Some fruits are bought at 15 for Rs. 140 and an equal number of fruits at 10 for Rs. 120. If all the fruits are sold at Rs. 132 per dozen, then what is the profit percent in the entire transaction?
Calculating Profit Percent in Fruit Transactions
This problem involves calculating the overall profit percentage when buying fruits from two different sources at different rates and selling them at a single rate. We need to determine the total cost price (CP) and the total selling price (SP) for a specific number of fruits to find the profit and subsequently the profit percent.
Step 1: Determine the Cost Price per Fruit for Each Lot
Fruits are bought from two different lots:
- Lot 1: 15 fruits for Rs. 140.
- Lot 2: 10 fruits for Rs. 120.
Let's find the cost of one fruit from each lot:
- Cost of 1 fruit from Lot 1 = \(\frac{140}{15}\) Rupees.
- Cost of 1 fruit from Lot 2 = \(\frac{120}{10} = 12\) Rupees.
Step 2: Find a Convenient Number of Fruits to Calculate Total CP and SP
The problem states that an "equal number" of fruits are bought from both lots. To simplify calculations, we can choose a number of fruits that is a multiple of the quantities given in the buying rates (15 and 10). The least common multiple (LCM) of 15 and 10 is 30. Let's assume 30 fruits are bought from Lot 1 and 30 fruits are bought from Lot 2.
Step 3: Calculate the Total Cost Price (CP)
We assumed 30 fruits from each lot, making a total of \(30 + 30 = 60\) fruits.
- Cost of 30 fruits from Lot 1 = \(30 \times \frac{140}{15} = 2 \times 140 = 280\) Rupees.
- Cost of 30 fruits from Lot 2 = \(30 \times 12 = 360\) Rupees.
Total Cost Price (CP) for 60 fruits = Cost from Lot 1 + Cost from Lot 2
Total CP = \(280 + 360 = 640\) Rupees.
Step 4: Calculate the Total Selling Price (SP)
All the fruits (60 fruits in total) are sold at Rs. 132 per dozen.
First, find out how many dozens are there in 60 fruits:
Number of dozens = \(\frac{\text{Total number of fruits}}{\text{Number of fruits in a dozen}} = \frac{60}{12} = 5\) dozens.
Now, calculate the total selling price:
Total Selling Price (SP) for 60 fruits = Number of dozens \(\times\) Selling price per dozen
Total SP = \(5 \times 132 = 660\) Rupees.
Step 5: Calculate the Profit
Profit is the difference between the Total Selling Price and the Total Cost Price.
Profit = Total SP - Total CP
Profit = \(660 - 640 = 20\) Rupees.
Step 6: Calculate the Profit Percent
The profit percent is calculated using the formula:
\(\text{Profit Percent} = \left( \frac{\text{Profit}}{\text{Total CP}} \right) \times 100\)
Profit Percent = \(\left( \frac{20}{640} \right) \times 100\)
Simplify the fraction \(\frac{20}{640}\):
\(\frac{20}{640} = \frac{2}{64} = \frac{1}{32}\)
Now, calculate the percentage:
\(\text{Profit Percent} = \frac{1}{32} \times 100 = \frac{100}{32}\)
Simplify the fraction \(\frac{100}{32}\) by dividing both numerator and denominator by their greatest common divisor, which is 4:
\(\frac{100 \div 4}{32 \div 4} = \frac{25}{8}\)
Convert the improper fraction \(\frac{25}{8}\) into a mixed fraction:
\(25 \div 8 = 3\) with a remainder of \(1\).
So, \(\frac{25}{8} = 3 \frac{1}{8}\).
The profit percent in the entire transaction is \(3 \frac{1}{8}\).
| Item | Quantity | Rate | Total Cost |
|---|---|---|---|
| Fruits (Lot 1) | 15 | Rs. 140 | - |
| Fruits (Lot 2) | 10 | Rs. 120 | - |
| Assumed Quantity (Each Lot) | 30 | - | - |
| Cost of 30 from Lot 1 | 30 | @ \(\frac{140}{15}\)/fruit | Rs. 280 |
| Cost of 30 from Lot 2 | 30 | @ Rs. 12/fruit | Rs. 360 |
| Total Fruits Bought | 60 | - | - |
| Total Cost Price (CP) | 60 | - | Rs. 640 |
| Selling Rate | 1 dozen (12 fruits) | Rs. 132 | - |
| Total Fruits Sold | 60 | (= 5 dozen) | - |
| Total Selling Price (SP) | 60 | @ Rs. 132/dozen | Rs. 660 |
| Profit | - | - | Rs. 20 |
| Profit Percent | - | - | \(3 \frac{1}{8}\)% |
Revision Table: Profit and Loss Concepts
| Concept | Definition | Formula |
|---|---|---|
| Cost Price (CP) | The price at which an article is bought. | - |
| Selling Price (SP) | The price at which an article is sold. | - |
| Profit | When SP > CP. | Profit = SP - CP |
| Loss | When SP < CP. | Loss = CP - SP |
| Profit Percent | Profit expressed as a percentage of the Cost Price. | \(\left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\) |
| Loss Percent | Loss expressed as a percentage of the Cost Price. | \(\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100\) |
Additional Information: Solving Profit and Loss Problems
When solving profit and loss problems involving different purchase rates and a single selling rate, especially when equal quantities are involved, it's helpful to:
- Calculate the cost price per unit for each purchase lot.
- Assume a convenient number of units for the calculation. Choosing the LCM of the quantities involved in the purchase rates simplifies finding the total cost.
- Calculate the total cost price for the assumed number of units.
- Calculate the selling price per unit based on the given selling rate (e.g., per dozen, per item).
- Calculate the total selling price for the same assumed number of units.
- Find the profit or loss by comparing the total SP and total CP.
- Calculate the profit or loss percentage based on the total CP.
Using the LCM helps ensure we deal with whole numbers or simpler fractions for total costs, making the calculations smoother.
Important Questions from Successive Selling
A dealer sold an article at a loss of 2%. Had he sold it for Rs. 44 more, he would have gained 20%. Find the cost price of the article
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If selling price of 75 articles is equal to cost price of 60 articles, then the approximate loss or gain percent is :
A.T.V. is sold at 8% gain. Had it been sold for Rs.2553 less; there would have been loss of 15%. To gain 18%, the selling price (in Rs.) of T.V. would be:
A trader sells an article at 16% below its cost price. Had he sold it for Rs. 192.20 more, he would have gained 15%. The cost price (in Rs.) of the article is:
