Percentage & Profit, Loss and Discount

Percentage

Percentage is a concept used to express proportions or rates as fractions of 100.
For example, if we have a whole pizza and someone asks for half (1/2), we are giving them 50% of the pizza. Similarly, if they ask for one-third (1/3), we will give them approximately 33.33% of the pizza. Percentages allow us to compare different quantities relative to a whole or each other.
Percentages are denoted using the symbol "%," with 100% representing the entirety or completeness of something. They are widely used in various fields such as mathematics, finance, statistics, and everyday life for expressing ratios, proportions, discounts, interest rates, and more.
In mathematical terms, to convert a fraction to a percentage, you multiply the fraction by 100. Conversely, to convert a percentage to a fraction, you divide the percentage by 100 and simplify if needed.
Understanding percentages is fundamental in areas like business, economics, education, and data analysis, as they provide a standardized way to express and compare relative values.

Common Percentages

10% = $\frac{10}{100}$ = $\frac{1}{10}$
20% = $\frac{20}{100}$ = $\frac{1}{5}$
30% = $\frac{30}{100}$ = $\frac{3}{10}$
40% = $\frac{40}{100}$ = $\frac{2}{5}$
100% = $\frac{100}{100}$ = $1$
1% = $\frac{1}{100}$

Calculating Percentage

Find 10% of 260:
Long way: $\frac{10}{100}$ * 260 = 26
Quick way: For 10%, move the decimal point one place to the left. So, 26.0 becomes 26 (answer).
Find 10% of 4556:
Quick way: Move the decimal point one place to the left for 10%, giving us 455.6.
So, 10% of 4556 is 455.6.
Find 1% of 4556:
Quick way: Move the decimal point two place to the left for 1%, giving us 45.56.
So, 1% of 4556 is 45.56.
Find 20% of 260:
We can express 20% as 2 * 10%.
So, 10% of 260 = 26.
Now, we multiply this by 2 to get 52.
Find 40% of 260:
We can express 40% as 4 * 10%.
So, 10% of 260 = 26.
Now, we multiply this by 4 to get 104.
Find 43% of 260:
To find 43%, we can break it down into 40% and 3% since we know how to find 40% easily and also how to find 1%.
40% of 260 = 104
1% of 260 = 2.6
3% = 3 * 1% = 3 * 2.6 = 7.8
Now, add 40% and 3%: 104 + 7.8 = 111.8
So, 43% of 260 is 111.8.
Find 58% of 260:
To find 58%, we can break it down into 60% and 2% (since 60 - 2 = 58), as we know how to find 60% easily and also how to find 1%.
60% of 260 = 156
1% of 260 = 2.6
2% = 2 * 1% = 2 * 2.6 = 5.2
Now, subtract 2% from 60%: 156 - 5.2 = 150.8
So, 58% of 260 is 150.8.

Mix Percentage:

33 $\frac{1}{3}$ % = $\frac{100}{3}$ % = $\frac{100}{3}$ * $\frac{1}{100}$ = $\frac{1}{3}$
66 $\frac{2}{3}$ % = $\frac{198 + 2}{3}$ % = $\frac{200}{3}$ * $\frac{1}{100}$ = $\frac{2}{3}$
16 $\frac{2}{3}$ % = $\frac{1}{6}$
8 $\frac{1}{3}$ % = $\frac{1}{12}$

Some concepts

18% of 24 + 7% of 24
- we take 24 as common
  24(18% + 7%)
  it become of 25% of 24 = 6
Now 24% of 18 + 7% of 24
Concept: a% of b = b% of a
so we can write 24% of 18 + 24% 0f 7
24% of (18 + 7)
24% of 25
we can also write 25% of 24 = 6

Percentage Increase

Increase 800 by 20% or add 20% of 800 to itself.

800 + 20% of 800 = 800 + 160 = 960
When it says "increase," we multiply by 100% plus that percentage, so here we multiply by 120%.
$\frac{120}{100}$ * 800 = 960

Percentage Decrease

Decrease 800 by 25% or subtract 25% of 800 from itself.

800 - 25% of 800 = 800 - 200 = 600
When it says "decrease," we multiply by 100% minus that percentage, so here we multiply by 75%.
$\frac{75}{100}$ * 800 = 600

Fraction to Percentage and Vice Versa

If we want to convert a percentage to a fraction, we multiply it by $\frac{1}{100}$ . For example, 25% as a fraction will be 25 * $\frac{1}{100}$ = $\frac{1}{4}$ .
If we want to convert a fraction to a percentage, we multiply it by 100. For example, $\frac{1}{4}$ as a percentage will be $\frac{1}{4}$ * 100 = 25%.

