Figure it Out — Applications and Challenges — Fractions in Disguise

1. 12 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   The population of Bengaluru in 2025 is about 250% of its population in 2000. If the population in 2000 was 50 lakhs, what is the population in 2025?

   Hint. 250% means two and a half times — note this is 250% OF the old figure, not an increase of 250%.

   Step 1 — Read the statement carefully. The 2025 population is 250% of the 2000 population — not 250% more than it. So we simply take 250% of 50 lakhs.

   Step 2 — Compute. 250% = 2.5 Population in 2025 = 2.5 × 50 lakhs = 125 lakhs

   That is 1.25 crore, or 1,25,00,000 people.

   Step 3 — Express the change. The increase is 125 − 50 = 75 lakhs, so the population rose by 150% while becoming 250% of its former size.

   The distinction that matters.

   Phrase	Meaning	Result from 50 lakhs
   "250% of"	multiply by 2.5	125 lakhs ✓
   "increased by 250%"	multiply by 3.5	175 lakhs

   The question says "of", so the answer is 125 lakhs. Reading it as "increased by" is the error the wording is designed to test — and the same trap as "fallen by 85%" earlier in the chapter.

   A note on percentages above 100. There is nothing unusual about 250%: a percentage over 100 simply means more than the whole. Here it means the city grew to two and a half times its size in twenty-five years.

   Sanity check: 100% of 50 lakhs is 50 lakhs, 200% is 100 lakhs, and 50% is 25 lakhs. So 250% = 100 + 25 = 125 lakhs ✓

   ✦ 125 lakhs (1.25 crore), since 250% of 50 lakhs means 2.5 × 50 — a rise of 150%, not of 250%.

2. 23 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   The world population in 2025 is about 8.2 billion. Match the given country populations with their approximate percentage share of the world population: Germany 83 million, India 1.46 billion, Bangladesh 175 million, USA 347 million. Options include 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1%.

   Hint. Put everything in the same unit first — millions is easiest — then divide by the world total and multiply by 100.

   Step 1 — Put every figure in millions. World population = 8.2 billion = 8,200 million Germany = 83 million India = 1.46 billion = 1,460 million Bangladesh = 175 million USA = 347 million

   Step 2 — Compute each share as (country ÷ 8200) × 100.

   Country	Population	Share	Nearest option
   India	1,460 m	(1460 ÷ 8200) × 100 = 17.8%	18% ✓
   USA	347 m	(347 ÷ 8200) × 100 = 4.23%	see note
   Bangladesh	175 m	(175 ÷ 8200) × 100 = 2.13%	2% ✓
   Germany	83 m	(83 ÷ 8200) × 100 = 1.01%	1% ✓

   Note on the USA. Its share works out at about 4.2%, and no 4% option appears among those readable in the text. The printed page carries more countries than the four listed here, so the remaining options (13%, 8%, 10%, 35%, 0.1%) belong to those. Match the USA to whichever option nearest 4% appears on your page.

   Step 3 — The estimating shortcut the hint points to. Writing the world total as 8.2 × 10⁹, note that 1% of the world is 82 million. Then every share can be read off almost instantly: • Germany is 83 million ≈ 82 million → about 1% • Bangladesh is 175 million ≈ 2 × 82 → about 2% • USA is 347 million ≈ 4 × 82 → about 4% • India is 1,460 million ≈ 18 × 82 = 1,476 → about 18%

   Finding 1% of the total first turns four divisions into four quick comparisons — exactly the technique used throughout this chapter.

   A striking fact worth noting: India alone accounts for roughly one person in every six on Earth.

   ✦ India ≈ 18%, USA ≈ 4.2%, Bangladesh ≈ 2%, Germany ≈ 1% — most quickly found by noting that 1% of the world population is 82 million.

3. 32 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   The price of a mobile phone is ₹8,250 and GST of 18% is added. Which of the following gives the final price including GST? (i) 8250 + 18 (ii) 8250 + 1800 (iii) 8250 + 18/100 (iv) 8250 × 18 (v) 8250 × 1.18 (vi) 8250 + 8250 × 0.18 (vii) 1.8 × 8250

   Hint. More than one option is correct — look for every expression that equals the price plus 18% of it.

   Step 1 — Work out what the final price must be. GST is 18% of the price, so final price = 8250 + (18% of 8250) = 8250 + 1,485 = ₹9,735

   Step 2 — Test each option.

   (v) 8250 × 1.18 = 9,735 ✓ CORRECT Adding 18% is the same as multiplying by 1.18.

   (vi) 8250 + 8250 × 0.18 = 8250 + 1485 = 9,735 ✓ CORRECT This is the same calculation written the long way.

   Both (v) and (vi) are right — they are algebraically identical, since 8250 × 1.18 = 8250(1 + 0.18) = 8250 + 8250 × 0.18.

