Understanding the Furniture Price Problem

This problem involves finding the price of the cheapest item among three furniture pieces. We are given the average price of these three items and the ratio of their individual prices. Our goal is to use this information to calculate the exact cost of the least expensive item.

Calculating the Item Prices Using Ratio

Let the prices of the three furniture items be represented according to the given ratio 3 : 5 : 7. We can denote these prices as:

• Price 1: $3x$
• Price 2: $5x$
• Price 3: $7x$

Here, '$x$' is a common multiplier that helps maintain the ratio between the prices.

Using the Average Price

The average price of the three items is given as ₹16,050. The formula for the average price is the sum of the prices divided by the number of items.

Sum of prices = $3x + 5x + 7x = 15x$

Number of items = 3

Average price = $\frac{\text{Sum of prices}}{\text{Number of items}}$

Substituting the values we have:

$16,050 = \frac{15x}{3}$

Solving for the Multiplier 'x'

Now, we simplify the equation to find the value of '$x$':

$16,050 = 5x$

To find '$x$', we divide the average price by 5:

$x = \frac{16,050}{5}$

$x = 3,210$

Determining the Cheapest Item Price

The prices of the items are $3x$, $5x$, and $7x$. The cheapest item corresponds to the smallest part of the ratio, which is 3.

Price of the cheapest item = $3x$

Substitute the value of '$x$' we found:

Price of the cheapest item = $3 \times 3,210$

Price of the cheapest item = $9,630$

Summary of Prices

Let's verify the prices:

Average Price Check: $\frac{9,630 + 16,050 + 22,470}{3} = \frac{48,150}{3} = 16,050$. This matches the given average price.

Therefore, the price of the cheapest furniture item is ₹9,630.