SSC CGL - Average, Profit and Loss E-Book Study Material

Average, Profit and Loss

Average

The concept of average, often referred to as the arithmetic mean, is fundamental in quantitative aptitude. It represents a central or typical value in a set of numbers. Calculating the average helps us to understand the overall trend or performance of a group of data points.

The formula for calculating the average is quite straightforward: Average = Sum of all observations / Number of observations

Let's break this down. 'Sum of all observations' means you add up every single number in your dataset. 'Number of observations' is simply how many numbers you have in total.

Types of Average Problems and Solutions

Average problems can range from simple calculations to more complex scenarios involving weighted averages or changes in the dataset.

1. Basic Average Calculation

Example: The marks obtained by a student in five subjects are 60, 75, 80, 55, and 90. Find the average marks.

Solution: Sum of marks = 60 + 75 + 80 + 55 + 90 = 360 Number of subjects = 5 Average marks = 360 / 5 = 72

2. Average of Consecutive Numbers

When dealing with consecutive numbers (e.g., 1, 2, 3, 4, 5 or 10, 12, 14, 16), the average is simply the middle number if the count of numbers is odd. If the count is even, the average is the mean of the two middle numbers.

Example: Find the average of the first 10 natural numbers.

Solution: The first 10 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. This is an even count. The two middle numbers are 5 and 6. Average = (5 + 6) / 2 = 11 / 2 = 5.5 Alternatively, for consecutive numbers, the average is also (First term + Last term) / 2. Average = (1 + 10) / 2 = 11 / 2 = 5.5

3. Average when a New Member is Added

When a new member joins a group, the average might increase, decrease, or stay the same. We need to calculate the new total sum and the new count.

Example: The average weight of 5 men is increased by 2 kg when one man weighing 60 kg is replaced by a new man. Find the weight of the new man.

Solution: Let the original total weight of 5 men be $S$. Original average = $S/5$. Let the new man's weight be $W$. The old man's weight is 60 kg. New total weight = $S - 60 + W$. New number of men = 5. New average = $(S - 60 + W) / 5$. We are given that the new average is 2 kg more than the original average. $(S - 60 + W) / 5 = S/5 + 2$ $S/5 - 60/5 + W/5 = S/5 + 2$ $S/5 - 12 + W/5 = S/5 + 2$ $-12 + W/5 = 2$ $W/5 = 14$ $W = 14 \times 5 = 70$ kg. The weight of the new man is 70 kg.

Shortcut: When a new member is added/replaced, the change in the total sum is equal to the change in average multiplied by the new total number of members. In the example above: Change in average = +2 kg. New number of members = 5. Total increase in weight = $2 \times 5 = 10$ kg. This 10 kg increase is the difference between the new man's weight and the old man's weight. New man's weight - Old man's weight = 10 kg. New man's weight - 60 kg = 10 kg. New man's weight = 70 kg.

4. Weighted Average

A weighted average is used when different observations have different levels of importance or frequency. The formula is: Weighted Average = Sum of (Weight × Value) / Sum of Weights

Example: A teacher takes a test for 30 students. The average score of 10 students is 80, and the average score of the remaining 20 students is 70. Find the overall average score.

Solution: Here, the number of students in each group acts as the weight. Sum of (Weight × Value) = $(10 \times 80) + (20 \times 70) = 800 + 1400 = 2200$ Sum of Weights = $10 + 20 = 30$ Weighted Average = $2200 / 30 = 220 / 3 = 73.33$ (approx.)

5. Average of Errors

Sometimes, a calculation error might occur, like entering a wrong number or using a wrong value. These problems involve finding the correct average after identifying the mistake.

Example: The average of 5 numbers is calculated as 30. Later, it was found that one number, 48, was wrongly entered as 24. Find the correct average.

Solution: Original sum = Average × Number of observations = $30 \times 5 = 150$. The error in the sum = Wrong number - Correct number = $24 - 48 = -24$. Correct sum = Original sum - Error = $150 - (-24) = 150 + 24 = 174$. Correct average = Correct sum / Number of observations = $174 / 5 = 34.8$.

