Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 2 · Chapter companion part
Build compound amounts stage by stage, distinguish amount from profit, and apply the model with clear periods and rounding.
Class 8 English Version: part C of the chapter 2 companion. Eight teaching sections, 30 written tasks and a notebook challenge. Selected companion coverage, not complete textbook coverage; use all four parts and ask a teacher to review explanations and open solutions.
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For annual compounding, the profit at each year end is added before the next year. Start with P = 2000 and decimal annual rate r = 0.10. Year one adds 200, giving 2200. Year two adds 220 on the new base, giving 2420. Year three adds 242, giving 2662. The extra amounts grow because the base grows.
Let A0 = P and Ak = A(k−1)(1 + r) for integer annual periods k. Repeating the multiplier gives An = P(1 + r)^n. Compound profit is An − P. At 2000 and 10% for three years, amount is 2662 and profit is 662. The source uses C for compound principal; C here would equal An, not the original principal P.
With P = 5000 and r = 0.085 for two years, amount is 5000 × 1.085² = 5886.125. Rounding half-up to two decimal places gives Tk 5886.13 and profit Tk 886.13. This task explicitly rounds only the final result. Rounding each year can change a later total; real account rules must specify their convention.
At P = 2000, r = 10% and three years, simple profit is 600 and compound profit is 662, a difference of 62. After one year the two models agree under the same annual rate and timing. At zero rate both give the original principal. For positive rate and more than one full annual period, reinvested profit creates the difference.
For an explicitly nominal annual rate r split equally into m periods per year, each rate is r/m and the number of periods over t years is mt. Amount = P(1 + r/m)^(mt), for whole completed periods. At nominal 12%, half-yearly means 6% twice: 20000 × 1.06² = 22472 after one year. This differs from a single 12% annual multiplication. Do not divide an effective annual rate by two and call it the same offer.
With no deposits or withdrawals and the same annual rate, A2/A1 = 1 + r. If amounts are 5500 after one year and 6050 after two, r = 6050/5500 − 1 = 0.10. Then P = 5500/1.10 = 5000. Check both years. The difference divided by the original principal would not give this adjacent-year rate.
A stated growth of 20 per thousand annually is 20/1000 = 0.02 = 2%. A fictional town of 50000 under this constant model reaches 50000 × 1.02² = 52020 after two years. This is a mathematical projection under assumed constant growth, not measured evidence or a guarantee. Round a non-integer population only at the requested final stage.
Sample question 3, printed 28, gives 66 for three-year compound profit on 200 at 10%. Exact calculation is 200 × 1.1³ − 200 = 66.20; the exact amount is 266.20 and the difference from simple profit is 6.20. The source’s whole-taka values are approximations only if rounding is explicitly intended. Our checks never treat an unstated approximation as exact.
Half-yearly compounding at a nominal annual 12% has rate 0.06 per period. On 20000 the first closing amount is 21200 and the second is 22472; profit is 2472. There are two completed periods, not one period at 12%.
Step through the three annual stages, or read all static stages. Play never starts automatically; pause and replay remain available, and reduced motion disables timed playback.
Every population value in this companion is invented. Constant proportional growth is an assumption for calculation, not a forecast about a named real city.
The source sample gives Tk 200 at 10% annual compound profit for 3 years, then calls the profit Tk 66. Exact amount is 200 × 1.1³ = Tk 266.20; exact profit is Tk 66.20. Simple profit is Tk 60, making the exact difference Tk 6.20. Only its first claim is exact, so no printed combination is exact. If whole-taka rounding is explicitly requested, 266 and 6 are rounded answers. Explain the rounding; do not mark those as exact values.
For the source creative amounts Tk 19500 after year one and Tk 20280 after year two, annual growth factor is 20280/19500 = 1.04. Principal is 19500/1.04 = Tk 18750. After 3 years, compound profit is 18750(1.04³ − 1) = Tk 2341.20; simple profit is 18750 × 0.04 × 3 = Tk 2250. Their difference is Tk 91.20. Keep the same principal, rate and period in both calculations.
The closing amount becomes the next year’s opening amount.
A0 = P Ak = A(k−1)(1 + r) An = P(1 + r)^n Compound profit = An − P
This form assumes no other cash flows and a constant rate. The exponent counts completed compounding periods.
Invented principal 2000; both annual rates are 10%; values are exact.
| Year | Simple profit (Tk) | Simple amount (Tk) | Compound profit (Tk) | Compound amount (Tk) |
|---|---|---|---|---|
| 0 | 0 | 2000 | 0 | 2000 |
| 1 | 200 | 2200 | 200 | 2200 |
| 2 | 400 | 2400 | 420 | 2420 |
| 3 | 600 | 2600 | 662 | 2662 |
At year three the difference of profits is 662 − 600 = 62. The same difference appears between amounts because both include the same initial principal.
