Using properties of proportion solve:

$\dfrac{x^2 - x + 1}{x^2 + x + 1} = \dfrac{112(1 - x)}{104(1 + x)}$

If x ≠ y and $\dfrac{x^2 - x + 1}{y^2 - y + 1} = \dfrac{x^2 + x + 1}{y^2 + y + 1}$, prove that xy = 1. Use properties of proportion.

Case study:
Amit runs a grocery store in a busy neighbourhood market. He stocks two varieties of lentils.

The premium quality sells at ₹ 280 per kg.

The standard quality sells at ₹ 240 per kg.

Customers in his area are price-conscious, so Amit wants to create a mixture of the two varieties that appears high quality but is also affordable. He decides to sell the mixture at ₹ 265 per kg.

In what ratio should the ₹ 240/kg and ₹ 280/kg varieties be mixed to obtain the mixture costing ₹ 265 per kg?

Case study:
In a school chemistry lab, the lab assistant has two bottles of acid solution on his shelf:

Solution A: 55% acid concentration

Solution B: 80% acid concentration

The science teacher wants a 60% acid solution for tomorrow’s experiment that is to be used by the students.

In what ratio should the 55% solution and the 80% solution be mixed to obtain a 60% acid solution?