Shortcut: The difference in the sum is the difference between the wrong value and the correct value. Difference in sum = $24 - 48 = -24$. This difference is distributed over the number of observations to find the change in average. Change in average = Difference in sum / Number of observations = $-24 / 5 = -4.8$. Correct average = Original average + Change in average = $30 + (-4.8) = 25.2$. Wait, let's re-check the calculation. Original sum = $30 \times 5 = 150$. Wrong number entered = 24. Correct number = 48. The sum was calculated using 24 instead of 48. So the calculated sum is less than the actual sum. The difference that was *subtracted* incorrectly is $48 - 24 = 24$. So, the correct sum should be $150 + 24 = 174$. Correct average = $174 / 5 = 34.8$. My shortcut logic was flawed. Let's refine the shortcut: The error in sum = Correct value - Incorrect value. Error in sum = $48 - 24 = 24$. Change in average = Error in sum / Number of observations = $24 / 5 = 4.8$. Correct average = Original average + Change in average = $30 + 4.8 = 34.8$. This is correct.

Profit and Loss

Profit and Loss are concepts crucial for understanding business transactions and financial dealings. They help determine the profitability of a venture.

Key terms:

Cost Price (CP): The price at which an article is purchased.
Selling Price (SP): The price at which an article is sold.
Profit: Occurs when SP is greater than CP. Profit = SP - CP.
Loss: Occurs when CP is greater than SP. Loss = CP - SP.
Overhead Charges: Expenses incurred beyond the purchase price, such as transportation, repair, etc. These are added to the CP.

Formulas for Profit and Loss

1. Profit = SP - CP 2. Loss = CP - SP 3. Profit Percentage = (Profit / CP) × 100 4. Loss Percentage = (Loss / CP) × 100

It is important to note that profit and loss percentages are always calculated on the Cost Price (CP), unless otherwise specified.

Calculating SP from CP and Profit/Loss Percentage

1. If there is a profit of $P\%$, then $SP = CP \times (100 + P) / 100$. 2. If there is a loss of $L\%$, then $SP = CP \times (100 - L) / 100$.

Calculating CP from SP and Profit/Loss Percentage

1. If there is a profit of $P\%$, then $CP = SP \times 100 / (100 + P)$. 2. If there is a loss of $L\%$, then $CP = SP \times 100 / (100 - L)$.

Types of Profit and Loss Problems

1. Basic Profit/Loss Calculation

Example: A shopkeeper buys an article for ₹400 and sells it for ₹480. Find his profit or loss percentage.

Solution: CP = ₹400 SP = ₹480 Since SP > CP, there is a profit. Profit = SP - CP = ₹480 - ₹400 = ₹80. Profit Percentage = (Profit / CP) × 100 = (80 / 400) × 100 = (1/5) × 100 = 20%.

Example: A vendor buys 10 lemons for ₹50 and sells them at 6 lemons for ₹30. Find his profit or loss percentage.

Solution: CP of 10 lemons = ₹50. SP of 10 lemons: SP of 6 lemons = ₹30. SP of 1 lemon = ₹30 / 6 = ₹5. SP of 10 lemons = ₹5 × 10 = ₹50. Since CP = SP (₹50 = ₹50), there is neither profit nor loss. The profit/loss percentage is 0%.

Shortcut: To compare prices when quantities are different, always try to find the price per unit (e.g., price per lemon, price per kg) or make the quantities equal. In the above example, we found the SP of 10 lemons to match the CP of 10 lemons. Alternatively, we could find the CP and SP for a common number of lemons, like LCM(10, 6) = 30 lemons. CP of 30 lemons = (50/10) * 30 = 5 * 30 = ₹150. SP of 30 lemons = (30/6) * 30 = 5 * 30 = ₹150. CP = SP, so 0% profit/loss.

