 4.732 * 0.006 = 0.028392. Adding these parts: 1.4196 + 0.28392 + 0.028392 ≈ 1.7319, which is extremely close to the original numerator 1.732, confirming that 0.366 is a very good approximation of the quotient.

Why Other Options Are Wrong:

• 0.566, 0.356, 0.346, and 0.376 yield products with 4.732 that are noticeably different from 1.732.
• For example, 4.732 * 0.356 is approximately 1.685, which is too small.
• 0.566 would give a product above 2.6, which is far too large.
• Only 0.366 reproduces the numerator accurately when multiplied back by the denominator.

Common Pitfalls:

• Forgetting to add 3 to √3 in the denominator and instead computing 1.732 / 3.
• Incorrect rounding, such as cutting off at two decimal places instead of three.
• Basic calculation errors during long division of 1.732 by 4.732.

Final Answer:
0.366