Percentages: Concepts and Calculations
Percentage: Definition and Basic Concept
The term percentage is derived from the Latin words 'per centum', meaning 'by the hundred' or 'for every hundred'. A percentage is essentially a fraction where the denominator is fixed at 100. It is a way of expressing a part of a whole as a proportion of 100. The symbol used to denote percentage is '%'.
For instance, if we say "20%", it means 20 out of 100 parts of the whole. This can be directly written as a fraction $\frac{20}{100}$.
In general, any percentage 'x%' can be written as the fraction $\frac{x}{100}$.
$\boldsymbol{\text{x % } = \frac{x}{100}}$
... (i)
Percentages provide a standardized way to compare proportions of different quantities or from different totals. Comparing, for example, securing 30 marks out of 50 in one test and 45 marks out of 75 in another test can be difficult directly. Converting them to percentages simplifies the comparison: $\frac{30}{50} = \frac{60}{100} = 60\%$ and $\frac{45}{75} = \frac{\cancel{45}^3}{\cancel{75}_5} = \frac{3}{5} = \frac{60}{100} = 60\%$. In this case, the performance is equally good in both tests, scoring 60%.
Understanding the Concept with Examples
The concept of percentage is about representing a fraction of a total as if the total were 100.
- If you say "50% of a quantity", it means $\frac{50}{100} = \frac{1}{2}$ (half) of that quantity.
- If you say "25% of a quantity", it means $\frac{25}{100} = \frac{1}{4}$ (one-quarter) of that quantity.
- If you say "100% of a quantity", it means $\frac{100}{100} = 1$ times the quantity, i.e., the entire quantity.
It is possible for a percentage to be greater than 100%. This represents a quantity that is larger than the original whole or base amount.
- If you say "150% of a quantity", it means $\frac{150}{100} = 1.5$ times the quantity. This could represent, for example, an increase of 50% over the original quantity (Original 100% + Increase 50% = Total 150%).
- If you say "200% of a quantity", it means $\frac{200}{100} = 2$ times the quantity (i.e., double the quantity).
Example 1. What does 30% mean?
Answer:
30% means 30 parts out of 100 equal parts of a whole.
As a fraction, $30\% = \frac{30}{100} = \frac{3}{10}$.
As a decimal, $30\% = \frac{30}{100} = 0.30$ or $0.3$.
Competitive Exam Notes:
Percentage is a fundamental concept in quantitative aptitude. It is widely used in topics like Profit & Loss, Simple & Compound Interest, Data Interpretation, etc.
- Definition: Percentage always means 'out of 100'. The symbol '%' is a shorthand for $\frac{1}{100}$.
- Basic Conversion: $x\% = \frac{x}{100}$. This is the most important conversion formula.
- Understanding Magnitude: Be comfortable with percentages less than 100% (representing a part less than the whole), 100% (representing the whole), and more than 100% (representing more than the whole, often used to show increases).
- Comparison: Percentages are excellent for comparing proportions from different base amounts. Always convert to percentages for comparison unless the base amounts are the same.
Conversion between Percentage, Fractions, Ratios, and Decimals
To effectively solve problems involving percentages, it is crucial to be able to convert values seamlessly between percentages, fractions, ratios, and decimals. These are just different ways of representing the same proportional value.
1. Percentage to Fraction
To convert a percentage to a fraction, simply divide the percentage value by 100 and simplify the resulting fraction to its lowest terms.
$\boldsymbol{\text{x % } = \frac{x}{100}}$
Examples:
- Convert 50% to a fraction:
- Convert 75% to a fraction:
- Convert 120% to a fraction:
- Convert $33\frac{1}{3}\%$ to a fraction:
$\text{50 %} = \frac{50}{100} = \frac{\cancel{50}^1}{\cancel{100}_2} = \boldsymbol{\frac{1}{2}}$
$\text{75 %} = \frac{75}{100} = \frac{\cancel{75}^3}{\cancel{100}_4} = \boldsymbol{\frac{3}{4}}$
$\text{120 %} = \frac{120}{100} = \frac{\cancel{120}^{6}}{\cancel{100}_{5}} = \boldsymbol{\frac{6}{5}}$
$33\frac{1}{3}\% = \left(\frac{3 \times 33 + 1}{3}\right)\% = \frac{100}{3}\%$
$\frac{100}{3}\% = \frac{\left(\frac{100}{3}\right)}{100} = \frac{100}{3 \times 100} = \frac{\cancel{100}^1}{3 \times \cancel{100}_1} = \boldsymbol{\frac{1}{3}}$
2. Fraction to Percentage
To convert a fraction to a percentage, multiply the fraction by 100 and append the '%' symbol.