Base Concept

Percentage is based on what it is compared with; the base goes in the denominator.

2 is what % of 5?
Here, 5 is the base, so it goes in the denominator:
$\frac{2}{5}$ * 100 = 40%
5 is what % of 2?
Here, 2 is the base, so it goes in the denominator:
$\frac{5}{2}$ * 100 = 250%
5 is what % more than 2?
Here, the base is 2, and since it's "more," the numerator is the difference (5 - 2 = 3):
$\frac{3}{2}$ * 100 = 150%
2 is what % less than 5?
Here, the base is 5, and since it's "less," the numerator is the difference (5 - 2 = 3):
$\frac{3}{5}$ * 100 = 60%

Set:1 Questions:

What is 56% of Y if it equals 182?
What percentage of 42kg is 336?
If 15% of Y is equal to 21% of Z, then what percentage of Z is equal to 12.5% of Y?
If the price of rice is 30% less than the price of wheat, then by how much percent is the price of wheat more than that of rice?
If the price of an apple is first increased by 10% and then decreased by 10%, what is the net change in the price of the apple?
If the price of sugar is raised by 25%, by what percentage should a person reduce his consumption of sugar so that his expenditure remains the same?
Y needs to score 40% marks to pass an exam. He scores 20 marks and fails by 40 marks. What are the maximum marks for the exam?
A scores 10% and fails by 30 marks, while B scores 40% marks and gets 30 marks more than the minimum marks needed to pass the exam. What are the maximum marks for the exam?
In a class, 15% of the total number of students failed in science, 25% failed in math, and 10% failed in both. What percentage of students passed in both math and science?
By a 20% decrease in the price of rice, people can buy 10kg more rice for Rs 100. What is the original price of 1kg of rice?
In an election contested by 2 candidates, one candidate got 40% of the total votes and lost by 1000 votes. What is the total number of votes cast in the election?
In a country where 55% of the population is female and 80% of the male population is literate, what percentage of females are literate if the overall literacy rate is 58%?
If 20% of an electricity bill is deducted, Rs 100 is still to be paid. What was the original bill?
A's salary is 50% more than B's salary. By what percentage is B's salary less than A's?
Two numbers are less than a third number by 30% and 37%, respectively. By what percentage is the second number less than the first?
10% of the inhabitants of a village died of cholera, and during a panic, 25% of the remaining inhabitants left the village. The population then reduced to 4050. What was the original population of the village?

Set 2: Questions

If Ram's salary is 20% more than Shyam's, what percentage is Shyam's salary less than Ram's?
The salary of a family is 8750 rupees, in which the ratio of Ram's salary to Shyam's salary is 2:3. Find what percentage Ram's salary is of Shyam's salary. (concept percentage doesn't need actual or real values)
If X is 20% less then Y, then find the value of 2X + 3Y 5X - 2Y
1. 2.3
2. 2:8
3. 2:7
4. 2:1
5. NOT
If 60 is subtracted from 60% of a number the result is 60, so find out the number?
1. 120
2. 200
3. 250
4. 210
5. NOT
If salary of Mattu is 25% more than Bittu, by how much % salary of Bittu is less than Mattu?
1. 20%
2. 16 $\frac{2}{3}$ %
3. 25%
4. 33 $\frac{1}{3}$ %
5. NOT
The Population of a town is 189000. 4/9 of them are males and rest females. 50% males are married.
Find % of married population.
1. 44 $\frac{4}{9}$ %
2. Cant be determined
3. 50%
4. None of these
A fraction is increased by 20% and after that numerator increases by 240% and denominator increased to 150% so that resultant becomes 1 $\frac{1}{5}$ . What is the original fraction.
The sum of square of two positive numbers is 628 and one number is 45 $\frac{5}{11}$ % less than other number. Find the smaller number.
Rama's expenditures and savings are in the ratio 5 : 3. If her income increases by 12% and expenditure by 15%, then by how much percent do her savings increase?
1. 12%
2. 7%
3. 8%
4. 13%
5. Not
An amount is divided among A, B & C. The share of B is 20% more than share of C, while share of B is 16.66% more than the share of A. Find ratio of difference between shares of B & C?
A man deposit 10% of his salary in Bank. He saves 30% of the remaining. The ratio of his expense on transport and education is 3 : 4 of the remaining salary after saving. If his expense on the transport was Rs 2700. Find the annual salary.
1. Rs 10,000
2. Rs 120,000
3. Rs 130,000
4. Rs 20,000
5. None of these
In an election Modi and Rahul participated. 20% voters did not vote. 12 1 2 % votes declared invalid and Modi gets 60% of the valid votes and won by 5600 votes. Find the number of voting list.
1. 10000
2. 22000
3. 30000
4. 40000
5. 50000

