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Percentage is a fundamental concept in mathematics that means "per hundred". Whenever you see the term percentage, it refers to a part out of 100. Imagine you have a cake cut into 100 equal pieces; the number of pieces you have is your percentage of the whole cake. Percentages are everywhere-calculating exam scores, discounts during sales, interest rates on bank deposits, or battery levels on your phone. In competitive exams, understanding percentages is critical because many real-life problems are framed using them.
Since percentages are closely related to fractions and decimals, grasping these connections helps solve problems more easily. In this section, we will begin from first principles, learn the meaning of percentages, and gradually move towards solving practical problems involving money (INR) and metric measurements.
The word percentage literally means per hundred. It is a way of expressing a number as a fraction of 100. The symbol for percentage is %.
For example, 25% means 25 out of 100.
To understand this visually, imagine a circle representing a whole (100%). Shaded parts of this circle represent the percentage.
Percentages can also be converted into fractions and decimals easily:
Often we are asked to find a certain percentage of a given number, like "What is 12% of 5000 INR?". This means finding how many parts out of the whole number correspond to the given percentage.
The general method is:
graph TD A[Start] --> B[Convert percentage p% to decimal: p/100] B --> C[Multiply decimal by the number N] C --> D[Result is p% of N]
Why does this work? Because percentage means parts per 100, converting p% to a decimal gives a fraction that when multiplied by N gives exactly the portion of N that represents p%.
Step 1: Convert 12.5% to decimal.
12.5% = \( \frac{12.5}{100} = 0.125 \)
Step 2: Multiply decimal by 3200.
\( 0.125 \times 3200 = 400 \)
Answer: 12.5% of 3200 INR is 400 INR.
Percentages are commonly used to describe how much a quantity increases or decreases compared to its original value. This is especially useful when dealing with prices, populations, or measurements.
Percentage increase means the new value is more than the original. Percentage decrease means the new value is less.
| Type | Original Amount | Percentage Change | Formula for New Amount |
|---|---|---|---|
| Increase | \( O \) | \( p\% \) increase | \( \text{New Amount} = O \times \left(1 + \frac{p}{100}\right) \) |
| Decrease | \( O \) | \( p\% \) decrease | \( \text{New Amount} = O \times \left(1 - \frac{p}{100}\right) \) |
Think of it this way: For an increase, you add the percentage to 100%, and for a decrease, you subtract it from 100%, then multiply with the original amount.
Step 1: Convert percentage increase to multiplier.
Multiplier = \( 1 + \frac{15}{100} = 1 + 0.15 = 1.15 \)
Step 2: Multiply original price by multiplier.
\( 1200 \times 1.15 = 1380 \)
Answer: New price after 15% increase = 1380 INR.
Sometimes, a quantity is changed by a percentage multiple times in succession. For example, a price can increase by 10% one month and decrease by 5% the next. Successive percentage changes do not add up simply; instead, you multiply the effects.
This is because each change applies to the new value after the previous change.
graph TD A[Start with Original Value] A --> B[Apply first change: Multiply by (1 + p1/100)] B --> C[Apply second change: Multiply by (1 + p2/100)] C --> D[Continue for each change] D --> E[Net Change = Product of multipliers - 1]
If \( p_1 \), \( p_2 \), ... are percentage changes (use negative values for decreases), then:
Step 1: Express changes as multipliers.
Increase by 10% = \( 1 + \frac{10}{100} = 1.10 \)
Decrease by 5% = \( 1 - \frac{5}{100} = 0.95 \)
Step 2: Multiply multipliers to get net change factor.
\( 1.10 \times 0.95 = 1.045 \)
Step 3: Net change = 1.045 - 1 = 0.045 = 4.5%
Answer: Net effect is a 4.5% increase.
Percentage calculations appear often in commerce and trading scenarios such as profit, loss, discount, and commission. Understanding how to apply percentages in these contexts is essential especially for exams related to business mathematics.
| Term | Formula | Explanation |
|---|---|---|
| Profit | Profit = Selling Price (SP) - Cost Price (CP) | Gain made on selling an item |
| Loss | Loss = Cost Price (CP) - Selling Price (SP) | Amount lost when selling price is less than cost price |
| Profit Percentage | Profit% = \( \frac{\text{Profit}}{\text{Cost Price}} \times 100 \) | Profit expressed as a percentage of Cost Price |
| Loss Percentage | Loss% = \( \frac{\text{Loss}}{\text{Cost Price}} \times 100 \) | Loss expressed as a percentage of Cost Price |
| Discount | Discount = Marked Price - Selling Price | Reduction from the marked price |
| Discount Percentage | Discount% = \( \frac{\text{Discount}}{\text{Marked Price}} \times 100 \) | Discount as a percentage of the Marked Price |
| Selling Price after Discount | Selling Price = Marked Price \( \times \left(1 - \frac{\text{Discount%}}{100}\right) \) | Price buyer pays after discount |
Step 1: Calculate Profit.
Profit = SP - CP = 575 - 500 = 75 INR
Step 2: Calculate profit percentage.
Profit% = \( \frac{75}{500} \times 100 = 15\% \)
Answer: Profit percentage is 15%.
Step 1: Convert discount percentage to multiplier.
Multiplier = \( 1 - \frac{10}{100} = 0.90 \)
Step 2: Calculate selling price.
Selling Price = 2000 \( \times \) 0.90 = 1800 INR
Answer: Selling price after discount is 1800 INR.
Step 1: Calculate Loss.
Loss = CP - SP = 25,000 - 23,500 = 1,500 INR
Step 2: Calculate Loss %.
Loss% = \( \frac{1,500}{25,000} \times 100 = 6\% \)
Answer: Loss percentage is 6%.
Step 1: Calculate increase amount.
Increase = 100 - 80 = 20 litres
Step 2: Calculate percentage increase.
Percentage increase = \( \frac{20}{80} \times 100 = 25\% \)
Answer: Percentage increase is 25%.
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