NLNET Law-VerseThe Universe of Law
UGC NET PAPER 1 · UNIT 05 · MODULE 4: RATIO, PERCENTAGE AND MONEY · TOPIC 9

Percentage

The workhorse topic: percentage change, successive change, elections and the percent versus percentage-point trap.

  • 8 sub-topics
  • 4 min read
  • Topic 9 of 17

What this topic covers

8 sub-topics, in the order to study them. Tap one to jump to it.

  1. §OverviewPercent means "per hundred".
  2. 9.1Core formulasConcept Formula Percentage of a quantity (percent/100) x quantity What percent is A of B (A/B) x 100 Percentage change (new - old)/old x 100 Multiplier for an increase of r%…
  3. 9.2Geometry-type percentage changesIf the length rises a% and the breadth rises b%, the area changes by a + b + ab/100 percent. If each side of a square rises by x%, the area rises by 2x + x squared/100…
  4. 9.3Elections and pass marksCovers Votes, Pass marks.
    • Votes
    • Pass marks
  5. 9.4Percent versus percentage pointIf an interest rate rises from 5% to 6%, it has risen by 1 percentage point but by 20 percent (1/5).
    • 1 percentage point
    • 20 percent
  6. 9.5Two-set percentagesFor two groups: percentage in at least one = percentage in A + percentage in B - percentage in both.
    • 60%
  7. 9.6Worked examplesExample 1.
    • Example 1
    • 15%
    • Example 2
    • Example 3
    • 16.67%
  8. 9.7TricksCovers Use fractions, Flip when it helps, Multiplier method, Find 10% first, The equal-rise-and-fall result.
    • Use fractions
    • Flip when it helps
    • Multiplier method
    • Find 10% first
    • The equal-rise-and-fall result
  9. 9.8TrapsUsing the new value instead of the old value as the base for percentage change. Adding successive percentages without the cross term ab/100. Forgetting that a rise of 50%…

Revision mode hides the full explanations and keeps each section's key terms, lists and tables. Use it from your second read onward.

§Overview#

Percent means "per hundred". Percentage is the base of profit and loss, interest, data interpretation and population problems, so it repays careful study.

Back to topic map ↑

9.1Core formulas#

In brief

Concept Formula Percentage of a quantity (percent/100) x quantity What percent is A of B (A/B) x 100 Percentage change (new - old)/old x 100 Multiplier for an increase of r%…

Concept Formula
Percentage of a quantity (percent/100) x quantity
What percent is A of B (A/B) x 100
Percentage change (new - old)/old x 100
Multiplier for an increase of r% 1 + r/100 (a 20% rise means multiply by 1.2)
Multiplier for a decrease of r% 1 - r/100 (a 20% fall means multiply by 0.8)
x% of y y% of x (8% of 50 = 50% of 8 = 4)
Successive changes of a% and b% Net = a + b + ab/100 (use signs: a rise is positive, a fall is negative)
A is x% more than B, so B is less than A by x/(100 + x) x 100 %
A is x% less than B, so B is more than A by x/(100 - x) x 100 %
Price rises x%; consumption must fall by (to keep spending the same) x/(100 + x) x 100 %
Price falls x%; consumption can rise by x/(100 - x) x 100 %
Population growth for n years at r% P(1 + r/100)^n
Depreciation (value falling) for n years at r% P(1 - r/100)^n

Back to topic map ↑

9.2Geometry-type percentage changes#

In brief

If the length rises a% and the breadth rises b%, the area changes by a + b + ab/100 percent. If each side of a square rises by x%, the area rises by 2x + x squared/100…

  • If the length rises a% and the breadth rises b%, the area changes by a + b + ab/100 percent.

  • If each side of a square rises by x%, the area rises by 2x + x squared/100 percent. A 10% rise in the side gives a 21% rise in the area.

  • If the radius of a circle rises by x%, the area changes by the same formula (2x + x squared/100)%.

Back to topic map ↑

9.3Elections and pass marks#

In brief

Covers Votes, Pass marks.

  • Votes
  • Pass marks

  • Votes: the winner with x% of the valid votes (x above 50) wins by (2x - 100)% of the valid votes. Invalid votes are not counted.