$\boldsymbol{\frac{a}{b} = \left(\frac{a}{b} \times 100\right)\%}$
Examples:
- Convert $\frac{1}{4}$ to a percentage:
- Convert $\frac{3}{5}$ to a percentage:
- Convert $\frac{7}{2}$ to a percentage:
$\frac{1}{4} = \left(\frac{1}{4} \times 100\right)\% = (1 \times 25)\% = \boldsymbol{25\%}$
$\frac{3}{5} = \left(\frac{3}{5} \times 100\right)\% = (3 \times 20)\% = \boldsymbol{60\%}$
$\frac{7}{2} = \left(\frac{7}{2} \times 100\right)\% = (7 \times 50)\% = \boldsymbol{350\%}$
3. Percentage to Decimal
To convert a percentage to a decimal, divide the percentage value by 100. This is equivalent to moving the decimal point two places to the left.
$\boldsymbol{\text{x % } = \frac{x}{100} = \text{Decimal Value}}$
Examples:
- Convert 45% to a decimal:
- Convert 8% to a decimal:
- Convert 17.5% to a decimal:
- Convert 300% to a decimal:
$\text{45 %} = \frac{45}{100} = \boldsymbol{0.45}$
$\text{8 %} = \frac{8}{100} = \boldsymbol{0.08}$
$\text{17.5 %} = \frac{17.5}{100} = \boldsymbol{0.175}$
$\text{300 %} = \frac{300}{100} = \boldsymbol{3}$
4. Decimal to Percentage
To convert a decimal to a percentage, multiply the decimal value by 100 and append the '%' symbol. This is equivalent to moving the decimal point two places to the right.
$\boldsymbol{\text{Decimal Value } = (\text{Decimal Value } \times 100)\%}$
Examples:
- Convert 0.65 to a percentage:
- Convert 0.03 to a percentage:
- Convert 1.25 to a percentage:
$\boldsymbol{0.65 = (0.65 \times 100)\% = 65\%}$
$\boldsymbol{0.03 = (0.03 \times 100)\% = 3\%}$
$\boldsymbol{1.25 = (1.25 \times 100)\% = 125\%}$
5. Ratio to Percentage and Percentage to Ratio
A ratio $a:b$ is equivalent to the fraction $\frac{a}{b}$. Therefore, to convert a ratio to a percentage, first write it as a fraction and then convert the fraction to a percentage by multiplying by 100%.
$\boldsymbol{\text{Ratio } a:b = \frac{a}{b} = \left(\frac{a}{b} \times 100\right)\%}$
Conversely, to convert a percentage to a ratio, first write the percentage as a fraction (divide by 100), simplify the fraction to its lowest terms, and then express the simplified fraction as a ratio.
$\boldsymbol{\text{x % } = \frac{x}{100} = \frac{a}{b} \text{ (in simplest form)} = a:b}$
Examples:
- Express the ratio $1:5$ as a percentage:
- Express 80% as a ratio in simplest form:
Ratio $1:5 = \frac{1}{5}$
Percentage $= \left(\frac{1}{5} \times 100\right)\% = (1 \times 20)\% = \boldsymbol{20\%}$
Percentage $80\% = \frac{80}{100}$
Simplify the fraction by dividing the numerator and denominator by their GCD, which is 20:
$\frac{80}{100} = \frac{\cancel{80}^4}{\cancel{100}_5} = \frac{4}{5}$
Ratio form $= \boldsymbol{4:5}$
Summary Table of Common Conversions
It is helpful to memorize some common conversions for quick calculations in competitive exams.
