Let's go through each type of percentage problem in detail with examples:
1. Finding the Percentage of a Number
Concept
To find a certain percentage of a given number, you use the formula:
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]
Example
Question: What is 20% of 500?
Solution
\[ \text{Percentage} = \left( \frac{20}{100} \right) \times 500 \]
\[ \text{Percentage} = 0.20 \times 500 \]
\[ \text{Percentage} = 100 \]
So, 20% of 500 is 100.
2. Percentage Increase or Decrease
Concept
To find the percentage increase or decrease from one value to another, use the formula:
\[ \text{Percentage Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100 \]
Example
Question: If the price of a product increases from $50 to $60, what is the percentage increase?
Solution
\[ \text{Percentage Change} = \left( \frac{60 - 50}{50} \right) \times 100 \]
\[ \text{Percentage Change} = \left( \frac{10}{50} \right) \times 100 \]
\[ \text{Percentage Change} = 0.20 \times 100 \]
\[ \text{Percentage Change} = 20\% \]
So, the price increase is 20%.
3. Discounts and Markups
Concept
To calculate the discounted price or selling price after a markup, use:
- For discount: Selling Price = Original Price × (1 - Discount Percentage)
- For markup: Selling Price = Original Price × (1 + Markup Percentage)
Example
Question: If an item is marked up by 25%, what is the selling price if the original price was $80?
Solution
\[ \text{Selling Price} = 80 \times (1 + 0.25) \]
\[ \text{Selling Price} = 80 \times 1.25 \]
\[ \text{Selling Price} = 100 \]
So, the selling price after a 25% markup is $100.
4. Profit and Loss
Concept
To find the profit or loss percentage based on the cost price (CP) and selling price (SP), use:
\[ \text{Profit Percentage} = \left( \frac{\text{SP} - \text{CP}}{\text{CP}} \right) \times 100 \]
Example
Question: If an item is sold for $120, and the cost price was $100, what is the profit percentage?
Solution
\[ \text{Profit Percentage} = \left( \frac{120 - 100}{100} \right) \times 100 \]
\[ \text{Profit Percentage} = \left( \frac{20}{100} \right) \times 100 \]
\[ \text{Profit Percentage} = 20\% \]
So, the profit percentage is 20%.
5. Simple Interest
Concept
To calculate simple interest (SI) earned on a principal amount (P) over time (t) at a specified interest rate (r), use:
\[ \text{Simple Interest} = \frac{P \times r \times t}{100} \]
Example
Question: If $2000 is invested at an interest rate of 5% per annum, what is the simple interest earned in 3 years?
Solution
\[ \text{Simple Interest} = \frac{2000 \times 5 \times 3}{100} \]
\[ \text{Simple Interest} = \frac{300}{100} \]
\[ \text{Simple Interest} = 30 \]
So, the simple interest earned is $30.
6. Successive Percentage Changes
Concept
To find the overall percentage change when there are successive percentage increases or decreases, use:
\[ \text{Overall Percentage Change} = \left( \frac{\text{Total Change}}{\text{Original Value}} \right) \times 100 \]
Example
Question: If the price of a product increases by 20% and then decreases by 10%, what is the overall percentage change?
Solution
First, calculate the new price after each change:
- Increase by 20%: New price = Original price × (1 + 0.20)
- Decrease by 10%: Final price = New price × (1 - 0.10)
Compute the overall percentage change from the original price.
7. Percentage of Another Quantity
Concept
To find one quantity as a percentage of another quantity, use:
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]
Example
Question: If 25 students in a class of 40 passed an exam, what percentage of the class passed?
Solution
So, 62.5% of the class passed the exam.
These examples cover the major types of percentage problems you may encounter in exams like SSC, UPSC, CAPF, CDS, railways, IBPS bank.
