Calculate total price of the items

Since the average price of the 3 items of furniture is $\text{Rs } 16860$, we multiply the average by the number of items ($3$) to find the total sum.

$\text{Total Sum} = 16860 \times 3 = 50580$

Set up the ratio equation

Let the prices of the three items be $3x$, $5x$, and $7x$. The sum of all parts is $3x + 5x + 7x = 15x$. Equate this to the total sum.

$15x = 50580$

Find the value of one unit

Divide the total sum by the total ratio parts ($15$) to solve for $x$.

$x = \frac{50580}{15} = 3372$

Determine the price of the cheapest item

The cheapest item corresponds to the smallest ratio term, which is $3x$. Multiply $x$ by $3$.

$\text{Cheapest Price} = 3 \times 3372 = 10116$