SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

For three terms x, y, and z, if their sum x + y + z = 0, then the sum of their cubes satisfies the identity: x³ + y³ + z³ = 3xyz

Applying this identity to the numerator of the expression: Let x = a - b, y = b - c, and z = c - a. Since x + y + z = (a - b) + (b - c) + (c - a) = 0, we can write the sum of their cubes as: (a - b)³ + (b - c)³ + (c - a)³ = 3(a - b)(b - c)(c - a)

Similarly, applying this identity to the denominator of the expression: Let X = a² - b², Y = b² - c², and Z = c² - a². Since X + Y + Z = (a² - b²) + (b² - c²) + (c² - a²) = 0, the sum of their cubes is: (a² - b²)³ + (b² - c²)³ + (c² - a²)³ = 3(a² - b²)(b² - c²)(c² - a²)

Using the difference of squares identity: a² - b² = (a - b)(a + b) b² - c² = (b - c)(b + c) c² - a² = (c - a)(c + a)

The denominator simplifies to: 3(a - b)(a + b)(b - c)(b + c)(c - a)(c + a)

Now, dividing the simplified numerator by the simplified denominator: [3(a - b)(b - c)(c - a)] / [3(a - b)(a + b)(b - c)(b + c)(c - a)(c + a)]

Canceling out the common terms 3(a - b)(b - c)(c - a) from both the numerator and the denominator yields: 1 / ((a + b)(b + c)(c + a))

Explanation

When two items are sold at the same selling price, one at a gain of x% and the other at a loss of x%, there is always an overall loss. This overall loss percentage can be directly calculated using the standard formula:

Overall Loss % = (x / 10)² %

Given that the gain and loss percentage x is 20%:

Overall Loss % = (20 / 10)² % = 2² % = 4%