| Fraction | Decimal | Percentage | Ratio |
|---|---|---|---|
| 1 | 1.0 | 100% | 1:1 |
| $\frac{1}{2}$ | 0.5 | 50% | 1:2 |
| $\frac{1}{3}$ | $0.\overline{3}$ | $33\frac{1}{3}\%$ or $\approx 33.33\%$ | 1:3 |
| $\frac{2}{3}$ | $0.\overline{6}$ | $66\frac{2}{3}\%$ or $\approx 66.67\%$ | 2:3 |
| $\frac{1}{4}$ | 0.25 | 25% | 1:4 |
| $\frac{3}{4}$ | 0.75 | 75% | 3:4 |
| $\frac{1}{5}$ | 0.2 | 20% | 1:5 |
| $\frac{2}{5}$ | 0.4 | 40% | 2:5 |
| $\frac{3}{5}$ | 0.6 | 60% | 3:5 |
| $\frac{4}{5}$ | 0.8 | 80% | 4:5 |
| $\frac{1}{6}$ | $0.1\overline{6}$ | $16\frac{2}{3}\%$ or $\approx 16.67\%$ | 1:6 |
| $\frac{5}{6}$ | $0.8\overline{3}$ | $83\frac{1}{3}\%$ or $\approx 83.33\%$ | 5:6 |
| $\frac{1}{8}$ | 0.125 | 12.5% | 1:8 |
| $\frac{3}{8}$ | 0.375 | 37.5% | 3:8 |
| $\frac{5}{8}$ | 0.625 | 62.5% | 5:8 |
| $\frac{7}{8}$ | 0.875 | 87.5% | 7:8 |
| $\frac{1}{9}$ | $0.\overline{1}$ | $11\frac{1}{9}\%$ or $\approx 11.11\%$ | 1:9 |
| $\frac{1}{10}$ | 0.1 | 10% | 1:10 |
| $\frac{1}{11}$ | $0.\overline{09}$ | $9\frac{1}{11}\%$ or $\approx 9.09\%$ | 1:11 |
| $\frac{1}{12}$ | $0.08\overline{3}$ | $8\frac{1}{3}\%$ or $\approx 8.33\%$ | 1:12 |
| $\frac{1}{100}$ | 0.01 | 1% | 1:100 |
Competitive Exam Notes:
Proficiency in converting between these forms is critical for speed and accuracy. Often, solving a problem is simpler when working with fractions or decimals than with percentages directly.
- Percentage to Fraction is Key: Many calculations become easier by converting percentages to their fractional equivalents (e.g., $25\% = 1/4$, $33.33\% = 1/3$, $12.5\% = 1/8$). Memorize the common conversions.
- Decimal for Calculations: For calculations involving multiplications or divisions, converting percentages to decimals is often convenient (e.g., to find 20% of 50, calculate $0.20 \times 50$).
- Ratios as Fractions: Treat ratios as fractions for conversion purposes.
- Avoid Rounding Too Early: When dealing with repeating decimals (like $33.33...\%$), it's usually best to work with their exact fraction form ($1/3$) to avoid rounding errors until the final step.
Finding the Percentage of a Given Quantity
A common application of percentages is calculating a specific percentage value of a given quantity or a total amount. To do this, we first convert the percentage into its equivalent fractional or decimal form and then multiply it by the given quantity.
The formula to find 'x%' of a quantity $Q$ is:
$\boldsymbol{\text{x % of Quantity } Q = \frac{x}{100} \times Q}$
... (i)
Alternatively, using the decimal form:
$\boldsymbol{\text{x % of Quantity } Q = (\text{Decimal equivalent of x%}) \times Q}$
Since the decimal equivalent of x% is $\frac{x}{100}$, these two formulas are the same.
Example 1. Find 25% of $\textsf{₹ } 1200$.
Answer:
We need to calculate 25% of $\textsf{₹ } 1200$.
Using the formula $\frac{x}{100} \times Q$ with $x=25$ and $Q=1200$:
25% of $\textsf{₹ } 1200 = \frac{25}{100} \times 1200$
Convert 25% to its fractional form and simplify:
$\frac{25}{100} = \frac{1}{4}$
So, the calculation becomes:
25% of $\textsf{₹ } 1200 = \frac{1}{4} \times 1200$
Perform the multiplication:
$\frac{1}{4} \times 1200 = \frac{\cancel{1200}^{300}}{\cancel{4}^{1}} = 300$
$\boldsymbol{= \textsf{₹ } 300}$
So, 25% of $\textsf{₹ } 1200$ is $\boldsymbol{\textsf{₹ } 300}$.
Using Decimal Form:
Convert 25% to a decimal: $25\% = \frac{25}{100} = 0.25$.
25% of $\textsf{₹ } 1200 = 0.25 \times 1200$
Performing the multiplication:
$\begin{array}{ccccccc}& & & 1 & 2 & 0 & 0 \\ \times & & & & 0 & . & 2 & 5 \\ \hline &&& 6 & 0 & 0 & 0 \\ && 2 & 4 & 0 & 0 & \times \\ \hline && 3 & 0 & 0 .& 0 & 0 \\ \hline \end{array}$
$\boldsymbol{= \textsf{₹ } 300}$
Both methods yield the same result.
Example 2. What is 12% of 50 litres?
Answer:
We need to calculate 12% of 50 litres.
Using the formula $\frac{x}{100} \times Q$ with $x=12$ and $Q=50$:
12% of 50 litres $= \frac{12}{100} \times 50$ litres
Simplify the fraction and multiply:
$\frac{12}{100} \times 50 = \frac{12}{\cancel{100}^{\normalsize 2}} \times \cancel{50}^{\normalsize 1} = \frac{12}{2} = 6$
$\boldsymbol{= 6}$ litres
So, 12% of 50 litres is $\boldsymbol{6}$ litres.
