Compounding: why repeated growth multiplies instead of adding

What to assume they know

• Percentage increase and decrease, and choosing the right base — that a change of k% is a multiplication by (1 + k/100), and that the base is the value you started from
• What a percentage greater than 100 does and does not mean — percentages above 100, and reading 133.1% as 1.331
• The FDP trio: fraction, decimal and percentage as one object — moving between a percentage and its decimal multiplier
• Multiplying several decimals in sequence, and repeated multiplication written as a power
• Substituting into an algebraic expression with three letters

What they should be able to do

• Read a rate given as "k% p.a." and compute one period's interest on a stated principal
• Trace both kinds of fixed deposit year by year, stating the principal at the start of each year
• Explain why the interest is equal every year in one case and grows in the other, by naming the base each time
• Express the total received as a percentage of the amount deposited, and as a multiplier
• Account for the difference between the two totals as interest earned on interest, and show it summing to the printed gap
• Write and use p(1 + rt) and p(1 + r)^t, and say which corresponds to which option
• Apply the same two forms to a quantity that shrinks — depreciation and a declining population
• Combine several successive percentage changes by multiplying their factors, and explain why adding the percentages is wrong
• Recognise compounding as growth that curves and non-compounding as growth that runs straight, and compare doubling times

Where it usually goes wrong

• "10% a year for 3 years is 30%, so the two options must agree." They agree only in Option 1. Adding the rate three times and multiplying by the factor three times are different operations, and the chapter prints both expressions on Part II p.22 precisely so they can be compared.
• "The interest rate goes up when you compound." It does not. The rate is 10% every single year in both options. What changes is the amount it is taken of.
• "Compounding always wins." Over three years, 12% without compounding beats 10% with — that is Q6 and it is set to break exactly this belief. Compounding is a mechanism, not a magic advantage; whether it wins depends on the rate and the number of periods.
• "A rise of 5% then falls of 2% and 3% cancel out." They very nearly do, and that is the trap. The product 0.99813 is under 1, so the population is smaller than it started. The percentages sum to zero and the factors do not multiply to one.
• "3% then 4% is 7%." It is 7.12%. The 0.12 is the 4% acting on the 3% already added — the same interest-on-interest term as in section 7.
• "60% of the Class 8 students is 60% of the excursion." It is 24% of the excursion. Percentages of a subgroup nest, and the rough-diagram hint the item prints is the corrective.
• "911.25 and 912 mean somebody made a mistake." Both are right; one rounds at the end and one rounds three times. Say which and why, or students will distrust the method that gives fractional people.
• "Depreciation is a different formula." It is p(1 − r)^t, the same machine with the sign of the change turned round. The TV and the village are the same problem as the fixed deposit.
• "Interest is only about money." The chapter's own exercises apply the identical form to a city's population, a bacterial culture and a lab's mice.

Questions to check understanding

• Compute the amount at maturity with and without compounding for a stated principal, rate and number of years, and state the difference
• Identify which of several algebraic expressions gives the total under one of the two options — the board's standard multiple-choice construction here
• Compare a higher rate without compounding against a lower rate with it, over a stated period
• Find the total interest at maturity, as distinct from the total amount
• Apply p(1 − r)^t: depreciation of an asset, decline of a population
• Combine two or three successive percentage changes and give the overall percentage change
• Nested percentages: a percentage of a subgroup as a percentage of the whole
• Estimate a doubling time under each kind of growth, and name which is linear and which exponential
• Reasoning: explain why equal rt gives equal totals without compounding but unequal totals with it

