Figure it Out — Interest and Compounding — Fractions in Disguise

1. 13 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 22

   Bank of Yahapur offers 10% p.a. Compare how much one gets on a deposit of ₹20,000 for 2 years with compounding and without compounding annually.

   Hint. Without compounding the interest is the same each year; with compounding it is calculated on the new, larger balance.

   Without compounding (simple interest). The interest is paid out each year, so the principal stays at ₹20,000 throughout.

   Interest per year = 20000 × 0.10 = ₹2,000 Interest for 2 years = 2000 × 2 = ₹4,000 Total received = 20000 + 4000 = ₹24,000

   Using the chapter's formula, amount = p(1 + rt) = 20000(1 + 0.10 × 2) = 20000 × 1.2 = ₹24,000 ✓

   With compounding. The interest is added back each year, so the second year earns interest on a larger balance.

   	Starting balance	Interest at 10%	Ending balance
   Year 1	₹20,000	₹2,000	₹22,000
   Year 2	₹22,000	₹2,200	₹24,200

   Using the formula, amount = p(1 + r)ᵗ = 20000 × (1.1)² = 20000 × 1.21 = ₹24,200 ✓ Interest earned = ₹4,200

   Comparison. Compounding gives ₹200 more — ₹24,200 against ₹24,000.

   Where the extra ₹200 comes from. It is the interest earned on the first year's interest: 10% of ₹2,000 = ₹200. That is the whole of compounding in one line — you earn interest on your interest.

   As percentages of the deposit: • without compounding, the gain over 2 years is 20% • with compounding, it is 24200/20000 = 121%, a gain of 21%

   ✦ Without compounding ₹24,000; with compounding ₹24,200 — a difference of ₹200, which is precisely the 10% interest earned on the first year's ₹2,000 of interest.

2. 23 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 23

   Bank of Wahapur offers 5% p.a. Compare how much one gets on a deposit of ₹20,000 for 4 years with compounding and without compounding annually.

   Hint. Same principal as the previous question, but half the rate over twice the time.

   Without compounding — the flat-rate case. The principal stays at ₹20,000 all four years, so each year pays the same. Interest per year = 20000 × 0.05 = ₹1,000 Interest for 4 years = 1000 × 4 = ₹4,000 Total received = ₹24,000

   By formula: p(1 + rt) = 20000(1 + 0.05 × 4) = 20000 × 1.2 = ₹24,000 ✓

   With compounding.

   	Starting balance	Interest at 5%	Ending balance
   Year 1	₹20,000	₹1,000	₹21,000
   Year 2	₹21,000	₹1,050	₹22,050
   Year 3	₹22,050	₹1,102.50	₹23,152.50
   Year 4	₹23,152.50	₹1,157.63	₹24,310.13

   By formula: p(1 + r)ᵗ = 20000 × (1.05)⁴ = 20000 × 1.21550625 = ₹24,310.13 ✓ Interest earned = ₹4,310.13

   Comparison. Compounding gives ₹310.13 more — ₹24,310.13 against ₹24,000.

   Notice how the yearly interest grows under compounding: ₹1,000, then ₹1,050, ₹1,102.50, ₹1,157.63. Each year's interest is larger because it is charged on a balance that already contains the previous years' interest. Without compounding it would have stayed at ₹1,000 every year.

   ✦ Without compounding ₹24,000; with compounding ₹24,310.13 — a difference of ₹310.13, with the yearly interest growing from ₹1,000 to ₹1,157.63 as the balance builds.

3. 33 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 23

   Do you observe anything interesting in the solutions of the two questions above? Share and discuss.

   Hint. Compare the simple-interest totals, then compare the compound ones.

   Observation 1 — the simple interest is identical. Both deposits give exactly ₹4,000 without compounding, and both end at ₹24,000.

   The reason is that the product of rate and time is the same in each case: • Yahapur: 10% × 2 years = 20% • Wahapur: 5% × 4 years = 20%

   Since simple interest is p × r × t, only the product rt matters — not how the rate and time are split between themselves. Halving the rate while doubling the time leaves it unchanged.