Example 3. In a village of 2500 people, 40% are literate. Find the number of illiterate people.
Answer:
Total population of the village $= 2500$.
Percentage of literate people $= 40\%$.
Method 1: Find literate people first, then illiterate people.
Number of literate people $= 40\%$ of 2500
Number of literate people $= \frac{40}{100} \times 2500$
$= \frac{40}{\cancel{100}^{\normalsize 1}} \times \cancel{2500}^{\normalsize 25}$
$= 40 \times 25 = 1000$
Number of literate people $= 1000$.
Number of illiterate people = Total population - Number of literate people
Number of illiterate people $= 2500 - 1000 = \boldsymbol{1500}$
Method 2: Find percentage of illiterate people first.
If 40% of the people are literate, the remaining percentage are illiterate. Total percentage is 100%.
Percentage of illiterate people $= (100 - 40)\% = 60\%$
Now, find 60% of the total population (2500).
Number of illiterate people $= 60\%$ of 2500
$= \frac{60}{100} \times 2500$
$= \frac{60}{\cancel{100}^{\normalsize 1}} \times \cancel{2500}^{\normalsize 25}$
$= 60 \times 25 = 1500$
Number of illiterate people $= \boldsymbol{1500}$.
Both methods give the same result. There are $\boldsymbol{1500}$ illiterate people in the village.
Competitive Exam Notes:
Calculating a percentage of a quantity is a basic operation. Be efficient in using both fraction and decimal conversions.
- Formula: $x\%$ of $Q = \frac{x}{100} \times Q$.
- Fractional Method: Convert the percentage to a simplified fraction (e.g., $20\% = 1/5$). This is often faster for mental calculations or when the quantity is a multiple of the denominator.
- Decimal Method: Convert the percentage to a decimal (e.g., $15\% = 0.15$). This is useful when the numbers don't simplify easily as fractions or when using a calculator.
- Complementary Percentages: If a percentage of one part is given, the percentage of the remaining part is $100\%$ minus the given percentage. This can sometimes shorten the calculation (as seen in Example 3).
- "Of" means Multiply: In percentage problems, the word "of" typically implies multiplication.
Finding a Quantity when its Percentage is Given
Sometimes, instead of being asked to find the percentage of a quantity, we are given a part of the quantity as a percentage of the whole, and we need to find the original whole quantity. This is the inverse of the previous type of problem.
Suppose we are told that $x \%$ of an unknown quantity $Q$ is equal to a known value $V$. We can write this as an equation:
$\text{x % of } Q = V$
Convert the percentage to a fraction:
$\frac{x}{100} \times Q = V$
... (ii)
To find the original quantity $Q$, we can rearrange equation (ii):
$\boldsymbol{Q = V \times \frac{100}{x}}$
[Formula to find the Original Quantity]
This formula tells us that to find the whole quantity, divide the given value ($V$) by the percentage ($x$), and then multiply by 100. Equivalently, multiply the value ($V$) by the reciprocal of the percentage expressed as a fraction ($\frac{100}{x}$).
Example 1. If 30% of a number is 150, what is the number?
Answer:
Let the unknown number be $Q$.
We are given that 30% of $Q$ is 150.
Set up the equation based on the problem statement:
30% of $Q = 150$
Convert 30% to a fraction:
$\frac{30}{100} \times Q = 150$
Simplify the fraction $\frac{30}{100} = \frac{3}{10}$.
$\frac{3}{10} \times Q = 150$
... (iii)
To solve for $Q$ from equation (iii), multiply both sides by the reciprocal of $\frac{3}{10}$, which is $\frac{10}{3}$:
$\boldsymbol{Q = 150 \times \frac{10}{3}}$
Perform the calculation:
$\boldsymbol{Q = \cancel{150}^{\normalsize 50} \times \frac{10}{\cancel{3}^{\normalsize 1}}}$
$\boldsymbol{Q = 50 \times 10 = 500}$
The number is $\boldsymbol{500}$.
Using the Formula $Q = V \times \frac{100}{x}$:
Here, the percentage $x = 30$ and the value $V = 150$.
$\boldsymbol{Q = 150 \times \frac{100}{30}}$
Simplify the fraction $\frac{100}{30} = \frac{10}{3}$.
$\boldsymbol{Q = 150 \times \frac{10}{3}}$
$\boldsymbol{Q = \cancel{150}^{\normalsize 50} \times \frac{10}{\cancel{3}^{\normalsize 1}}}$
$\boldsymbol{Q = 50 \times 10 = 500}$
The number is $\boldsymbol{500}$.