Answers: 3-a, 4-b, 5-a, 6-a, 7-(15/34), 8-12, 9-b, 10-7, 11-b, 12-d

Profit, Loss, and Discount

Profit: When we buy something at a low price and sell it at a higher price, we have a profit.
Loss: When we buy something at a high price and sell it at a lower price, we have a loss.

Some Basic Terminologies

CP - Cost Price: The price at which a product is purchased, for example, the cost of acquiring inventory.
SP - Selling Price: The price at which a product is sold to customers, for example, the retail price.
If CP > SP, then it results in a loss because the selling price is less than the cost price, indicating a negative financial outcome.
If SP > CP, then it results in a profit because the selling price is higher than the cost price, leading to a positive financial outcome.

Percentage Change

Percentage change is calculated based on initial values.

Example: A buys a shirt for 100 and sells it to B for 110, resulting in a change of +10.
We know that percentage change is always calculated based on initial values, so:
$\frac{10}{100}$ * 100 = 10% change
Note: Whenever we need to calculate profit, we do so based on the cost price (CP).
Another example: A buys a shirt for 110 and sells it for 99, resulting in a change of -11.
$\frac{11}{110}$ * 100 = 10% loss.
Note: Again, we can see that loss is also calculated based on the cost price (CP).
Final note: Both profit and loss are always calculated based on the cost price (CP). Always calculate them based on the cost price unless stated otherwise.

Profit & Loss formula

Loss = CP > SP
Loss = CP - SP
Loss % = $\frac{Loss}{CP}$ * 100
Profit = SP > CP
Profit = SP - CP
Profit % = $\frac{Profit}{CP}$ * 100

Discount

If a shopkeeper wants to give a discount, he first increases the price from the cost price (CP).
When the price is increased, it is known as markup from the cost price (CP), and it becomes the Maximum Retail Price (MRP). After giving a discount on MRP, that price becomes the Selling Price (SP).
Markup = MRP - CP
Markup % = $\frac{Markup}{CP}$ * 100
Discount = MRP - SP
Discount % = $\frac{Discount}{MRP}$ * 100
Note: We always calculate the discount on MRP, and we always calculate the markup on CP.
Condition: If there is a markup but no discount is given, then MRP becomes SP, and the markup becomes profit.

Percentage to Fraction

When a percentage is given, we convert it into a fraction, and the denominator value is called the initial value ratio.

Example: 20% Profit = $\frac{1 (Profit)}{5 (Initial Value or CP)}$
CP + Profit = SP
5 + 1 = 6
CP : SP
5 : 6
20% Loss = $\frac{1}{5}$
CP (5) - Loss (1) = SP (4)
CP : SP
5 : 4
20% Markup = $\frac{1}{5}$
CP (5) + Markup (1) = MRP (6)
CP : MRP
5 : 6
20% Discount = $\frac{1}{5}$
MRP (5) - Discount (1) = SP (4)
MRP : SP
5 : 4
X buys a shirt for Rs 1100 with a 10% profit. Find the cost price (CP).
SP = 1100
Profit = 10% = $\frac{1}{10}$
CP = 10 and 1 is profit so SP = 11
CP : SP is 10 : 11
11 = 1100
1 = 100
10 * 100 = 1000
CP = 1000

Profit calculation of SP

Actual profit is always calculated on CP but sometime when stated we have to calculated it on SP
20% profit on SP = $\frac{1 (profit)}{5 (SP)}$
CP + Profit = SP
CP = SP - Profit
CP = 5 - 1
CP = 4
Now ratio will be CP : SP
4 : 5
20% loss on SP = $\frac{1 (loss)}{5 (SP)}$
CP - Loss = SP
CP = SP + Loss
CP = 5 + 1
CP = 6
Now ratio will be CP : SP
6 : 5
20% loss on SP, find actual loss percentage.
20% loss= $\frac{1 (loss)}{5 (SP)}$
CP : SP
6: 5
actual loss percentage = $\frac{1 (profit)}{6 (CP)}$ * 100 = 16.66%

