Percentage

Percent change, successive percentages

Percentage is one of the most important topics in quantitative aptitude, forming the foundation for many other concepts like profit and loss, simple and compound interest, and data interpretation. A percentage represents a fraction with 100 as its denominator, making it a standardised way to compare quantities and express proportions.

Key Formulas

\text{Percentage = (Part/Whole) x 100}

\text{Part = (Percentage/100) x Whole}

\text{Percentage Increase = [(New Value - Original Value)/Original Value] x 100}

\text{Percentage Decrease = [(Original Value - New Value)/Original Value] x 100}

\text{If a value increases by x\% and then decreases by x\%, net effect = -(x\textasciicircum{}2/100)\% (loss)}

\text{If a value increases by x\% and then by y\%, total increase = x + y + (xy/100)\%}

\text{If A is x\% more than B, then B is [x/(100+x)] x 100\% less than A}

\text{If A is x\% less than B, then B is [x/(100-x)] x 100\% more than A}

\text{Successive percentage change: a\% followed by b\% = a + b + (ab/100)\%}

\text{To reverse a percentage increase of x\%: multiply by 100/(100+x)}

\text{To reverse a percentage decrease of x\%: multiply by 100/(100-x)}

Key Concepts

Basic Percentage Calculations

Converting fractions to percentages: multiply by 100. Converting percentages to fractions: divide by 100. Common conversions to remember: 1/2 = 50%, 1/3 ~= 33.33%, 1/4 = 25%, 1/5 = 20%, 1/6 ~= 16.67%, 1/8 = 12.5%, 1/10 = 10%, 1/12 ~= 8.33%, 1/16 = 6.25%, 1/20 = 5%. These quick conversions save time during calculations.

Percentage Change

Percentage increase or decrease is always calculated with respect to the original value. The formula is: (Change/Original) x 100. A common mistake is using the new value as the base. Remember: increase of 50% followed by decrease of 50% does NOT bring you back to the original value - it results in a net loss of 25%.

Successive Percentages

When percentages are applied one after another (successive), they compound rather than simply add. For example, a 20% increase followed by a 30% increase is NOT 50%, but rather: 20 + 30 + (20x30)/100 = 56%. The formula for successive changes of a% and b% is: a + b + (ab/100). This applies to both increases and decreases (use negative for decreases).

Reverse Percentage (Finding Original Value)

When given a value after percentage change and asked to find the original: If final value is after x% increase, original = Final x 100/(100+x). If final value is after x% decrease, original = Final x 100/(100-x). This is crucial for problems stating 'after a 20% discount, the price is 400, find original price'.

Comparing Percentages

When A is x% more than B, B is NOT x% less than A. Use the formula: If A is x% more than B, then B is [x/(100+x)] x 100% less than A. Example: If A is 25% more than B, B is (25/125) x 100 = 20% less than A. This asymmetry is commonly tested in exams.

Percentage to Ratio and Vice Versa

Percentage can be easily converted to ratio and vice versa. 25% = 25/100 = 1/4 = 1:4. Similarly, ratio 3:5 can be written as (3/5) x 100 = 60%. This conversion is useful when comparing multiple quantities expressed in different formats.

Solved Examples

Problem 1:

A number is increased by 20% and then decreased by 20%. What is the net percentage change?

Solution:

Let the original number be 100.
After 20% increase: 100 + (20/100)x100 = 120
After 20% decrease on 120: 120 - (20/100)x120 = 120 - 24 = 96
Net change = 100 - 96 = 4 (decrease)
Using formula: Net change = -(x^2/100)% = -(20^2/100)% = -400/100% = -4%
Answer: 4% decrease

Problem 2:

If the price of sugar increases by 25%, by what percentage should a family reduce consumption to keep expenditure the same?

Solution:

Let original price = 100 per kg, original consumption = 1 kg
Original expenditure = 100
New price = 125 per kg (25% increase)
For same expenditure of 100:
New consumption = 100/125 = 0.8 kg = 4/5 kg
Reduction = 1 - 4/5 = 1/5 = 0.2 kg
Percentage reduction = (0.2/1) x 100 = 20%
Using shortcut formula: [x/(100+x)] x 100 = [25/(100+25)] x 100 = 25/125 x 100 = 20%
Answer: 20%

Problem 3:

A student's marks in Mathematics increased from 60 to 75. What is the percentage increase?

Solution:

Original marks = 60
New marks = 75
Increase = 75 - 60 = 15
Percentage increase = (Increase/Original) x 100
= (15/60) x 100
= (1/4) x 100
= 25%
Answer: 25% increase

Problem 4:

After a 15% discount, an article costs 340. What was its original price?

Solution:

Let original price = x
After 15% discount, price = 85% of x = 0.85x
Given: 0.85x = 340
x = 340/0.85
x = 340 x (100/85)
x = 340 x (20/17)
x = 20 x 20 = 400
Answer: 400

Tips & Tricks

Always identify the base (original) value before calculating percentage change - this is the most common source of errors
Memorise common fraction-percentage conversions (1/2 to 1/20) to speed up calculations significantly
For successive percentage changes, remember they compound - never simply add or subtract percentages directly
When a value changes by x% and then by y%, the net effect is always x + y + (xy/100) - use this to verify your answers
For 'maintaining expenditure' problems, use the formula: percentage decrease in consumption = [100x/(100+x)]% where x is the price increase percentage
Always check if your answer makes logical sense - a number cannot decrease by more than 100% (it would become negative)
When comparing two values where one is x% more than the other, the reverse relationship is never the same percentage - use x/(100+/-x) formula

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