Example 2. A student scored 65% marks in an exam, which is equal to 520 marks. Find the maximum marks for the exam.
Answer:
Let the maximum marks for the exam be $M$. The student's score (520 marks) represents 65% of the maximum marks.
We are given that 65% of $M$ is 520.
Set up the equation:
65% of $M = 520$
Convert 65% to a fraction:
$\frac{65}{100} \times M = 520$
Simplify the fraction $\frac{65}{100}$ by dividing numerator and denominator by 5:
$\frac{13}{20} \times M = 520$
... (iv)
To solve for $M$ from equation (iv), multiply both sides by the reciprocal of $\frac{13}{20}$, which is $\frac{20}{13}$:
$\boldsymbol{M = 520 \times \frac{20}{13}}$
Perform the calculation. Note that $520$ is a multiple of $13$ ($13 \times 4 = 52$):
$\boldsymbol{M = \cancel{520}^{\normalsize 40} \times \frac{20}{\cancel{13}^{\normalsize 1}}}$
$\boldsymbol{M = 40 \times 20 = 800}$
The maximum marks for the exam are $\boldsymbol{800}$.
Using the Formula $Q = V \times \frac{100}{x}$:
Here, the percentage $x = 65$ and the value $V = 520$. The quantity $Q$ is the maximum marks $M$.
$\boldsymbol{M = V \times \frac{100}{x} = 520 \times \frac{100}{65}}$
Simplify the fraction $\frac{100}{65}$ by dividing numerator and denominator by 5: $\frac{\cancel{100}^{20}}{\cancel{65}_{13}} = \frac{20}{13}$.
$\boldsymbol{M = 520 \times \frac{20}{13}}$
Now multiply:
$\boldsymbol{M = \cancel{520}^{\normalsize 40} \times \frac{20}{\cancel{13}^{\normalsize 1}}}$
$\boldsymbol{M = 40 \times 20 = 800}$
The maximum marks are $\boldsymbol{800}$.
Competitive Exam Notes:
Finding the original quantity from a given percentage is a core problem type. The key is setting up the equation correctly.
- Equation Setup: Always remember that "$x\%$ of $Q$ is $V$" translates to $\frac{x}{100} \times Q = V$.
- Formula: The derived formula $Q = V \times \frac{100}{x}$ is very useful for quick calculation.
- Reciprocal Thinking: Mentally, you are multiplying the given value by the reciprocal of the percentage expressed as a fraction (e.g., for 20% or 1/5, multiply by 5; for 75% or 3/4, multiply by 4/3).
- Unitary Method Logic: This problem is also solvable using the unitary method. If $x\%$ corresponds to value $V$, then $1\%$ corresponds to $\frac{V}{x}$, and $100\%$ (the whole quantity) corresponds to $\frac{V}{x} \times 100$. This leads back to the same formula.
Calculating Percentage Change (Increase/Decrease)
Percentage change is a measure that quantifies the relative change in a quantity over time or between two points of reference. It expresses the change as a percentage of the quantity's original value. This concept is widely used in economics (e.g., inflation rates, GDP growth), finance (e.g., stock price changes), and everyday situations.
There are two types of percentage change: percentage increase and percentage decrease.
Percentage Increase
Percentage increase occurs when a quantity's new value is greater than its original value. To calculate the percentage increase, we find the amount of increase, divide it by the original value, and then multiply the result by 100%.
The amount of Increase is calculated as: Increase = New Value - Original Value.
The formula for Percentage Increase is:
$\text{Percentage Increase } = \frac{\text{Increase}}{\text{Original Value}} \times 100 \%$
Substituting the formula for 'Increase', we get:
$\boldsymbol{\text{Percentage Increase } = \frac{\text{New Value - Original Value}}{\text{Original Value}} \times 100 \%}$
... (i)
For the percentage increase to be meaningful, the original value must be non-zero.
Percentage Decrease
Percentage decrease occurs when a quantity's new value is less than its original value. To calculate the percentage decrease, we find the amount of decrease, divide it by the original value, and then multiply the result by 100%.
The amount of Decrease is calculated as: Decrease = Original Value - New Value.
The formula for Percentage Decrease is:
$\text{Percentage Decrease } = \frac{\text{Decrease}}{\text{Original Value}} \times 100 \%$
Substituting the formula for 'Decrease', we get:
$\boldsymbol{\text{Percentage Decrease } = \frac{\text{Original Value - New Value}}{\text{Original Value}} \times 100 \%}$
... (ii)
For percentage change calculations, it is crucial to remember that the denominator is always the original value or the starting value from which the change occurred.
