Average price of 20 pens is Rs. 18. If 12 new pens are added, then the average price rise by Rs. 4. What is the total price of the new pens?
Rs. 344
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.
| Item | Number | Average Price (Rs.) | Total Price (Rs.) |
|---|---|---|---|
| Initial Pens | 20 | 18 | $$20 \times 18 = 360$$ |
| New Pens | 12 | ? | ? |
| Total Pens (After Addition) | $$20 + 12 = 32$$ | $$18 + 4 = 22$$ | $$32 \times 22 = 704$$ |
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
| Concept | Formula | Explanation |
|---|---|---|
| Average | $$\text{Average} = \frac{\text{Sum of Quantities}}{\text{Number of Quantities}}$$ | A measure of the central value of a set of numbers. |
| Total Sum | $$\text{Total Sum} = \text{Average} \times \text{Number of Quantities}$$ | The sum of all values in the set. |
| Change in Average | Affects the Total Sum | Adding items with a different average value changes the overall average and total sum. |
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.
Important Questions from Successive Selling
- During a sale 35% of the goods are sold at 22% profit, 40% of the remaining goods are sold at 37% profit and the still remaining goods are sold at a loss of 38%. If there is an overall profit of x%, then what is the value of x?
- A dealer buys two articles X and Y for ₹1,000 each. He marks each of them at the same price. He sells X by giving two successive discounts of 15% and 30% and still earns ₹337 as profit. If he sells Y at a single discount of 49%, then what is the profit percentage on Y?
- Find the single equivalent discount (rounded off to two decimal places) for successive discounts of 18%, 6% and 25%.
A shopkeeper marks the price of three identical articles, first, 60% above the Cost Price; second, 45% above the Cost Price and the third, 40% above the cost price. If he gives discount of 25% on first article, 20% on the second article and 15% on the third one, what is his overall profit percentage?
A dealer bought a six-seater Dining Table after getting successive discounts of 15% and 12%. Thereafter, he spent 10% on transport, etc. and finally sold it for Rs. 24,169.75, thereby earning a profit of 25%. What was the marked price of the Dining Table?
