Using the approximation √42 ≈ 6.480, estimate the value of the square root √(6/7) correct to three decimal places.

Introduction / Context:
This question checks understanding of square roots, manipulation of fractions under a square root, and use of a given approximate value to estimate another root. Such problems are common in exams where calculators are not allowed, and candidates must use number sense and algebraic relationships between related square roots. Here we are given an approximate value of √42 and asked to estimate √(6/7).

Given Data / Assumptions:

• Given approximation: √42 ≈ 6.480.

• Target expression: √(6/7).

• We assume positive square roots only, since we are dealing with magnitudes.

• Required answer is an approximate decimal value, chosen from the options.

Concept / Approach:
The idea is to rewrite 6/7 in a form that involves 42, because we are given √42. Notice that 6/7 = 36/42. This allows us to write √(6/7) as √(36/42). Then we can separate numerator and denominator using the property √(a/b) = √a / √b for positive a and b. Since √36 is exactly 6 and we know an approximation for √42, we can form a ratio 6 / √42 and use the given decimal to estimate the value.

Step-by-Step Solution:
Step 1: Rewrite the fraction: 6/7 = 36/42. Step 2: Express the root: √(6/7) = √(36/42). Step 3: Use the property of roots: √(36/42) = √36 / √42 for positive numbers. Step 4: Compute √36 = 6 exactly. Step 5: Substitute the approximation √42 ≈ 6.480. Step 6: Form the ratio: √(6/7) ≈ 6 ÷ 6.480. Step 7: Evaluate 6 ÷ 6.480 ≈ 0.9259, which rounds to approximately 0.926.

Verification / Alternative check:
We know that 6/7 is less than 1, so its square root must also be less than 1. Among the options, only 0.915 and 0.925 are significantly below 1. A quick mental calculation can bound the value. Since √(6/7) is close to √(0.857), and √0.81 = 0.9 while √0.9 ≈ 0.949, we expect the answer somewhere in the low 0.9 range. The approximate calculation 6 ÷ 6.480 gives about 0.926, which is closer to 0.925 than to 0.935 or other options. This confirms the chosen option.

Why Other Options Are Wrong:
Option 0.935 and 0.945 are somewhat larger than the calculated value 0.926 and would correspond to a slightly larger fraction under the root. Option 0.955 is even larger and would match a value much closer to 1 or above. Option 0.915 is a little smaller than 0.926 and therefore underestimates the root. Given the approximation provided, 0.925 is the most accurate among the listed answers.

Common Pitfalls:
Learners may directly divide 6 by 7 and then try to approximate the square root of the resulting decimal, which is slower and more error prone. Others may forget to use the relationship between 6/7 and 36/42, losing the benefit of the given √42 approximation. Rounding too early, for example using 6.5 instead of 6.480, can push the estimate away from the correct option. Careful algebraic manipulation followed by precise division avoids these mistakes.

Final Answer:
The approximate value of √(6/7) using √42 ≈ 6.480 is 0.925.