Example 1. The price of a bicycle increased from $\textsf{₹ } 5000$ to $\textsf{₹ } 6000$. Find the percentage increase in price.
Answer:
Original Price $= \textsf{₹ } 5000$.
New Price $= \textsf{₹ } 6000$.
Since the new price is greater than the original price, this is a percentage increase.
Calculate the increase in price:
Increase $= \text{New Price} - \text{Original Price} = \textsf{₹ } 6000 - \textsf{₹ } 5000 = \textsf{₹ } 1000$
Using the percentage increase formula (i):
$\text{Percentage Increase } = \frac{\text{Increase}}{\text{Original Price}} \times 100 \%$
$\text{Percentage Increase } = \frac{\textsf{₹ } 1000}{\textsf{₹ } 5000} \times 100 \%$
Simplify the fraction and calculate:
$\frac{1000}{5000} = \frac{1}{5}$
$\text{Percentage Increase } = \frac{1}{5} \times 100 \% = \boldsymbol{20 \%}$
The percentage increase in the price of the bicycle is $\boldsymbol{20\%}$.
Example 2. The number of students in a school decreased from 800 to 760 in a year. Find the percentage decrease in the number of students.
Answer:
Original Number of Students $= 800$.
New Number of Students $= 760$.
Since the new number is less than the original number, this is a percentage decrease.
Calculate the decrease in the number of students:
Decrease $= \text{Original Number} - \text{New Number} = 800 - 760 = 40$
Using the percentage decrease formula (ii):
$\text{Percentage Decrease } = \frac{\text{Decrease}}{\text{Original Number}} \times 100 \%$
$\text{Percentage Decrease } = \frac{40}{800} \times 100 \%$
Simplify the fraction and calculate:
$\frac{40}{800} = \frac{4}{80} = \frac{1}{20}$
$\text{Percentage Decrease } = \frac{1}{20} \times 100 \% = \boldsymbol{5 \%}$
The percentage decrease in the number of students is $\boldsymbol{5\%}$.
Competitive Exam Notes:
Percentage change is a very common calculation. Pay close attention to whether it's an increase or decrease and always use the original value as the base (denominator).
- Formula for Increase: $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$. The numerator is positive.
- Formula for Decrease: $\frac{\text{Original} - \text{New}}{\text{Original}} \times 100\%$. The numerator is positive. Alternatively, use $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$ and the negative sign indicates a decrease.
- Identifying Original Value: The original value is the value *before* the change happened.
- Base for Calculation: The base (denominator) is always the original value, not the new value or the amount of change.
- Fractional/Decimal Equivalents: Converting the fraction $\frac{\text{Change}}{\text{Original Value}}$ to its simplified form or decimal equivalent can simplify the calculation before multiplying by 100%.
- Practical Applications: This concept is applied in calculating profit/loss percentage, discount percentage, growth/depreciation rates, etc.
Successive Percentage Change
Successive percentage change refers to situations where a quantity undergoes a percentage change, and then the resulting new quantity undergoes another percentage change (or multiple percentage changes). The key characteristic is that each subsequent percentage change is applied to the result of the previous change, not to the original value.
Calculating Successive Changes
There are several methods to calculate the final value and the net percentage change after successive percentage changes:
Method 1: Step-by-step Calculation
This is the most intuitive method. Calculate the first percentage change and find the resulting value. Then, use this resulting value as the new base for the second percentage change calculation, and so on. Finally, compare the final value to the original value to find the net change and net percentage change.
Method 2: Using Multipliers
A percentage change can be represented by a multiplier. If a quantity increases by $x\%$, the new value is $100\% + x\% = (100+x)\%$ of the original value. The multiplier is $\frac{100+x}{100} = 1 + \frac{x}{100}$. If a quantity decreases by $y\%$, the new value is $100\% - y\% = (100-y)\%$ of the original value. The multiplier is $\frac{100-y}{100} = 1 - \frac{y}{100}$.
- For an increase of $x\%$, the multiplier is $\left(1 + \frac{x}{100}\right)$.
- For a decrease of $y\%$, the multiplier is $\left(1 - \frac{y}{100}\right)$.
To find the final value after successive changes, multiply the original quantity by the multiplier for each change in sequence.
$\boldsymbol{\text{Final Value } = \text{Original Value} \times \left(1 \pm \frac{\text{Change 1 %}}{100}\right) \times \left(1 \pm \frac{\text{Change 2 %}}{100}\right) \times \left(1 \pm \frac{\text{Change 3 %}}{100}\right) \times \dots}$
... (iii)
(Use '+' for a percentage increase and '-' for a percentage decrease).