Examples worth working on the board

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

• What interest is (Part II, §1.3, p.20). The section opens on two ordinary advertisements — a fixed deposit paying 6% per annum, a savings account paying 2.5% per annum — and defines interest as the extra money institutions such as banks and post offices pay on money kept with them, and that people and institutions pay when they borrow.
• The fixed-deposit box (Part II, §1.3, p.20, tinted). A specific amount is deposited for a set length of time at a rate agreed in advance; the money stays locked for the chosen duration; the bank pays interest on it; withdrawing before the maturity date incurs a penalty; and at the end you get your deposit back together with the interest it has earned. That last clause is the one the whole topic turns on — both.
• One year on ₹6000 at 10% p.a. (Part II, §1.3, p.21). The page explains p.a. as per annum, meaning for every year, then works the single year: a ₹6000 deposit earns ₹600. It names the parts — 10% is the rate, and ₹6000 is the principal — and then computes the amount two ways: 6000 + (0.10 × 6000) = 6000 + 600 = ₹6600; and, equivalently, the deposit becomes 110% of itself, 6000 × (110/100) = 6000 × 1.1 = 6600. One rough bar sits beside it: a blue segment labelled as the principal, an amount scale above it, a percentage scale below it running to 100% with a short red segment past that mark carrying a 10% arrow, and an "amount after 1 year" span arrow beneath the whole. There is no second bar — what looks like one is the lower scale and that span arrow. A printed error sits on this figure: the amount-scale tick above the end of the principal reads 1000, where the example's principal is ₹6000. Redraw it with 6000 at that tick and the unknown at 6600; anyone copying the page as printed will contradict the text three lines above it. A displayed line states it in words: the amount after one year is the principal plus the rate of interest of the principal.
• Example 7, the same ₹6000 for three years, two ways (Part II, §1.3, pp.21–22). The page says outright that the answer depends on which fixed deposit was chosen.
  • Option 1 — each year's interest is handed straight back to the depositor, and the principal returns at maturity. The flow table on Part II p.21 has two columns, Interest returned (10% p.a.) and Amount in the FD, and three year-blocks each split into Beginning and Ending. Every year begins at ₹6000, a ×0.1 arrow gives ₹600 returned, and the FD ends the year still at ₹6000. Total received: ₹1800 + ₹6000 = ₹7800.
  • Option 2 — the interest is added back, raising the principal for the next period; the page names this compounding. The flow table on Part II p.22 has the columns Interest added back (10% p.a.) and Amount in the FD, with a ×1.10 arrow running down the right-hand side. Year 1 begins at ₹6000, adds ₹600, ends at ₹6600. Year 2 begins at ₹6600, adds ₹660, ends at ₹7260. Year 3 begins at ₹7260, adds ₹726, ends at ₹7986, and ₹7986 is what is returned. Total received: ₹7986. The page's own conclusion: the closing figure is larger once the interest has been compounded. ₹600, ₹660, ₹726 are all 10% — of ₹6000, ₹6600, ₹7260. Nothing about the rate changed. Only the base moved.
• Example 8, the gap as a percentage (Part II, §1.3, p.22). Total received as a percentage of the amount deposited, computed for both options: 7800/6000 × 100 = 130% = 1.3 and 7986/6000 × 100 = 133.1% = 1.331. The page then rewrites each as a product: 6000 × (1 + 0.1 + 0.1 + 0.1) = 6000 × 1.3, against 6000 × 1.1 × 1.1 × 1.1 = 6000 × 1.331. And it states the gains: 30% over three years without compounding, 33.1% with. Show the two rewritings side by side — one adds the rate three times, the other multiplies by the factor three times, and that is the entire difference between the two options.
• Where the 3.1% comes from (section 7, an added accounting). Arithmetic added here: the gap is ₹7986 − ₹7800 = ₹186, and it can be found in the flow table itself. Year 1 pays ₹600 either way. Year 2 pays ₹660 instead of ₹600 — ₹60 more, which is 10% of the first year's ₹600. Year 3 pays ₹726 instead of ₹600 — ₹126 more. Together, ₹60 + ₹126 = ₹186. Expanding the factor gives the same breakdown symbolically: (1.1)³ = 1 + 3(0.1) + 3(0.1)² + (0.1)³, so on ₹6000 the three parts past the principal are ₹1800, ₹180 and ₹6 — and ₹180 + ₹6 = ₹186. This is the section that turns compounding from a formula into an explanation, and the chapter does not print it.
• Example 9, the generalisation (Part II, §1.3, p.23). Printed as two panels, each with the ₹6000 instance on the left and the general form on the right.
  • No compounding: the interest gained in one term is 6000 × 0.1, or p × r; in three terms 6000 × 0.1 × 3, or in t terms p × r × t; and the total at the end is 6000 + (6000 × 0.1 × 3), generally p + prt = p(1 + rt). The stated reason: the interest is paid back each term, so the principal — and hence each term's interest — never changes.
  • With compounding: after Year 1, 6000 × 1.1, generally p(1 + r); after Year 2, (6000 × 1.1) × 1.1, generally p(1 + r)(1 + r); after Year 3, (6000 × 1.1 × 1.1) × 1.1; and after t years, 6000 × (1.1)^t, generally p(1 + r)^t. Each line labels the bracketed part as the principal for that year, which is the figure's whole pedagogic point. The stated reason: the interest is added back, so the principal rises every term and the interest rises proportionately. A Math Talk on Part II p.24 then asks for the formula for the total interest at maturity under each option, which is the total minus p in both cases.