   Why the others fail: • (i) 8250 + 18 — adds 18 rupees, not 18 per cent. A percentage is never a fixed amount. • (ii) 8250 + 1800 — the tax is ₹1,485, not ₹1,800. This treats 18% of 10,000 as though it applied. • (iii) 8250 + 18/100 — adds 0.18 rupees, that is 18 paise. • (iv) 8250 × 18 — multiplies the price eighteenfold, giving ₹1,48,500. • (vii) 1.8 × 8250 — this is an 80% increase, since 1.8 = 1 + 0.8. The decimal point has slipped: 18% is 0.18, not 0.8.

   The rule. To add r% to a quantity, multiply by (1 + r/100). For 18% that is 1.18 — and the two correct options are simply the compact and expanded forms of the same thing.

   Sanity check: the tax should be a little under a fifth of the price. A fifth of 8,250 is 1,650, and the actual GST of 1,485 is slightly less ✓

   ✦ Both (v) 8250 × 1.18 and (vi) 8250 + 8250 × 0.18 are correct, each giving ₹9,735.

4. 43 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   The monthly percentage change in a lab's mouse population is +5%, then −2%, then −3%. With initial population p, which statements are true? (i) after three months it is p × 0.05 × 0.02 × 0.03 (ii) p × 1.05 × 0.98 × 0.97 (iii) p + 0.05 − 0.02 − 0.03 (iv) the population after three months was p (v) it was more than p (vi) it was less than p

   Hint. Percentage changes multiply, they do not add. Work out the combined multiplier and compare it with 1.

   Step 1 — Turn each change into a multiplier. • +5% → × 1.05 • −2% → × 0.98 • −3% → × 0.97

   Since each month's change applies to the population at that moment, the multipliers are applied one after another:

   population after 3 months = p × 1.05 × 0.98 × 0.97

   So (ii) is TRUE ✓

   Step 2 — Evaluate the combined multiplier. 1.05 × 0.98 = 1.029 1.029 × 0.97 = 0.99813

   So the final population is 0.99813p — very slightly less than p.

   Therefore (vi) is TRUE ✓ (and (iv) and (v) are false).

   Step 3 — Why the others fail.

   • (i) p × 0.05 × 0.02 × 0.03 — uses the changes rather than the multipliers. This would shrink the population to 0.00003p, essentially wiping it out. • (iii) p + 0.05 − 0.02 − 0.03 — adds fixed numbers to a population, which is not what percentages do. It also happens to give exactly p, which is why it tempts. • (iv) the population was p — this is the trap. Adding the percentages gives 5 − 2 − 3 = 0, suggesting no net change. But percentage changes do not add, because each one is taken on a different base. • (v) more than p — false, since 0.99813 < 1.

   Why the population ends slightly lower. The +5% was applied to the original population, but the −2% and −3% were applied to a larger population, so the falls removed slightly more than the rise added. In numbers, starting from 1,000 mice: 1,050 → 1,029 → 998.13.

   The general principle. A rise of r% followed by a fall of r% never returns you to the start — it always leaves you slightly lower, since the fall acts on a bigger base.

   ✦ (ii) and (vi) are true. The combined multiplier is 1.05 × 0.98 × 0.97 = 0.99813, so the population ends slightly below p — percentage changes multiply rather than add.

5. 53 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   A shopkeeper initially set the price of a product with a 35% profit margin. Due to poor sales, he offered a 30% discount on the selling price. Will he make a profit or a loss? Give reasons.

   Hint. Take the cost price as ₹100 — it makes every step readable.

   Step 1 — Take a convenient cost price. Let the cost price be ₹100. Since the question asks only about profit or loss as a proportion, any starting value gives the same conclusion, and 100 makes the percentages read directly.

   Step 2 — Apply the 35% profit margin. Marked selling price = 100 × 1.35 = ₹135

   Step 3 — Apply the 30% discount on that selling price. Amount received = 135 × 0.70 = ₹94.50

   Step 4 — Compare with the cost. He receives ₹94.50 but paid ₹100, so

   Loss = 100 − 94.50 = ₹5.50, that is a loss of 5.5%

   Why the intuition misleads. A 35% profit followed by a 30% discount looks as if it should still leave a gain, since 35 > 30. But the two percentages are taken on different bases: the 35% is added to the cost (₹100), while the 30% is taken off the higher selling price (₹135). Thirty per cent of 135 is ₹40.50 — more than the ₹35 that was added.

   The combined multiplier makes it immediate: 1.35 × 0.70 = 0.945 Since 0.945 < 1, the shopkeeper ends below his cost whatever the actual price was.

   How large a discount could he have afforded? To break even he needs 1.35 × (1 − d) = 1, giving d = 1 − 1/1.35 = 0.2593. So any discount above about 25.9% puts him into loss — a 25% discount would have kept him just in profit.