Method 3: Net Percentage Change Formula (for two successive changes)
If a quantity undergoes two successive percentage changes, say a change of $a\%$ followed by a change of $b\%$, the net percentage change can be calculated using a single formula:
$\boldsymbol{\text{Net Percentage Change } = \left(a + b + \frac{ab}{100}\right) \%}$
... (iv)
In this formula, 'a' and 'b' represent the percentage values of the individual changes. It is crucial to use the correct sign for 'a' and 'b': use a positive value for a percentage increase and a negative value for a percentage decrease.
The result of the formula (iv) gives the overall percentage change relative to the original value. If the result is positive, it's a net increase; if negative, it's a net decrease.
Example 1. The population of a town was 10,000. It increased by 10% in the first year and then increased by 20% in the second year. Find the population after two years and the net percentage increase.
Answer:
Original Population $= 10,000$.
Method 1: Step-by-step Calculation
First year increase: 10% increase on 10,000.
Increase in year 1 $= 10\%$ of $10,000 = \frac{10}{100} \times 10000 = 10 \times 100 = 1000$
Population after 1st year $= 10,000 + 1000 = 11,000$
Second year increase: 20% increase on the population after the first year (11,000).
Increase in year 2 $= 20\%$ of $11,000 = \frac{20}{100} \times 11000 = \frac{1}{5} \times 11000 = 2200$
Population after 2nd year $= 11,000 + 2200 = \boldsymbol{13,200}$
Population after two years is 13,200.
Now calculate the net percentage change relative to the original population.
Net Change $= \text{Final Population} - \text{Original Population} = 13,200 - 10,000 = 3200$
Net Percentage Change $= \frac{\text{Net Change}}{\text{Original Population}} \times 100 \%$
$= \frac{3200}{10000} \times 100 \%$
$= \frac{\cancel{3200}^{\normalsize 32}}{\cancel{10000}^{\normalsize 100}} \times 100 \% = \frac{32}{100} \times 100 \% = \boldsymbol{32 \%}$
The net percentage increase is $\boldsymbol{32\%}$.
Method 2: Using Multipliers
Original Population $= 10,000$.
First change: 10% increase. Multiplier $= \left(1 + \frac{10}{100}\right) = \left(1 + 0.1\right) = 1.10$ or $\frac{110}{100} = \frac{11}{10}$.
Second change: 20% increase. Multiplier $= \left(1 + \frac{20}{100}\right) = \left(1 + 0.2\right) = 1.20$ or $\frac{120}{100} = \frac{6}{5}$.
Final Population = Original Population $\times$ Multiplier 1 $\times$ Multiplier 2
Final Population $= 10000 \times \frac{11}{10} \times \frac{6}{5}$
Simplify the calculation:
Final Population $= \cancel{10000}^{\normalsize 2000} \times \frac{11}{\cancel{10}^{\normalsize 1}} \times \frac{6}{5}$
Final Population $= \cancel{2000}^{\normalsize 400} \times 11 \times \frac{6}{\cancel{5}^{\normalsize 1}}$
Final Population $= 400 \times 11 \times 6 = 400 \times 66 = 26400$
Final Population $= 10000 \times \frac{11}{10} \times \frac{6}{5} = \cancel{10000}^{1000} \times \frac{11}{\cancel{10}} \times \frac{6}{5} = 1000 \times 11 \times \frac{6}{5} = \cancel{1000}^{200} \times 11 \times \frac{6}{\cancel{5}} = 200 \times 11 \times 6 = 200 \times 66 = 13200$
Final Population $= 13,200$.
This matches the step-by-step result.
To find the net percentage change using multipliers, calculate the overall multiplier relative to the original value:
Overall Multiplier $= \left(1 + \frac{10}{100}\right) \times \left(1 + \frac{20}{100}\right) = 1.10 \times 1.20 = 1.32$
An overall multiplier of 1.32 means the final value is 1.32 times the original value.
To convert this multiplier back to a percentage change, subtract 1 and multiply by 100%:
Net Percentage Change $= (1.32 - 1) \times 100 \% = 0.32 \times 100 \% = \boldsymbol{32 \%}$
The net percentage increase is $\boldsymbol{32\%}$.
Method 3: Net Percentage Change Formula
This applies directly for two successive changes. Let $a = +10\%$ (increase) and $b = +20\%$ (increase).
Net Percentage Change $= \left(a + b + \frac{ab}{100}\right) \%$
$= \left(10 + 20 + \frac{10 \times 20}{100}\right) \%$
$= \left(30 + \frac{200}{100}\right) \%$
$= \left(30 + 2\right) \%$
$= \boldsymbol{32 \%}$
The net percentage change is a $\boldsymbol{32\%}$ increase. This formula is the most efficient for finding the net percentage change involving exactly two successive changes.
