Simontaneous equation and profit and loss

The total cost of 2 chairs and 3 tables is $180.

$$2c + 3t = 180 \quad (*)$$

A 20% profit on a chair means its selling price is $c + 0.20c = 1.20c$.
A 10% loss on a table means its selling price is $t - 0.10t = 0.90t$.
The total selling price for 2 chairs and 3 tables is $192.

$$2(1.20c) + 3(0.90t) = 192$$ $$2.40c + 2.70t = 192 \quad (**)$$

From equation $(*)$, express $c$ in terms of $t$:

$$2c = 180 - 3t$$ $$c = 90 - \frac{3}{2}t$$

Substitute this expression for $c$ into equation $(**)$:

$$2.40\left(90 - \frac{3}{2}t\right) + 2.70t = 192$$ $$216 - 3.6t + 2.7t = 192$$ $$216 - 0.9t = 192$$ $$0.9t = 216 - 192$$ $$0.9t = 24$$ $$t = \frac{24}{0.9} = \frac{240}{9} = \frac{80}{3}$$

Now substitute the value of $t$ back into the expression for $c$:

$$c = 90 - \frac{3}{2}\left(\frac{80}{3}\right)$$ $$c = 90 - \frac{80}{2}$$ $$c = 90 - 40$$ $$c = 50$$

The cost price of one chair is $50 and the cost price of one table is$\frac{80}{3}$(approximately$26.67).

The cost price of one chair is \boxed{\50}$and the cost price of one table is$\boxed{$ \frac{80}{3}}$.