• Decline (Part II, §1.3, p.24, subheading). Things lose financial value with time. The example given is a bike bought at ₹1,00,000 and worth less when it is resold, by an amount depending on how many years have passed, how many kilometres it has run, whether it was damaged, and whether parts were replaced. The page names this depreciation — a reduction in value from use and age.
• Example 10, the TV (Part II, §1.3, p.25). Bought at ₹21,000; after one year it depreciates by 5%. Two printed methods, exactly the pair from Profit, loss and taxes as percentages of a stated amount's Example 6: the reduction is 5% of 21,000 = 0.05 × 21,000 = 1050, so the value is 21,000 − 1050 = ₹19,950; or the value is 95% of the current value, 0.95 × 21,000 = 19,950.
• Example 11, the village (Part II, §1.3, p.25). Population falling by about 10% every decade, currently 1250; find it after 3 decades. Two columns are printed, and they do not agree, which is the interesting part.
  • Left column, factors: each decade multiplies by 0.9, so after one decade 1250 × 0.9, after two 1250 × 0.9 × 0.9, and after three 1250 × 0.9 × 0.9 × 0.9 = 911.25.
  • Right column, subtract and round: first decade's decrease 0.1 × 1250 = 125, leaving 1125; second decade 0.1 × 1125 = 112.5, rounded to 112, leaving 1013; third decade 0.1 × 1013 = 101.3, rounded to 101, leaving 912. The page reconciles them by rounding off to around 910. Section 10 must say why they differ: 912 is not 911.25 rounded — it is the output of a different procedure, because the right-hand column rounds mid-way, at the end of each decade, so its trajectory diverges from the exact one. Whole people cannot be halved, so the rounding is honest — but it must be done once, at the end, or acknowledged each time.
• Figure it Out, the compounding set (Part II, §1.3, pp.22–24).
  • Q1: Bank of Yahapur pays 10% p.a.; compare ₹20,000 for 2 years with and without annual compounding. Arithmetic added here: ₹24,000 against ₹24,200.
  • Q2: Bank of Wahapur pays 5% p.a.; compare ₹20,000 for 4 years the same way. Arithmetic added here: ₹24,000 against ₹24,310.13.
  • Q3: is there anything interesting in the two solutions above — share and discuss. Arithmetic added here, and it is a genuinely good observation: both non-compounding totals are the same ₹24,000, because 10% twice and 5% four times both come to rt = 0.2. The compounded totals are not the same, and the smaller rate over more periods compounds further ahead. That is the exercise's payload.
  • Q4: Jasmine invests p for 4 years at 6% p.a.; which expressions give the total without compounding. Seven options are printed: p × 6 × 4; p × 0.6 × 4; p × (0.6/100) × 4; p × (0.06/100) × 4; p × 1.6 × 4; p × 1.06 × 4; p + (p × 0.06 × 4). Arithmetic added here: only the last is right, and every other option is a specific slip — rate not converted, converted twice, or the principal left out of the sum.
  • Q5: a post office pays 7% p.a.; the interest on ₹50,000 for 3 years without compounding, and how much more with. Arithmetic added here: ₹10,500 against about ₹11,252, so about ₹752 more.
  • Q6: Giridhar takes a loan of ₹12,500 at 12% p.a. over 3 years, with no compounding; Raghava takes an identical sum over an identical period at 10%, compounded annually. Which of the two ends up paying more interest, and what is the gap. Arithmetic added here: Giridhar ₹4,500, Raghava ₹4,137.50 — Giridhar pays ₹362.50 more, so the higher rate still beats compounding over three years. A properly instructive item: compounding is not automatically the bigger number.
  • Q7, flagged Math Talk: ₹1000 growing at 10% p.a. — how long to double with and without compounding; and is compounding an instance of exponential growth while non-compounding is linear. Arithmetic added here: without compounding, ten years exactly; with compounding, the eighth year is the first to clear ₹2000.
  • Q8: a city's population rising about 3% every year, currently 1.5 crore — the expected population after 3 years. Arithmetic added here: about 1.639 crore.
  • Q9: bacteria increasing at 2.5% per hour from an initial count of 5,06,000 — the count after 2 hours. Arithmetic added here: about 5,31,616.
• Successive percentage changes, from the chapter-end set. These are the same mathematics with the periods relabelled, and section 11 should treat them together.
  • Part II p.28, Q4: monthly percentage change in a lab's mice population against the previous month — +5%, then −2%, then −3%, from an initial population p. Six statements are offered, including that the result is p × 0.05 × 0.02 × 0.03, that it is p × 1.05 × 0.98 × 0.97, that it is p + 0.05 − 0.02 − 0.03, that it is p, that it is more than p, and that it is less than p. Arithmetic added here: 1.05 × 0.98 × 0.97 = 0.99813, so the population ends slightly below p — even though the percentages sum to zero. The best single item in the chapter for this topic.
  • Part II p.29, Q10: bus fares raised 3% last year and 4% this year — the overall increase across the two years. Arithmetic added here: 1.03 × 1.04 = 1.0712, so 7.12%, not 7%.
  • Part II p.29, Q8: an excursion in which 40% of the students are from Class 8 and the rest from Class 9, and 60% of those Class 8 students are girls; (i) what percentage of all the students are Class 8 girls; (ii) if 160 students are going, how many Class 8 girls. A printed hint suggests drawing a rough diagram. Arithmetic added here: 60% of 40% is 24%, because the second percentage is taken of the first group and not of the whole — the same nesting as compounding. Part (ii) does not come out whole: see Notes.