Example 2. The price of a television set is $\textsf{₹ } 25,000$. It is first decreased by 10% due to festive offer and then increased by 5% due to increase in material cost. Find the final price of the television set.
Answer:
Original Price $= \textsf{₹ } 25,000$.
Method 1: Step-by-step Calculation
First change: 10% decrease on $\textsf{₹ } 25,000$.
Decrease amount $= 10\%$ of $\textsf{₹ } 25,000 = \frac{10}{100} \times 25000 = \frac{1}{10} \times 25000 = \textsf{₹ } 2500$
Price after 1st decrease $= \textsf{₹ } 25,000 - \textsf{₹ } 2500 = \textsf{₹ } 22,500$
Second change: 5% increase on the price after the first change ($\textsf{₹ } 22,500$).
Increase amount $= 5\%$ of $\textsf{₹ } 22,500 = \frac{5}{100} \times 22500 = \frac{1}{20} \times 22500$
Calculate the increase:
$\frac{22500}{20} = \frac{2250}{2} = 1125$
Increase amount $= \textsf{₹ } 1125$
Final Price $= \text{Price after 1st change} + \text{Increase amount} = \textsf{₹ } 22,500 + \textsf{₹ } 1125 = \boldsymbol{\textsf{₹ } 23,625}$
The final price of the television set is $\textsf{₹ } 23,625$.
Method 2: Using Multipliers
Original Price $= \textsf{₹ } 25,000$.
First change: 10% decrease. Multiplier $= \left(1 - \frac{10}{100}\right) = \left(1 - 0.10\right) = 0.90$ or $\frac{90}{100} = \frac{9}{10}$.
Second change: 5% increase. Multiplier $= \left(1 + \frac{5}{100}\right) = \left(1 + 0.05\right) = 1.05$ or $\frac{105}{100} = \frac{21}{20}$.
Final Price = Original Price $\times$ Multiplier 1 $\times$ Multiplier 2
Final Price $= \textsf{₹ } 25000 \times \frac{9}{10} \times \frac{21}{20}$
Simplify the calculation:
Final Price $= \textsf{₹ } \cancel{25000}^{\normalsize 2500} \times \frac{9}{\cancel{10}^{\normalsize 1}} \times \frac{21}{20}$
Final Price $= \textsf{₹ } \cancel{2500}^{\normalsize 125} \times 9 \times \frac{21}{\cancel{20}^{\normalsize 1}}$
Final Price $= \textsf{₹ } 125 \times 9 \times 21$
$125 \times 9 = 1125$
$\begin{array}{cccc}& & 1 & 1 & 2 & 5 \\ \times & & & & 2 & 1 \\ \hline & & 1 & 1 & 2 & 5 \\ 2 & 2 & 5 & 0 & \times \\ \hline 2 & 3 & 6 & 2 & 5 \\ \hline \end{array}$
Final Price $= \boldsymbol{\textsf{₹ } 23,625}$
This matches the step-by-step result.
Net Percentage Change (Using Formula):
First change $a = -10\%$ (decrease).
Second change $b = +5\%$ (increase).
Net Percentage Change $= \left(a + b + \frac{ab}{100}\right) \%$
$= \left(-10 + 5 + \frac{(-10) \times 5}{100}\right) \%$
$= \left(-5 + \frac{-50}{100}\right) \%$
$= \left(-5 - 0.5\right) \%$
$= \boldsymbol{-5.5 \%}$
The net percentage change is a decrease of $\boldsymbol{5.5\%}$. (Note: The question only asked for the final price, but calculating the net percentage change is good practice).
Competitive Exam Notes:
Successive percentage changes are a frequent topic. Be comfortable with all three methods.
- Step-by-step: Reliable but can be time-consuming for multiple changes. Always calculate the next change on the *new* value.
- Multipliers: Most versatile method for multiple changes. For $n$ changes, multiply the original value by $n$ multipliers. Final Value = Original $\times M_1 \times M_2 \times \dots \times M_n$. $M = (1 + \frac{\text{Increase %}}{100})$ or $M = (1 - \frac{\text{Decrease %}}{100})$.
- Formula for Two Changes: Net change $= \left(a + b + \frac{ab}{100}\right) \%$. Fastest for exactly two changes. Remember signs: + for increase, - for decrease.
- Identifying the Base: The first change is on the original value. The second change is on the result of the first change, the third on the result of the second, and so on.
- Applications: Population growth, depreciation of assets, changes in prices after discounts and taxes, etc., are common applications.