30 pens and 75 pencils were purchased for Rs. 390. If the average price of a pencil is Rs. 2.00, then calculate the average price (in Rs.) of a pen. A. 6 B. 4 C. 8 D. 12
C
Understanding the Problem: Calculating Average Price
The question asks us to find the average price of a pen. We are given the total number of pens and pencils purchased, their combined total cost, and the average price of a single pencil. To find the average price of a pen, we first need to determine the total cost specifically for the pens.
Step-by-Step Solution for Average Price
Here's how we can break down the problem and solve it:
- Calculate the total cost of all the pencils.
- Calculate the total cost of all the pens.
- Calculate the average price of one pen.
1. Calculate the Total Cost of Pencils
We know the number of pencils and the average price per pencil.
- Number of pencils = 75
- Average price of a pencil = Rs. 2.00
The total cost of pencils is the product of the number of pencils and the average price per pencil.
Total cost of pencils = Number of pencils $\times$ Average price per pencil
Total cost of pencils = $75 \times 2.00$
Total cost of pencils = Rs. 150
2. Calculate the Total Cost of Pens
The total cost of both pens and pencils combined is Rs. 390. We just calculated the total cost of pencils as Rs. 150.
- Total cost of pens and pencils = Rs. 390
- Total cost of pencils = Rs. 150
The total cost of pens is the total combined cost minus the total cost of pencils.
Total cost of pens = Total combined cost - Total cost of pencils
Total cost of pens = $390 - 150$
Total cost of pens = Rs. 240
3. Calculate the Average Price of a Pen
We now know the total cost of pens and the number of pens.
- Total cost of pens = Rs. 240
- Number of pens = 30
The average price of a pen is the total cost of pens divided by the number of pens.
Average price of a pen = $\frac{\text{Total cost of pens}}{\text{Number of pens}}$
Average price of a pen = $\frac{240}{30}$
Average price of a pen = 8
So, the average price of a pen is Rs. 8.
Verification of the Average Price Calculation
Let's quickly check our calculation:
- Cost of 30 pens at Rs. 8 each = $30 \times 8 = 240$
- Cost of 75 pencils at Rs. 2 each = $75 \times 2 = 150$
- Total cost = $240 + 150 = 390$
This matches the total given in the problem, confirming our calculation is correct.
Summary of Calculations
| Item | Quantity | Average Price (Rs.) | Total Cost (Rs.) |
|---|---|---|---|
| Pencils | 75 | 2.00 | $75 \times 2.00 = 150$ |
| Pens | 30 | ? | $390 - 150 = 240$ |
| Total | 390 |
From the table, the total cost of 30 pens is Rs. 240. Average price per pen = $\frac{240}{30} = 8$ Rs.
Conclusion on Average Price of a Pen
Based on our calculations, the average price of a pen is Rs. 8.
Revision Table: Key Concepts
| Concept | Formula | Application in this Problem |
|---|---|---|
| Total Cost | Average Price $\times$ Quantity | Used to find the total cost of pencils: $2 \times 75 = 150$ |
| Total Cost of One Type of Item | Overall Total Cost - Total Cost of Other Items | Used to find the total cost of pens: $390 - 150 = 240$ |
| Average Price | $\frac{\text{Total Cost}}{\text{Quantity}}$ | Used to find the average price of a pen: $\frac{240}{30} = 8$ |
Additional Information: Understanding Averages
The average (or arithmetic mean) is a central value of a set of numbers. It is calculated by summing up all the values in the set and then dividing by the count of those values. In this problem, the "average price" refers to the total cost of a group of identical items divided by the number of items in that group.
- If you buy 5 apples for a total of Rs. 50, the average price per apple is $\frac{50}{5} = 10$.
- If items are of different types but purchased together, their individual total costs contribute to the overall total cost.
- Finding the average price of one type of item requires isolating its total cost first.
This type of problem tests your ability to work with total cost, quantity, and average price relationships.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
