Calculate Total Price

Since the average price of the three items is given as $16845$, the total price of all three items together is obtained by multiplying the average by the number of items ($3$).

$\text{Total Sum} = 16845 \times 3 = 50535$

Use Ratio to Find Parts

The prices are in the ratio $3: 5: 7$. Let the prices be $3x$, $5x$, and $7x$. The sum of these ratio parts represents the total price.

$3x + 5x + 7x = 15x$

Solve for x

Equating the total ratio sum to the total price calculated in step 1, we find the value of $x$.

$15x = 50535 \implies x = \frac{50535}{15} = 3369$

Calculate the Cheapest Item

The cheapest item corresponds to the smallest ratio term, which is $3$. Substituting $x = 3369$, we get the price of the cheapest item.

$\text{Cheapest Price} = 3x = 3 \times 3369 = 10107$