  • Pass marks: equate the two descriptions. If a student scoring 40% fails by 20 marks and one scoring 50% passes by 40 marks, then 10% of the total = 60, so the total is 600 and the pass mark is 260.

Back to topic map ↑

9.4Percent versus percentage point#

In brief

If an interest rate rises from 5% to 6%, it has risen by 1 percentage point but by 20 percent (1/5).

  • 1 percentage point
  • 20 percent

If an interest rate rises from 5% to 6%, it has risen by 1 percentage point but by 20 percent (1/5). Exam writers use this difference as a trap.

Back to topic map ↑

9.5Two-set percentages#

In brief

For two groups: percentage in at least one = percentage in A + percentage in B - percentage in both.

  • 60%

For two groups: percentage in at least one = percentage in A + percentage in B - percentage in both. So if 80% pass Hindi, 70% pass English and 10% fail both, then 90% pass at least one subject and 80 + 70 - 90 = 60% pass both.

Back to topic map ↑

9.6Worked examples#

In brief

Example 1.

  • Example 1
  • 15%
  • Example 2
  • Example 3
  • 16.67%
  • Example 4
  • 19,800
  • Example 5

Example 1. The price of a shirt rises from 800 to 920. Find the percentage increase.

  • Change = 120. Percentage = 120/800 x 100 = 15%.

Example 2. A number is increased by 20% and then decreased by 20%. What is the net change?

  • Net = 20 - 20 + (20 x -20)/100 = -4. The number falls by 4%.

Example 3. A's salary is 20% more than B's. By what percent is B's salary less than A's?

  • 20/(100 + 20) x 100 = 16.67% (that is, 1/6).

Example 4. A town has 20,000 people. The population rises 10% in the first year and falls 10% in the second. Find the population after two years.

  • 20,000 x 1.1 x 0.9 = 19,800.

Example 5. The side of a square increases by 20%. How much does the area increase?

  • 20 + 20 + (20 x 20)/100 = 44. The area rises by 44%.

Example 6. A candidate gets 45% of the votes and loses by 1500 votes. Find the total number of votes.

  • The winner has 55%, so the margin is 10% of the total. 10% of total = 1500, so the total = 15,000.

Example 7. A population of 10,000 grows 10% a year for two years.

  • 10,000 x 1.1 x 1.1 = 12,100.

Back to topic map ↑

9.7Tricks#

In brief

Covers Use fractions, Flip when it helps, Multiplier method, Find 10% first, The equal-rise-and-fall result.

  • Use fractions
  • Flip when it helps
  • Multiplier method
  • Find 10% first
  • The equal-rise-and-fall result
  • loss of x squared/100 percent

  • Use fractions: 12.5% = 1/8, 16.67% = 1/6, 33.33% = 1/3, 37.5% = 3/8, 62.5% = 5/8, 87.5% = 7/8.

  • Flip when it helps: to find 4% of 75, find 75% of 4.

  • Multiplier method: for chained increases or decreases, multiply the multipliers (1.2 x 0.8 = 0.96 means a 4% fall).

  • Find 10% first: move the decimal one place left, then build 5% (half of 10%), 15% (10% + 5%) and so on.

  • The equal-rise-and-fall result: a rise of x% followed by a fall of x% always leaves a net loss of x squared/100 percent.

Back to topic map ↑

9.8Traps#

In brief

Using the new value instead of the old value as the base for percentage change. Adding successive percentages without the cross term ab/100. Forgetting that a rise of 50%…

  • Using the new value instead of the old value as the base for percentage change.

  • Adding successive percentages without the cross term ab/100.

  • Forgetting that a rise of 50% needs a fall of only 33.33% to return to the start (and a fall of 50% needs a rise of 100%).

  • Confusing percent with percentage points.

Back to topic map ↑

Test yourself on Percentage

Practise with free MCQs and answer keys, then return here to fix what you got wrong.

↑ Module 4: Ratio, Percentage and Money · All 18 topics in Mathematical Reasoning · Unit overview and sample MCQs

Last updated: . Notes are a revision aid; check official UGC sources for any rule or scheme that changes over time. Spotted a mistake? Write to netlawverse@gmail.com.