*Friday, August 7th, 2020*

Dishonest Dealers and faulty weights is an important topic in profit and loss generally asked in all competitive exams. In this topic, we come across such type of problems in which shopkeeper sells the items at cost price but still, he makes a profit on those items. This can be only possible if he gives less quantity to the customer then the indicated on the product. A dishonest merchant shows he is selling the product on cost price in the welfare of customer but in that he reduced its weight.

For example, if shopkeeper sells a 1-liter packet of milk at its cost price but he gives only 800 ml milk in that packet. In this case, he is giving 800 ml milk to a customer and charging amount for 1000 ml to the customer. Hence, we can say that 800 ml is cost price and 1000 ml is selling price. So, he is getting a profit of 200 ml of milk on selling 800 ml milk.

Percentage profit = ((Selling Price – Cost Price)/(Cost Price))×100

Percentage profit = ((1000-800)/800)×100 = 25%

Let’s take different examples to understand this concept with more clarity. Suppose a shopkeeper gives 25% less quantity then indicated but he gives 10% discount also on cost price to the customer. To calculate his profit percentage we assume shopkeeper purchased 100 units but while selling he is giving only 75 units and claims that he is giving 100 units at 10% discount means he is charging the amount for only 90 units. In this case, he is getting the amount of 90 units after discount but he is giving only 75 units to the customer. So he is getting a profit of 15 units on the sell of 75 units.

Percentage profit = ((90-75)/75)×100 = 20%

Sometimes, we come across such type of questions in which a shopkeeper makes two profits by selling the product at a higher price from cost price along with faulty weights or shopkeeper is using faulty weights at both the times while selling and purchasing the products. In this case, a shopkeeper is getting two types of profit one by selling the product at a different price other by giving the wrong weight to customer. This type of question can simply be solved with the concept of the successive (compounded) percentage change. Let us suppose shopkeeper is making x% profit while selling the product and making y% profit while selling the same product. So his overall profit can be directly calculated as {x + y + (xy/100)}. We can also use this formula id shopkeeper is making a loss because of any reason or he is giving any discount by putting a negative sign in front of percentage. We understand this concept by solving some examples based on it.

**Eg1:** A shopkeeper purchased 20% extra quantity than indicated and sells 20% less than indicated while selling the milk. Find the percentage profit if he sells the product at its cost price.

**Solution:**

We can solve this question with different methods.

**Method 1:** Let us assume cost price of milk is 1Rs/ml. On purchasing that milk shopkeeper purchases 120 ml and pays for only 100 ml i.e. 100 Rs. While selling he is giving only 80 ml and charging for 100 ml i.e. 100 Rs. So the total selling price of 120 ml milk that he purchased in 100 Rs is 100/80×120 = 150 Rs. Hence shopkeeper makes 50% profit on selling this milk.

**Method 2:** In this type of questions we can directly calculate the profit percentage earned by the shopkeeper while purchasing and selling the product we can combine both percentages by using the concept of successive as discussed above. In this question shopkeeper is purchasing 120 ml milk and paying for only 100 ml means he is making 20% profit and while selling he is charging the price of 100 ml but giving only 80 ml to customer means he is getting a profit of 20 ml on selling 80 ml milk means his percentage profit on selling is (20/80) × 100 = 25%. So the overall profit of the shopkeeper is 20 + 25 + (20×25/100) = 50%.

**Eg2:** A shopkeeper sells rice at 20 Rs/kg that he purchased in 16 Rs/kg and he is giving only 800 gm of rice instead of 1 kg while selling. Find the actual percentage profit of the shopkeeper?

**Solution:**

**Method 1:** Given that cost price of 1000 gm rice is 16 Rs but the shopkeeper is giving only 800 gm so the cost price of 800gm rice is 16/1000×800 = 12.8 Rs. And shopkeeper sells this quantity of rice to a customer claiming as 1 kg at price 20 Rs/kg. So he gains 7.2 Rs as a profit. Hence,

Percentage profit is = (20-12.8)/12.8×100 = 56.25%

**Method 2:** the shopkeeper is giving 800 gm instead of 1000 gm so he is making a profit of 200 gm on 800 gm so his percentage profit on selling is (200/800)×100 = 25% but charging 20 Rs/kg that he purchased in 16 Rs/kg percentage profit is (20-16)/16×100 = 25%. So the overall percentage profit is 25 + 25 + (25×25/100) = 56.25%.

