Let x = square root of 8 minus square root of 7, y = square root of 6 minus square root of 5, and z = square root of 10 minus 3; determine the correct order relation among x, y, and z from the given options.

Introduction / Context:
This problem asks us to compare three expressions involving square roots and differences. Such questions are designed to test the ability to estimate and compare irrational numbers without necessarily computing them to high precision. By using approximate values or algebraic reasoning, we can determine the correct inequality relation between x, y, and z.

Given Data / Assumptions:
- x is defined as square root of 8 minus square root of 7.
- y is defined as square root of 6 minus square root of 5.
- z is defined as square root of 10 minus 3.
- Square roots are taken in their principal (positive) values.
- All comparisons are made in the real number system.

Concept / Approach:
One approach is to approximate each square root using known nearby perfect squares and then compute rough decimal values for x, y, and z. Since all three expressions are small positive numbers, even modest accuracy is usually enough to determine their ordering. Another approach involves manipulating the expressions algebraically, but in this case numerical estimation is both faster and sufficiently precise for an aptitude question.

Step-by-Step Solution:
Step 1: Approximate square roots using known benchmarks. We know that 2.8 squared is 7.84 and 2.9 squared is 8.41, so square root of 8 is a little less than 2.83. Square root of 7 is a bit more than 2.64. Step 2: From these approximations, x = square root of 8 minus square root of 7 is roughly 2.83 - 2.65, giving about 0.18. Step 3: Approximate square root of 6 and square root of 5. Since 2.4 squared is 5.76 and 2.5 squared is 6.25, square root of 6 is around 2.45, and square root of 5 is around 2.24. Step 4: Then y = square root of 6 minus square root of 5 is approximately 2.45 - 2.24, giving about 0.21. Step 5: Approximate square root of 10. Since 3.1 squared is 9.61 and 3.2 squared is 10.24, square root of 10 is about 3.16. Step 6: Compute z = square root of 10 minus 3, which is about 3.16 - 3.00 = 0.16. Step 7: Comparing the approximate values, we have z around 0.16, x around 0.18, and y around 0.21. Step 8: Therefore z is the smallest, y is the largest, and x lies between them, so the relation is z < x < y.

Verification / Alternative check:
We can refine the approximations slightly. Using more precise estimates, square root of 8 is about 2.828, square root of 7 is about 2.646, so x is about 0.182. Square root of 6 is about 2.449 and square root of 5 is about 2.236, so y is about 0.213. Square root of 10 is about 3.162, so z is about 0.162. The refined values confirm the ordering z < x < y. This consistency between rough and refined estimates strengthens confidence in the inequality chosen.

Why Other Options Are Wrong:
Option x > y < z: This states that y is smaller than both x and z, which contradicts the calculations showing that y is actually the largest among the three.
Option x < y < z: This implies z is the largest, but z has the smallest approximate value, so this relation is incorrect.
Option x > y > z: This says x is the largest and z is the smallest, but in reality y is larger than x.
Option z < x < y: This matches our ordering and is correct, so the remaining option z < y < x is necessarily wrong.
Option z < y < x: This would place y between z and x, but we have already established that x lies between z and y, not the other way around.

Common Pitfalls:
A common error is to treat square root of n as roughly equal to n divided by 2 or some other inaccurate heuristic, which leads to wrong comparisons. Another pitfall is to compare the differences of the radicands directly, for example assuming that because 8 - 7 is 1 and 6 - 5 is also 1, the differences must be equal, which is not true for square roots. Careful approximation or algebraic reasoning is needed whenever square roots are involved.

Final Answer:
The correct order relation is z < x < y.