The problem asks us to find the total price of 12 new pens added to a group of 20 pens, given the initial average price and the new average price after the addition.

Understanding Average Price Concepts

The average price is calculated by dividing the total price of items by the number of items. Mathematically, this is:

$$\text{Average Price} = \frac{\text{Total Price}}{\text{Number of Items}}$$

From this formula, we can also find the total price if we know the average price and the number of items:

$$\text{Total Price} = \text{Average Price} \times \text{Number of Items}$$

Calculating Initial Total Price of Pens

We are given that the average price of 20 pens is Rs. 18.

• Number of initial pens = 20
• Initial average price = Rs. 18

Using the formula for total price:

$$\text{Initial Total Price} = \text{Initial Average Price} \times \text{Number of Initial Pens}$$

$$\text{Initial Total Price} = 18 \times 20 = \text{Rs. } 360$$

So, the total price of the initial 20 pens was Rs. 360.

Calculating New Average and Total Pens

When 12 new pens are added:

• Number of new pens added = 12
• Total number of pens = Initial number of pens + Number of new pens added
• Total number of pens = $$20 + 12 = 32$$

The average price rises by Rs. 4. The initial average price was Rs. 18.

• Rise in average price = Rs. 4
• New average price = Initial average price + Rise in average price
• New average price = $$18 + 4 = \text{Rs. } 22$$

Calculating New Total Price of Pens

Now we have the total number of pens and the new average price:

• Total number of pens = 32
• New average price = Rs. 22

Using the formula for total price with the new values:

$$\text{New Total Price} = \text{New Average Price} \times \text{Total Number of Pens}$$

$$\text{New Total Price} = 22 \times 32$$

Let's calculate $22 \times 32$:

$$22 \times 32 = 22 \times (30 + 2) = (22 \times 30) + (22 \times 2) = 660 + 44 = 704$$

So, the new total price of the 32 pens is Rs. 704.

Finding the Total Price of New Pens

The new total price is the sum of the initial total price and the total price of the new pens.

$$\text{New Total Price} = \text{Initial Total Price} + \text{Total Price of New Pens}$$

We want to find the Total Price of New Pens:

$$\text{Total Price of New Pens} = \text{New Total Price} - \text{Initial Total Price}$$

• New Total Price = Rs. 704
• Initial Total Price = Rs. 360

$$\text{Total Price of New Pens} = 704 - 360 = \text{Rs. } 344$$

Therefore, the total price of the 12 new pens is Rs. 344.

From the table and calculations, the total price of the new pens is the difference between the final total price and the initial total price: $$704 - 360 = 344$$.

Revision Table: Average Price Concepts

Additional Information: Solving Average Problems

When tackling average problems involving adding or removing items, the key is to always work with the total sum (or total price in this case). The average is just a derived value.

• Always calculate the initial total sum/price first.
• Determine the new number of items and the new average.
• Calculate the new total sum/price using the new number of items and new average.
• The difference between the new total sum and the initial total sum gives the sum/price of the items that were added or removed.

In this specific problem, the average price increased, which means the new pens added were, on average, more expensive than the initial pens. If the average had decreased, the new pens would have been, on average, less expensive.

The final answer is Rs. 344.