   Observation 2 — the compound interest is not identical. • Yahapur (10%, 2 years): ₹24,200 • Wahapur (5%, 4 years): ₹24,310.13

   The Wahapur deposit ends ₹110.13 richer, even though the two look equivalent under simple interest.

   Why the longer, slower option wins. Compounding depends on how many times the interest is added back, not just on the total rate. Wahapur compounds four times while Yahapur compounds only twice, so there are more opportunities for interest to start earning interest of its own.

   Comparing the multipliers directly: (1.10)² = 1.21 against (1.05)⁴ = 1.2155…

   The general principle. For the same product rt, more frequent compounding over a longer period beats a higher rate over a shorter one. This is exactly why banks advertise how often interest is compounded — quarterly beats annually at the same headline rate — and why long-term saving is so powerful.

   A caution the comparison also shows: the same logic applies to debt. A loan compounding over many periods grows faster than the headline rate suggests.

   ✦ The simple interest is identical (₹4,000 each) because rt = 20% in both cases. But the compound amounts differ — ₹24,200 against ₹24,310.13 — because compounding depends on how many times interest is added back, and Wahapur compounds four times to Yahapur's two.

4. 42 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 24

   Jasmine invests amount p for 4 years at 6% p.a. Which expression(s) describe the total amount she will get after 4 years when compounding is not done? (i) p × 6 × 4 (ii) p × 0.6 × 4 (iii) p × (0.6/100) × 4 (iv) p × (0.06/100) × 4 (v) p × 1.6 × 4 (vi) p × 1.06 × 4 (vii) p + (p × 0.06 × 4)

   Hint. The TOTAL amount is the principal plus the interest, not the interest alone.

   Step 1 — Build the correct expression. Without compounding, the interest each year is p × 0.06, so over 4 years the interest is p × 0.06 × 4

   The total amount is the principal plus that interest: p + (p × 0.06 × 4)

   So the answer is option (vii).

   (This equals p(1 + 0.24) = 1.24p, matching the chapter's formula p(1 + rt).)

   Step 2 — Why each of the others fails.

   • (i) p × 6 × 4 — uses 6 instead of 0.06, treating the rate as 600%. It also gives only interest, not the total. • (ii) p × 0.6 × 4 — uses 0.6, which is 60%, ten times too large. • (iii) p × (0.6/100) × 4 — this is 0.006, that is 0.6%, ten times too small. • (iv) p × (0.06/100) × 4 — this is 0.0006, a hundred times too small. Converting 6% to a decimal is done once: either 6/100 or 0.06, never both. • (v) p × 1.6 × 4 — a 60% one-off increase, then multiplied by 4 for no valid reason. • (vi) p × 1.06 × 4 — the multiplier 1.06 is right for one year's growth, but multiplying by 4 does not extend it correctly. Four years of compounding would be p × (1.06)⁴, and four years of simple interest is option (vii). Multiplying the multiplier by 4 is neither.

   The two traps here. First, converting the rate — 6% is 0.06, and dividing by 100 again is the commonest slip. Second, the question asks for the total amount, so the principal must be included; several options give only the interest.

   Check with numbers. If p = ₹1,000, then (vii) gives 1000 + 240 = ₹1,240 ✓ — sensible for 6% over 4 years. Option (vi) would give ₹4,240, which is absurd.

   ✦ (vii) p + (p × 0.06 × 4), which equals 1.24p — the principal plus four years of simple interest at 6%.

5. 53 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 24

   The post office offers 7% p.a. How much interest would one get on ₹50,000 invested for 3 years without compounding? How much more would one get if it were compounded?

   Hint. Work out both totals, then take the difference of the interest amounts.

   Without compounding. Interest = p × r × t = 50000 × 0.07 × 3 = ₹10,500

   Total amount = 50,000 + 10,500 = ₹60,500

   With compounding. Amount = p × (1 + r)ᵗ = 50000 × (1.07)³

   Building it year by year:

   	Starting balance	Interest at 7%	Ending balance
   Year 1	₹50,000	₹3,500	₹53,500
   Year 2	₹53,500	₹3,745	₹57,245
   Year 3	₹57,245	₹4,007.15	₹61,252.15

   By formula: 50000 × 1.225043 = ₹61,252.15 ✓ Interest earned = 61,252.15 − 50,000 = ₹11,252.15

   The difference. 11,252.15 − 10,500 = ₹752.15 more with compounding.