Questions

The cost price of 15 articles is same as the selling price of 10 articles. The profit percent is?
1. 30%
2. 50%
The ratio of cost price and selling price is 5 : 4, the loss percentage is?
1. 20%
2. 33%
A man purchased an article in Rs 10500. He sold it at a profit of 20%. Find the selling price of an article?
1. 10000
2. 12600
A man sold an article in Rs 12624 at a loss of 25%. Find the cost price of an article?
1. 16832
2. 14500
If an item is sold for rupees 720, there is a loss of 20%, then how much rupees should be sold to make a profit of 20%?
1. 2500
2. 2400
3. 1080
If there is a profit of 25% on the selling price by selling an article at a fixed price, find the actual profit percentage of the item.
1. 33.33%
2. 20%
3. 25%
If the loss on a product is 30% of the selling price then what is ratio of the cost price to loss?
1. 1:13
2. 3:13
3. 13:3
A shopkeeper sold his article at a 25% profit calculated on selling price. Find the selling price of article which cost is Rs 4500?
1. 5000
2. 7500
3. 6000
A shopkeeper sold his article at a 18.75% profit calculated on selling price. Find the selling price of article which cost is Rs 52000?
1. 65000
2. 64000
3. 54000
In selling an article for Rs. 76, there is a profit of 52%. If it is sold for Rs. 75, the profit percent will be?
1. 55%
2. 44%
3. 50%
By selling 250 pens a shopkeeper gain the SP of 50 pen. Find his gain?
1. 25%
2. 20%
After selling 63 article a man losses SP of 9 article find his loss?
1. 12.5%
2. 16.66%
By selling 60 pens a shopkeeper gains the CP of 12 pen. Find his gain?
1. 33%
2. 7%
3. 20%
By selling 18 articles a seller gains the S.P. of 3 articles then find the profit percentage?
1. 16%
2. 20%
A machine is sold at a profit of 18.18%. Had it been sold for Rs 75 less, there would have been a profit of 12.5%. Find the SP of the machine.
1. Rs 1569
2. Rs 1560
3. Rs 1540
A person sells a table at a 8% profit. If he had bought the table at 5% loss cost and sold for Rs 90 more he would have gained 20%. The cost price of table is.
1. Rs 2000
2. Rs 1500
3. Rs 1600
If books bought at prices from Rs. 150 to Rs. 300 are sold at prices ranging from Rs. 250 to Rs. 350, what is the greastest possible profit that might be made in selling 15 books?
1. Cannot be determind
2. Rs 750
3. Rs 3000
An article was sold at a gain of 10%. Had it been bought at 20% less and sold for Rs 20 more the gain would have been 30% more. Find the SP if the gain is 37.5%.
1. 1375
2. 1400
Two vehicles are sold for Rs. 1818 each. One is sold at a profit of 42.84% and another at a loss of 6.25%. What is the net profit/loss %?
1. 13.20%
2. 14.65%

Ans: 1-b, 2-a, 3-b, 4-a, 5-c, 6-a, 7-a, 8-c, 9-b, 10-c, 11-a, 12-a, 13-c, 14-b, 15-b, 16-b, 17-c, 18-a, 19-a,

Price and Article related questions:

Akash buys some toffees at 7 for a rupee and sells them at 5 for a ruppee. His profit % is?
1. 55%
2. 35%
3. 40%
A person bought some articles at 23 for Rs 20 and sold them at the rate of 20 for Rs 23 Find his profit/loss %?
1. 32.25%
2. 33.25%
3. 34.5%

Ans: 1-c, 2-a

Dishonest Shopkeeper

In this scenario, a shopkeeper employs fraudulent techniques to maximize profit. Some of these techniques include:

