Percentage increase and decrease, and choosing the right base

What to assume they know

• Why forcing every fraction onto a scale of 100 makes them comparable — a percentage is a fraction over 100
• Computing a percentage in your head by splitting it — a percentage of a quantity, and recovering the quantity from a part
• What a percentage greater than 100 does and does not mean — that a percentage may exceed 100, and that 1.65 and 165% are the same multiplier
• Subtracting to find a difference, and knowing which of two values is the earlier
• Solving a one-step equation of the form 1.2x = 90

What they should be able to do

• Compute a percentage increase and a percentage decrease from two values, putting the earlier value in the denominator
• Explain why the base is the original amount, and what goes wrong if the final amount is used instead
• Show by example that a rise and the fall that undoes it are different percentages
• Prove that a value being 165% of another is the same statement as its having increased by 65%, using the multiplier form
• Distinguish "is k% of", "has increased by k%", and "is k% more than"
• Recover the original value from the changed value and the percentage change
• Compute a percentage change exactly, as a fraction, when it does not come out to a terminating decimal
• Read the word inflation as a percentage increase in prices, and compute one

Where it usually goes wrong

• "Divide the change by the bigger number." No — by the earlier one. Sometimes that is the bigger number and sometimes it is not, which is why "bigger" is a rule that fails half the time.
• "A 37.5% fall is undone by a 37.5% rise." It takes 60%. This is the single most valuable thing in the topic and it should get a whole section, not a remark.
• "Increased by 65% and is 165% of are different." They are the same, and the chapter proves it. Students who cannot move between the additive and multiplier forms lose the compounding topic entirely.
• "Increased by 165% means it is now 165% of what it was." It means 265%. The two readings differ by the whole original amount.
• "The price he paid is the base for the discount." The discount is a percentage of the marked price, which is what he did not pay. The Samson item is designed to catch exactly this, and dividing by 0.85 rather than multiplying by 1.15 is the fix.
• "An awkward answer means I made a mistake." ₹5,17,647 and 9.0909…% are the correct answers to two of these items. An explanation that only ever lands on round numbers trains students to distrust correct work.
• "Inflation is a different calculation." It is a percentage increase with an economic name. The chapter says so in one parenthesis and moves on.
• "Percentage change works out the same whichever end you measure from, roughly." For small changes it nearly does, which is why the belief survives. For the chapter's own numbers — 37.5 against 60 — it plainly does not.

Questions to check understanding

• Compute a percentage increase or decrease from two dated values
• Multiple choice on percentage increase where the distractors include base errors — the board's standard construction; the petrol item has one, 40% measured against the final price
• Recover the original value from the changed value and the percentage change, including the case where the change was a discount
• Choose which of several statements are equivalent to a stated percentage rise or fall
• Express a change exactly as a fraction where the decimal does not terminate
• Given a fixed area or a fixed product, find the exact percentage change in one factor needed to offset a stated change in the other
• Identify inflation in a worded item as a percentage increase and compute it
• Reasoning: explain why a k% rise is not undone by a k% fall

Examples worth working on the board

Values marked arithmetic added here are added here; the chapter prints no answer key for Part II.