2. Problems involving Marked Price (MP) and Discount

When a shopkeeper displays goods with a price tag higher than the CP, this is the Marked Price (MP). The reduction given on the MP is called Discount.

Key terms:

Marked Price (MP): The price listed on the article.
Discount: The reduction offered on the MP.
Discount Percentage: Usually calculated on the MP. Discount % = (Discount / MP) × 100.

Relationships:

Discount = MP - SP
SP = MP - Discount
SP = MP × (100 - Discount %) / 100

Example: A shopkeeper marks his goods 40% above the cost price and then offers a discount of 10% on the marked price. Find the profit percentage.

Solution: Let the CP = ₹100. MP = CP + 40% of CP = $100 + (40/100) \times 100 = 100 + 40 = ₹140$. Discount = 10% of MP = $(10/100) \times 140 = ₹14$. SP = MP - Discount = $140 - 14 = ₹126$. Profit = SP - CP = $126 - 100 = ₹26$. Profit Percentage = (Profit / CP) × 100 = $(26 / 100) \times 100 = 26\%$.

Shortcut: This type of problem can be solved using a direct formula: $SP = CP \times (100 + Profit\%) / 100$ $SP = MP \times (100 - Discount\%) / 100$ Let MP be $x\%$ above CP, and discount be $y\%$. $MP = CP \times (100 + x) / 100$ $SP = MP \times (100 - y) / 100$ Substituting MP: $SP = [CP \times (100 + x) / 100] \times (100 - y) / 100$ $SP = CP \times [(100 + x)(100 - y) / 10000]$ So, $Profit\% = [(100 + x)(100 - y) / 100] - 100$. In the example: $x = 40$, $y = 10$. $Profit\% = [(100 + 40)(100 - 10) / 100] - 100$ $Profit\% = [140 \times 90 / 100] - 100$ $Profit\% = [12600 / 100] - 100$ $Profit\% = 126 - 100 = 26\%$.

3. Problems involving Two Articles Sold at the Same Price

This is a common scenario. If two articles are sold at the same selling price, and one yields a profit of $x\%$ and the other a loss of $x\%$, then there is always a loss in the overall transaction.

Formula for Loss: Loss % = $(x^2 / 100) \%$

Example: A man sells two horses for ₹19,200 each. On one, he gains 25%, and on the other, he loses 25%. Find his overall profit or loss percentage.

Solution: Since both horses are sold at the same price, and there's an equal percentage of profit and loss, we apply the formula for loss. Here, $x = 25$. Loss % = $(25^2 / 100) \% = (625 / 100) \% = 6.25\%$. There is an overall loss of 6.25%.

Important Note: If the question stated that the CP of both items was the same, and one was sold at a profit of $x\%$ and the other at a loss of $x\%$, then there would be no overall profit or loss. The scenario of equal SP is key here.

4. Problems involving False Weights

These problems involve a dishonest dealer who uses faulty weights to cheat customers. For example, a dealer might claim to sell 1 kg but actually sell less (e.g., 900g).

Example: A dishonest dealer professes to sell his goods at cost price, but he uses a weight of 950 grams instead of 1 kilogram. Find his gain percentage.

Solution: The dealer claims to sell at CP. This means if the CP of 1 kg (1000g) is ₹100, he sells 1000g for ₹100. However, he actually gives only 950g for ₹100. So, the CP for 950g is $(100 / 1000) \times 950 = ₹95$. The dealer sells 950g for ₹100 (which is the CP of 1000g). Profit = SP - CP = $100 - 95 = ₹5$. Profit Percentage = (Profit / CP) × 100 = $(5 / 95) \times 100 = (1/19) \times 100 = 100/19 \% \approx 5.26\%$.

Shortcut: When a dealer cheats on weight: Gain % = [(True weight - False weight) / False weight] × 100 In the example: True weight = 1000g, False weight = 950g. Gain % = [(1000 - 950) / 950] × 100 = (50 / 950) × 100 = (1/19) × 100 = 100/19 %. This shortcut directly calculates the profit percentage based on the amount cheated relative to the amount actually given.

5. Successive Selling

In successive selling, an article is sold multiple times by different people. The profit or loss percentage for each transaction is calculated based on the price at which it was bought in that particular transaction.

Example: A sells an article to B for ₹2000, gaining 25%. B then sells it to C for ₹2500. Find B's profit percentage.

Solution: A sells to B for ₹2000. This is B's Cost Price (CP). CP for B = ₹2000. B sells to C for ₹2500. This is B's Selling Price (SP). SP for B = ₹2500. B's Profit = SP for B - CP for B = $2500 - 2000 = ₹500$. B's Profit Percentage = (B's Profit / B's CP) × 100 = $(500 / 2000) \times 100 = (1/4) \times 100 = 25\%$.

Example: A sells an article to B at a profit of 10%. B sells it to C at a profit of 20%. If C pays ₹6600 for it, what was A's cost price?

Solution: Let A's Cost Price (CP) be $x$. A sells to B at 10% profit. SP of A = B's CP = $x \times (100 + 10) / 100 = 1.1x$. B sells to C at 20% profit. SP of B = C's CP = $(1.1x) \times (100 + 20) / 100 = 1.1x \times 1.2 = 1.32x$. We are given that C pays ₹6600. So, $1.32x = 6600$. $x = 6600 / 1.32 = 660000 / 132$. $x = 5000$. A's cost price was ₹5000.

Shortcut: For successive sales, you can write the final price in terms of the initial price: Final Price = Initial Price $\times \frac{(100 \pm P_1\%)}{100} \times \frac{(100 \pm P_2\%)}{100} \times \dots$ Where $P_n\%$ is the profit percentage (use +) or loss percentage (use -) at the $n^{th}$ stage. In the example: C's price = A's CP $\times \frac{(100 + 10\%)}{100} \times \frac{(100 + 20\%)}{100}$ $6600 = A's CP \times \frac{110}{100} \times \frac{120}{100}$ $6600 = A's CP \times 1.1 \times 1.2$ $6600 = A's CP \times 1.32$ $A's CP = 6600 / 1.32 = 5000$.

Mixed Problems (Average and Profit/Loss Combined)

Sometimes, problems combine these concepts. For instance, you might need to find the average profit or average selling price.

Example: A trader buys 5 articles for ₹100 each. He sells the first two at a profit of 10% and the remaining three at a loss of 10%. What is his overall profit or loss percentage?

Solution: CP of each article = ₹100. Total CP = $5 \times 100 = ₹500$. For the first two articles: CP of 2 articles = $2 \times 100 = ₹200$. Profit = 10%. SP of 2 articles = $200 \times (100 + 10) / 100 = 200 \times 1.1 = ₹220$. For the remaining three articles: CP of 3 articles = $3 \times 100 = ₹300$. Loss = 10%. SP of 3 articles = $300 \times (100 - 10) / 100 = 300 \times 0.9 = ₹270$. Total SP = SP of first two + SP of remaining three = $220 + 270 = ₹490$. Overall Profit/Loss = Total SP - Total CP = $490 - 500 = -₹10$. Since the result is negative, it's a loss. Overall Loss Percentage = (Loss / Total CP) × 100 = $(10 / 500) \times 100 = (1/50) \times 100 = 2\%$.

Shortcut: You can think of this as a weighted average of profit/loss. Average profit/loss % = $\frac{(\text{Number of items}_1 \times \text{Profit/Loss}_1\%) + (\text{Number of items}_2 \times \text{Profit/Loss}_2\%)}{\text{Total Number of items}}$ Average profit/loss % = $\frac{(2 \times +10\%) + (3 \times -10\%)}{5}$ Average profit/loss % = $\frac{20\% - 30\%}{5} = \frac{-10\%}{5} = -2\%$. The negative sign indicates a loss of 2%.