Selling items above their Cost Price (CP).
- For example, selling a product that costs $50 at $70.
Selling items at CP but reducing the actual weight.
- For example, selling a kilogram of vegetables at $5 but providing only 800 grams.
- For example, if 1 gram of a product costs Rs. 1 and the shopkeeper manipulates the weight so that 1000 grams of the same product only weigh 900 grams when sold, the calculation would be as follows:
  - Actual cost of 1000 grams = 1000 grams * Rs. 1 per gram = Rs. 1000
  - But the shopkeeper sells only 900 grams at Rs. 1 per gram, which should be Rs. 900 based on the actual weight.
  - Profit = Rs. 1000 - Rs. 900 = Rs. 100
  - Profit percentage = (Profit / Actual cost) * 100 = (100 / 900) * 100 ≈ 11.11%
- 900 grams of the product only weigh 800 grams when sold, the calculation would be as follows:
  - Actual cost of 900 grams = 900 grams * Rs. 1 per gram = Rs. 900
  - But the shopkeeper sells the 800-gram product for Rs. 900, resulting in a profit.
  - Profit = Rs. 900 (selling price) - Rs. 800 (actual cost) = Rs. 100
  - Profit percentage = (Profit / Actual cost) * 100 = (100 / 800) * 100 = 12.5%
- 1000 grams of the product only weigh 700 grams when sold, the calculation would be as follows:
  - Actual cost of 1000 grams = 1000 grams * Rs. 1 per gram = Rs. 1000
  - But the shopkeeper sells only 700 grams at Rs. 1 per gram, which should be Rs. 700 based on the actual weight.
  - Profit = Rs. 1000 (selling price) - Rs. 700 (actual cost) = Rs. 300
  - Profit percentage = (Profit / Actual cost) * 100 = (300 / 700) * 100 ≈ 42.86%
- 800 grams of the product only weigh 300 grams when sold, the calculation would be as follows:
  - Actual cost of 800 grams = 800 grams * Rs. 1 per gram = Rs. 800
  - The shopkeeper sells 300 grams but claims it as 800 grams, so the claimed selling price is Rs. 800
  - Actual cost for 300 grams = 300 grams * Rs. 1 per gram = Rs. 300
  - Profit = Rs. 800 (claimed selling price) - Rs. 300 (actual cost) = Rs. 500
  - Profit percentage = (Profit / Actual cost for 300 grams) * 100 = (500 / 300) * 100 ≈ 166.67%
Selling items above CP while also reducing the weight.
- For example, selling a product that costs $30 at $40 but providing only 80% of the advertised quantity.
Using inflated weights during purchases and using lesser weights during sales.
- For example, showing that he bought only 1 kilogram of fruit, but in reality, he reports it as 1.5 kilograms during purchase, then sells 1 kilogram at the market price.
- The shopkeeper claims to buy 1100 grams of a product but records it as only 1000 grams. Then, the shopkeeper sells this reduced quantity of 900 grams.
  This means the shopkeeper is making a profit when selling by selling less than the stated quantity (1000 grams instead of 1100 grams).
  So, in this situation, the shopkeeper gains profit by misrepresenting the quantity during purchase and then selling less than what was stated, resulting in a profit both when buying and selling.
  This kind of behavior is called a successive scenario, where the shopkeeper benefits at both ends of the transaction.

These deceptive practices enable the shopkeeper to increase profits unfairly at the expense of customers.

Questions:

A dishonest shopkeeper professed to sell his goods at cost price but he uses a weight of 800 gram instead of a kilogram. Find his profit percentage?
1. 33%
2. 20%
3. 25%
A dishonest shopkeeper promises to sell his goods at its CP but he uses 40% less weigt. find his profit?
1. 44.44%
2. 45.55%
3. 66.66%
A dishonest shopkeeper promises to sell his good at 30% profit but he uses 800 gm weight instead of 1 kg. Find his actual profit?
1. 62.5%
2. 58.33%
3. 60%
A dishonest shopkeeper professes to sell his goods at CP but he cheats 10% at the time of purchasing as well as selling. Find his overall profit or loss percentage?
1. 21%
2. 22.22%
3. 25%
A man marks up his goods by 15% but he gives 920gm instead of 1 kg to his customer find his profit?
1. 23%
2. 22%
3. 25%
4. 55%
A shopkeeper marks his goods 16% above the CP. At the time of selling the goods he uses 845gm weigt instead of 1kg. Find his profit?
1. 38.46%
2. 35%
A trader cheats both his supplier and his customer by using faulty weights. When he bus from the supplier, he takes 20% more than the indicated weight. When he sells to his customer, he gives 20% less than the indicated weight. If he sells his articles at the cost price, what is his net profit?
1. 50%
2. 66.66%
3. 44.44%

Answers: 1-c, 2-c, 3-a, 4-b, 5-c, 6-a, 7-a