Figures to have open

• The two flow tables of Part II pp.21–22, redrawn as schematics. These are the chapter's own figures and the best pair of diagrams in the chapter: the columns and the Beginning/Ending split are what make the moving base visible. Keep the ×0.1 and ×1.10 arrows.
• The two-panel generalisation of Part II p.23, redrawn with the instance on the left and the general form on the right, and the "principal for this term" labels kept.
• A stacked accounting of the ₹186, for section 7. An added figure and the one that carries the argument.
• Two curves and a doubling line, for section 12. Not in the book; the chapter asks the question in Q7 and draws nothing.
• A nested rough diagram for the excursion item — the whole group, the 40% block, and the 60% of that block. Standard schematic, and the item's own hint asks for it.
• No photograph is needed.

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 subheadings "Growth and Compounding" (Part II p.20, running through Example 9 on Part II p.23) and "Decline" (Part II p.24, through Example 11 on Part II p.25). The fixed-deposit tinted box is on Part II p.20 and the Math Talk on the interest formulas is on Part II p.24.
• Exercise items: Part II pp.22–24 nos. 1–9; Part II p.28 no. 4; Part II p.29 nos. 8 and 10.
• The chapter's SUMMARY, Part II p.31, prints both closing forms — p(1 + rt) without compounding and p(1 + r)^t with — and annotates the second with the principal for each of terms 1, 2, 3 and t. It also notes that interest rates are the common example of compounding.
• The identity that a change of k% is a multiplication by (1 + k/100) is proved at Part II p.16, Example 3 (Percentage increase and decrease, and choosing the right base), and everything here rests on it.