• The section's opening claim (Part II, §1.3, p.15): a percentage is a natural way of reporting how fast something is changing. Both worked instances follow immediately, and they are printed as a numbered pair — one rise and one fall — so that the shared shape of the fraction is visible before any formula is used.
• Tomatoes (Part II, §1.3, p.15, item 1). One kilogram cost ₹30 three years ago and costs ₹42 now, so the increase is ₹12. The formula is displayed as the amount of increase over the original amount or base, times 100, and it gives (12/30) × 100 = 40%. The reading printed underneath: the price rose by 40% over the three years.
• The theatre (Part II, §1.3, p.16, item 2). Average footfall was 160 before COVID and is 100 now, so the decrease is 60. Same formula shape: (60/160) × 100 = 37.5%.
• The asymmetry, which the chapter does not spell out. (sections 4 and 5). Arithmetic added here: the theatre's gap of 60 read the other way — 100 climbing back to 160 — is (60/100) × 100 = 60%. One gap, two percentages, because 160 and 100 are different denominators. Generalise it: undoing a rise of k% takes a fall of k/(100 + k) as a percentage, so a 25% rise needs a 20% fall, a 50% rise needs a 33.33% fall, and a 100% rise needs a 50% fall. Do the numbers at least once; the general form on its own will not land.
• Example 3, the two statements (Part II, §1.3, p.16). Do these mean the same thing: (i) a state's 1991 population is 165% of its 1961 population; (ii) the population increased by 65% from 1961 to 1991. The page answers yes and proves it with p for 1961 and q for 1991, in two parallel columns:
  • from (i): q = 165% of p = (165/100) × p = **1.65 *p***;
  • from (ii): q = p + 65% of p = p + 0.65 × p = **1.65 *p***. Then the reading: the 1991 population is 1.65 times the 1961 population. This is the topic's central identity — a change of k% is a multiplication by (1 + k/100) — and it is what makes Compounding: why repeated growth multiplies instead of adding possible at all.
• The third phrasing (section 7, not in the book). "Increased by 165%" would mean q = p + 1.65p = 2.65p, which is not the same state and not the same population. Put the three bars side by side: 1.65p, 1.65p, 2.65p. The chapter proves two of them equal.
• Exercise items on the increase and decrease. All printed in §1.3's sets; hand the data over intact.
  • Part II p.19, Q5, multiple choice: petrol was ₹60 in 2015 and is ₹100 in 2025; the options offered are 50%, 40%, 60%, 66.66%, 140%, 160.66%. Arithmetic added here: 40/60 = 66.66%. Two of the distractors have a nameable origin — 40% is the change measured against the final price (40/100), which is the one genuine base error in the set, and 60% is the 2015 price as a percentage of the 2025 price (60/100). 50%, 140% and 160.66% have no stated derivation; say so rather than presenting the whole set as a pattern of base errors.
  • Part II p.20, Q5: rice was ₹38 per kg in 2024 and is ₹42 in 2025; find the rate of inflation, with inflation defined on the page as the percentage increase in prices. Arithmetic added here: 4/38 = 10.53%.
  • Part II p.20, Q6: a number increased by 20% becomes 90; find the number. Arithmetic added here: 90/1.2 = 75. The multiplier form makes this a one-step division; the additive form makes it an equation.
  • Part II p.20, Q3: Samson paid ₹4,40,000 for a car once the dealer had taken 15% off; what was the price before that. The base is the figure he did not pay. Arithmetic added here: 4,40,000 ÷ 0.85 ≈ ₹5,17,647 — not a round figure, so a student who expects a tidy answer will assume an error. Say so.
  • Part II p.20, Q8, multiple choice: an elephant population rose 5% over a decade; if it was p, it is now — with the options p × 0.5, p × 0.05, p × 1.5, p × 1.05, p + 1.50. Example 3's identity, examined directly.
  • Part II p.20, Q9: which statements mean the same as camera demand having fallen by 85% over a decade. Six options; the one that holds is that demand now is 15% of demand a decade ago. Two options invert which decade is the base and two use 185%.
  • Part II p.29, Q11: a rectangle's length grows by 10% while its area stays as it was; find exactly the percentage fall in its breadth. Arithmetic added here: the breadth must be multiplied by 1/1.1 = 10/11, a decrease of 1/11, which is 9.0909…% — so the printed word exactly is doing real work, and the answer is a fraction rather than a decimal. This is the best item in the chapter for section 11, and it is the asymmetry of section 4 in geometric form: a 10% rise is undone by a fall of less than 10%.
• Where the same idea reappears in the chapter.: percentage profit and percentage loss (Part II pp.16–19) are percentage changes whose base is the cost price, handled in Profit, loss and taxes as percentages of a stated amount; growth over several periods (Part II pp.20–25) is this identity applied repeatedly, in Compounding: why repeated growth multiplies instead of adding; and the marbles item at Part II p.27, where one child's count is 120% of the other's and the other must find the matching statement, is section 4's asymmetry as a puzzle — it is printed under "Tricky Percentages" and is handled in Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss.

Figures to have open

• Two bars of a shared 60-unit gap, one 100 long and one 160 long, for section 4. An added figure and the most important one here.
• A double-column proof panel for Example 3, reproducing the chapter's parallel layout as a schematic (not its typography).
• Three bars for the three phrasings in section 7. Not in the book.
• A rectangle pair of equal area, one stretched 10% in length and narrowed to compensate. Standard schematic.
• A discount bar for the Samson item: the unknown marked price at the top, a 15% strip removed, ₹4,40,000 labelled on what is left.
• No photograph is needed. The chapter's own tomato and theatre illustrations are decorative and carry no data.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 1, "Fractions in Disguise", §1.3 "Using Percentages", the unnumbered bold subheading "Percentage Increase or Decrease", Part II pp.15–16, through Example 3 on Part II p.16.
• Exercise items: Part II p.19 no. 5; Part II p.20 nos. 3, 5, 6, 8, 9; Part II p.29 no. 11.
• The chapter's SUMMARY, Part II p.31, states that an increase or decrease in a quantity can itself be expressed as a percentage.
• Forward pointers inside the same section: profit and loss, Part II pp.16–19 (Profit, loss and taxes as percentages of a stated amount); growth and compounding, Part II pp.20–25 (Compounding: why repeated growth multiplies instead of adding); the marbles item, Part II p.27 (Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss).