Understanding the Faulty Weighing Machine Problem

This problem involves calculating the selling price of almond packets when a faulty weighing machine is used. We are given the machine's error, the cost price of almonds, the size of the packets shown by the machine, and the desired profit percentage. Our goal is to find the selling price per packet.

Analyzing the Faulty Machine

The weighing machine shows a higher weight than the actual weight. This means the seller gives less quantity than what is displayed.

• The machine shows $\text{400 gm}$ when the actual weight is $\text{350 gm}$.
• This gives us a ratio of Shown weight to Actual weight: $\frac{\text{Shown weight}}{\text{Actual weight}} = \frac{\text{400 gm}}{\text{350 gm}} = \frac{40}{35} = \frac{8}{7}$.
• This ratio $\frac{8}{7}$ means that for every 8 units shown by the machine, the actual quantity is 7 units.

Calculating Actual Weight in Each Packet

Packets are made to show $\text{200 gm}$ using the faulty machine. We need to find the actual weight of almonds in such a packet.

• Let the shown weight be $\text{W}_{\text{shown}} = \text{200 gm}$.
• Using the ratio from the faulty machine: $\text{Actual weight} = \text{W}_{\text{shown}} \times \frac{7}{8}$.
• Actual weight in a packet $= \text{200 gm} \times \frac{7}{8} = \text{(25 \times 8) gm} \times \frac{7}{8} = \text{25 gm} \times 7 = \text{175 gm}$.

So, each packet that is shown as $\text{200 gm}$ actually contains only $\text{175 gm}$ of almonds.

Determining the Cost Price of Each Packet

The cost price of almonds is given as $\text{₹880}$ per kg ($\text{1000 gm}$). We need to find the cost of the actual quantity of almonds in each packet ($\text{175 gm}$).

• Cost price per gram $= \frac{\text{₹880}}{\text{1000 gm}} = \text{₹0.88}$ per gm.
• Cost price of $\text{175 gm} = \text{175 gm} \times \text{₹0.88}$/gm.
• Cost price of each packet $= \text{175} \times \text{0.88}$.
  • $\text{175} \times \text{0.88} = \text{175} \times \frac{88}{100} = \text{175} \times \frac{22}{25}$.
  • $= \text{(7 \times 25)} \times \frac{22}{25} = 7 \times 22 = \text{154}$.

The cost price of almonds in each packet is $\text{₹154}$.

Calculating the Selling Price for a 25% Profit

The desired profit is 25% on the cost price. We need to calculate the selling price per packet.

• Cost Price (CP) of each packet $= \text{₹154}$.
• Desired Profit $= \text{25% of CP}$.
• Profit amount $= \text{25% of ₹154} = \frac{25}{100} \times 154 = \frac{1}{4} \times 154 = \frac{154}{4}$.
• $\frac{154}{4} = \frac{77}{2} = \text{38.50}$.
• Profit amount $= \text{₹38.50}$.
• Selling Price (SP) = CP + Profit.
• SP $= \text{₹154} + \text{₹38.50} = \text{₹192.50}$.

Alternatively, Selling Price (SP) can be calculated directly:

• SP = CP $\times (1 + \frac{\text{Profit Percentage}}{100})$.
• SP $= \text{₹154} \times (1 + \frac{25}{100}) = \text{₹154} \times (1 + 0.25) = \text{₹154} \times 1.25$.
• $\text{154} \times \text{1.25} = \text{154} \times \frac{5}{4} = \frac{154 \times 5}{4} = \frac{770}{4} = \text{192.50}$.

The selling price of each packet should be $\text{₹192.50}$ to get a profit of 25%.

Here is a summary of the calculations:

The selling price of each packet needs to be $\text{₹192.50}$ to achieve the desired 25% profit margin, considering the faulty machine reduces the actual quantity of almonds in each packet.

Revision Table: Faulty Scale Profit

Additional Information: Faulty Weight Problems

Problems involving faulty weighing machines are common in profit and loss calculations. The key is to understand how the error affects the actual quantity being measured or sold.

• If the machine shows more than actual: The seller gives less quantity than marked. The actual quantity is less than the shown quantity. This usually benefits the seller if they calculate cost based on actual quantity and sell based on shown quantity.
• If the machine shows less than actual: The seller gives more quantity than marked. The actual quantity is more than the shown quantity. This usually causes a loss or reduced profit for the seller if not accounted for.
• Always calculate the cost price based on the actual weight or quantity of the goods.
• Calculate the selling price or profit/loss based on the price charged for the shown weight or quantity, or the total revenue received.
• In this problem, the cost is tied to the actual $\text{175 gm}$, but the packet size is advertised (implicitly by the machine setting) as $\text{200 gm}$. The profit is calculated on the actual cost incurred.