   Where the extra ₹752.15 comes from. Under simple interest each year pays a flat ₹3,500. Under compounding the yearly interest grows to ₹3,500, ₹3,745 and ₹4,007.15 — the extra amounts being ₹0, ₹245 and ₹507.15, which total ₹752.15 ✓

   As a percentage gain over 3 years: • without compounding: 10500/50000 = 21% • with compounding: 11252.15/50000 = 22.5%

   Why the gap is still modest. Over only 3 years compounding adds about 1.5 percentage points. Over 20 years at the same rate the effect is dramatic: (1.07)²⁰ ≈ 3.87, so the money nearly quadruples, against just 2.4 times under simple interest. Compounding rewards time above all else.

   ✦ Without compounding the interest is ₹10,500; with compounding it is ₹11,252.15 — ₹752.15 more, because each year's interest is charged on a balance that already includes the earlier interest.

6. 64 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 24

   Giridhar borrows ₹12,500 at 12% per annum for 3 years without compounding, and Raghava borrows the same amount for the same time at 10% per annum compounded annually. Who pays more interest, and by how much?

   Hint. Do not judge by the rates alone — a higher rate without compounding may still cost less than a lower rate with it.

   Giridhar — 12% simple, 3 years. Interest = p × r × t = 12500 × 0.12 × 3 = ₹4,500

   Total repayable = 12,500 + 4,500 = ₹17,000

   Raghava — 10% compounded annually, 3 years.

   	Starting balance	Interest at 10%	Ending balance
   Year 1	₹12,500	₹1,250	₹13,750
   Year 2	₹13,750	₹1,375	₹15,125
   Year 3	₹15,125	₹1,512.50	₹16,637.50

   By formula: 12500 × (1.1)³ = 12500 × 1.331 = ₹16,637.50 ✓ Interest = 16,637.50 − 12,500 = ₹4,137.50

   Comparison. Giridhar pays ₹4,500 and Raghava pays ₹4,137.50, so

   Giridhar pays more, by 4,500 − 4,137.50 = ₹362.50

   Why this is worth noticing. Raghava's loan compounds, which pushes the cost up — yet he still pays less, because his rate is 2 percentage points lower. Over three years the 2% rate advantage outweighs the compounding disadvantage.

   But the balance shifts with time. Compare the total cost as the term lengthens:

   Years	Giridhar at 12% simple	Raghava at 10% compound
   3	₹4,500	₹4,137.50
   5	₹7,500	₹7,628.13
   10	₹15,000	₹19,921.88

   By year 5 Raghava is paying more, and by year 10 he pays far more. Compounding grows without limit while simple interest grows in a straight line, so a compounded loan always overtakes eventually — the only question is when.

   The practical lesson: when comparing loans, the headline rate alone is not enough. You must also know whether, and how often, the interest compounds.

   ✦ Giridhar pays more, by ₹362.50 — ₹4,500 against Raghava's ₹4,137.50 — because over just 3 years his 2% higher rate outweighs Raghava's compounding. Over a longer term the compounded loan would become the costlier one.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp201.pdf), where this is Chapter 1 (pages 1-32) — the eighth chapter of the Class 8 course. The title refers to percentages being fractions in disguise, with denominator 100. IMPORTANT DIFFERENCE FROM PART 1: this PDF carries NO printed answer key, so every answer here was derived from first principles and independently recomputed in Python before being written — including all the compound-interest amounts, the two-buffalo profit-and-loss problem, the reverse-percentage car price, and the left-handed-people puzzle. FOUR ITEMS ARE FLAGGED IN PLACE where the printed figure cannot be recovered from text: the bar-model diagrams (Figure it Out 2, Q1), the runners' race picture (Q4 of the first block), the region-E area diagram and the chips-packet label, and the computer-ability bar graph. One genuine inconsistency in the book's own numbers is also flagged: 24% of 160 students is 38.4, which is not a whole number of people.. Questions are referenced from the NCERT textbook for identification.