**Eg3:** Anil sells sugar at cost price but uses false weight thus gain 20% profit. Find how many grams sugar he is giving in 1 kilogram?

**Solution:** The profit percentage of shopkeeper is calculated as

Percentage profit = (Indicated weight-Actual weight)/(Actual weight)×100

20 = (1000-Actual weight)/(Actual weight)×100 ⤇ Actual Weight = 5/6×1000 = 833.33 grams.

**Eg4:** A shopkeeper claims that he is selling sugar at 20 Rs/kg which cost him 25 Rs/kg but he is giving 800 grams instead of 1 kg. Find the percentage profit or loss of the shopkeeper?

**Solution:** Given that cost price of 1000 gm sugar is 25 Rs but the shopkeeper is giving only 800 gm so the cost price of 800gm rice is 25/1000×800 = 20 Rs. And shopkeeper selling this quantity of sugar claiming as 1 kg in 20 Rs/kg. So there is no profit and loss to the shopkeeper as he purchased this 800 grams sugar in 20 Rs. by selling this at the same price.

**Eg5:** A shopkeeper marks up the price of rice by 20% and gives a discount of 10%. Apart from this, he is selling 800 grams in 1000 grams. What is his net percentage profit?

**Solution:** Let us assume the cost price for 1000 grams rice is 100 Rs. The shopkeeper is marking this price as 120 Rs/kg and giving 10% discount means he is charging overall 108 Rs. For this but he is giving only 800 grams rice in this amount, which he purchased in 80 Rs as the cost of 1000 grams rice is 100 Rs. So the shopkeeper is gaining 28 Rs. So the percentage profit of the shopkeeper is (108-80)/80×100 = 35%.

**Eg6:** A dishonest merchant sells his grocery using weight 10 % less than the true weight and makes a profit of 20%. Find his total percentage profit?

**Solution:**

The merchant sells 900 grams in 1-kilogram packet. Let us assume the cost price is 1 Rs/gram. Then the cost price of each packet is 900 Rs. Selling price of this packet is 20% higher than its actual cost price i.e. 1.2×1000 = 1200 Rs. So the shopkeeper is gaining 1200 – 900 = 300 Rs as a profit. Hence the Percentage profit of merchant is 300/900×100 = 33.33%.

CAT Level Questions:

**Eg7:** A weight machine reads 1 kg when 800 gm actually weighted on it and the shopkeeper already marked up the price of his goods by 20%. One day a customer detects this and complains this issue to police, then police inspector orders him to repair his weight machine and sell the goods at 20% discount on cost price for a week from then onwards as a punishment. After that, a customer paid Rs. 16 for 1 kg sugar. Then find before the raid, how much amount he would have paid for the same quantity?

**Solution:**

It is given that initially, the shopkeeper gives only 800 gm instead of 1 kg means he earns 200 gm profit on selling 800 gm sugar. So, he earns 25% profit by altering the weight along with he marked up the price by 20% means his overall profit can be calculated by successive i.e. 25+20+(20×25)/100=50%. But now shopkeeper sells the goods at Rs 16 per kg after giving 20% discount means the actual cost price of 1 kg sugar is 16/80×100=20. And earlier shopkeeper was gaining 50% profit on this product means the price of this quantity before raid would be 20×1.5 = 30Rs.

**Eg8:** A shopkeeper normally makes a profit of 20% in a certain transaction; he weighted 900 gm instead of 1 kg due to an error in weight machine. If he charges 20% less what he normally charges, what is his actual profit or loss percentage?

**Solution:**

Let the cost of 1 kg of goods is Rs. 100 but shopkeeper charges Rs. 120 to the customer. But now shopkeeper is giving only 900 grams to a customer that cost him at Rs. 90 but he gives 20% discount on his previous selling price i. e. 80/100×120 = 96.

Profit Percentage = (96-90)/90×100=20/3%

As you can see the concept is quite easy but we have to decide approach before starting the question, you will face similar problems in the exam where you have to decide what is the Indicated weight and Actual weight. You will get more idea of these types of questions you need to practice through past year CAT papers. This is a very important concept in Profit and Loss. Hopefully, I am able to explain the concept clearly.

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