Step through invented principal 2000 at 10% annual compounding. Each new amount becomes the next base.
Play is optional and never automatic. Previous/Next and all three static stages remain available; reduced motion disables timed playback. Bar lengths share one money scale.
A nominal annual rate split equally between periods is an explicit condition, not a universal account rule.
A = P(1 + r/m)^(mt) From adjacent annual amounts: r = A2/A1 − 1 Original P = A1/(1 + r)
An effective annual rate must not be divided by m and treated as the same arrangement. The reverse ratio requires equal annual intervals and no cash flows.
A constant proportional-growth model reuses the same multiplier; it is a classroom assumption, not a real-city forecast.
g = b/1000 Population after n years = P0(1 + g)^n
The second addition is 1020 because its base is 51000. Thirty per thousand would be 3%, not 30%.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original invented values. Every bar shares a zero origin and a scale of Tk 10 per drawing unit. The sequence assumes no other transactions.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original fictional model. Boxes are not scaled population graphics; no forecast about a real city is implied.
Show your working on paper. Check your answer, then choose Next.
2200
2000 × 1.10 = 2200.
2420
2000 × 1.10² = 2420.
420
Amount 2420 minus principal 2000 gives 420.
2662
2000 × 1.10³ = 2662.
62
662 − 600 = 62.
5886.13
Exact amount 5886.125 rounds to 5886.13.
22472
20000 × 1.06² = 22472.
6
12/2 = 6, corresponding to decimal 0.06.
10
100 × (6050/5500 − 1) = 10.
5000
5500/1.10 = 5000.
52020
50000 × 1.02² = 52020.
2.5
25/1000 × 100 = 2.5.
5
100 × (5040/4800 − 1) = 5.
4000
The multiplier is 1 each period.
242
2420 × 0.10 = 242; this is not the cumulative profit 662.
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Bases are 2000, 2200 and 2420; additions are 200, 220 and 242; closing amounts are 2200, 2420 and 2662. Prior profit remains invested.
Connect each new base to the preceding closing amount.
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Year one gives P(1 + r). Year two adds rP(1 + r), so the total is P(1 + r)(1 + r) = P(1 + r)².
The second profit must use the enlarged base.
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Amount = 4000 × 1.05³ = 4630.50; profit = 630.50.
Require both quantities and defined periods.
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Simple profit is 1200; compound amount is 7260 and profit 1260; the excess is 60.
Compare profits with profits, not a profit with an amount.
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Expand P(1 + r)² − P = 2Pr + Pr². Simple profit is 2Pr, leaving Pr².
State same principal, annual rate, two years and no cash flows.
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Amount = 8000 × 1.075² = 9245; profit = 1245.
Use 0.075, not 7.5, and retain precision.
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Annual amount = 11000; half-yearly amount = 10000 × 1.05² = 11025. The difference is 25 under the stated nominal-rate convention.
Do not describe the two effective yearly gains as equal.
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The ratio is 1.04, giving rate 4%. Principal = 6240/1.04 = 6000. Three-year compound amount is 6749.184, so profit is 749.184; simple profit is 720. Their difference is 29.184, or 29.18 rounded half-up at the end.
State the no-other-transactions assumption and compare profits over the same three years.
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P = 10648/1.1³ = 8000.
Divide by the full growth factor, not by three times the rate.
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Rate = 0.025; amount = 80000 × 1.025² = 84050. The model assumes the rate persists.
State what the rate means and that the result is an assumed projection.
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30/1000 = 0.03 = 3%. A 1000-person base gains 30 under the stated annual model; 30% would gain 300.
Name the denominator and illustrate the tenfold difference.
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Exact compound profit is 66.20; exact amount is 266.20; simple profit is 60 and the difference is 6.20. Whole-taka rounding would need to be stated.
Accept an explicitly labelled approximation, never as an exact equality.
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That expression gives simple amount 2600. Annual compound amount applies three successive 1.10 multipliers and gives 2662.
Explain the changing base instead of only replacing a formula.
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The no-transaction closed formula alone is insufficient. Update the balance at the specified transaction time, then apply each subsequent period’s factor to that balance.
Require transaction timing; do not guess an ordering.
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One valid choice is 4000 → 4200 → 4410, with additions 200 and 210. Amount minus original principal gives profit 410.
Accept any positive choice with consistent calculations and an explicitly stated rounding rule.
Build an annual ledger and a half-yearly ledger, then explain why their bases change.
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This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 2, printed pages 12–28. It does not replace the complete chapter or its practical teaching.
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