   The general lesson: a p% markup followed by a p% discount always ends in a loss, because the discount acts on the larger, marked-up figure.

   ✦ A loss of 5.5%. The combined multiplier is 1.35 × 0.70 = 0.945, so he receives only 94.5% of his cost — the 30% discount is taken on the higher marked price, making it worth more than the 35% added to the cost.

6. 62 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 28

   What percentage of the area is occupied by the region marked 'E' in the figure?

   Hint. Find what fraction of the whole shape region E covers, then convert to a percentage.

   Note: the regions A to E are printed in the diagram on page 28, so the exact fraction depends on that figure. The method below is what the question tests.

   The method.

   1. Find the total area of the whole figure, counting in whatever unit the diagram uses — grid squares, or the area of one basic shape.
   2. Find the area of region E in the same unit.
   3. Convert to a percentage:

   $\text{percentage}=\frac{\text{area of E}}{\text{total area}}\times 100$

   A worked illustration. If the whole figure covers 40 grid squares and region E covers 6 of them: (6 ÷ 40) × 100 = 15%

   Two shortcuts worth using.

   • If the regions are equal, no counting is needed: five equal regions means each is 100 ÷ 5 = 20%.

   • If the other regions are easier to measure, find them instead and subtract. If A, B, C and D come to 82% between them, then E must be 100 − 82 = 18% — often much quicker when E is the awkward shape.

   The check that must always hold. The percentages of all five regions must total exactly 100%. If your figures come to 97% or 104%, a region has been mismeasured — and this check will catch it before you lose the mark.

   On counting part-squares: where region E cuts across grid squares, pair up the partial squares — two halves make one whole. Estimating each part to the nearest quarter is accurate enough for a percentage answer.

   ✦ Percentage of E = (area of E ÷ total area) × 100. Count in grid squares, or find the other four regions and subtract from 100% — and check that all five regions total exactly 100%.

7. 73 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

   What is 5% of 40? What is 40% of 5? What is 25% of 12? What is 12% of 25? What is 15% of 60? What is 60% of 15? What do you notice? Can you make a general statement and justify it using algebra, comparing x% of y and y% of x?

   Hint. Compute all six, then look at the pairs.

   Step 1 — Compute all six.

   Calculation	Working	Result
   5% of 40	0.05 × 40	2
   40% of 5	0.40 × 5	2
   25% of 12	0.25 × 12	3
   12% of 25	0.12 × 25	3
   15% of 60	0.15 × 60	9
   60% of 15	0.60 × 15	9

   Step 2 — What we notice. Each pair gives the same answer. Swapping the percentage and the quantity leaves the result unchanged.

   Step 3 — The general statement. For any numbers x and y, x% of y equals y% of x.

   Step 4 — The algebraic justification.

   $x\%\text{ of }y=\frac{x}{100}\times y=\frac{xy}{100}$

   $y\%\text{ of }x=\frac{y}{100}\times x=\frac{yx}{100}$

   Since multiplication is commutative, xy = yx, so the two expressions are identical. That single fact is the whole reason the pattern holds — and it holds for every x and y, not just the six examples.

   Why this is genuinely useful. It lets you swap an awkward calculation for an easy one: • 4% of 75 is hard, but 75% of 4 is three quarters of 4 = 3 ✓ • 8% of 50 is fiddly, but 50% of 8 is half of 8 = 4 ✓ • 18% of 50 becomes 50% of 18 = 9 ✓

   The habit to build: whenever a percentage calculation looks awkward, try swapping the two numbers. One of the two orders is almost always easier to do in your head.

   ✦ Each pair matches: 2, 2, 3, 3, 9, 9. In general x% of y = y% of x, because both equal xy/100 and multiplication is commutative — which makes 4% of 75 easy by computing 75% of 4 instead.

8. 83 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

   A school is organising an excursion. 40% of the students are Grade 8 and the rest Grade 9. Among the Grade 8 students, 60% are girls. (i) What percentage of the students going are Grade 8 girls? (ii) If the total number going is 160, how many are Grade 8 girls?

   Hint. The 60% is a percentage OF the Grade 8 students, not of everyone.

   (i) Percentage who are Grade 8 girls.

   The key is that the two percentages apply to different groups: 40% is of all students, but 60% is only of the Grade 8 students. So the second percentage is taken of the first.

   Grade 8 girls = 60% of 40% of the total = 0.60 × 0.40 = 0.24 = 24%

   A diagram makes this clear. Imagine 100 students:

   Group	Count
   Grade 8	40
   — of whom girls (60%)	24
   — of whom boys (40%)	16
   Grade 9	60

   So 24 out of every 100 students are Grade 8 girls ✓

   The error to avoid is adding or averaging the percentages. Grade 8 girls are not 40 + 60 = 100%, nor 50%. When one percentage is taken of another, they multiply.

   (ii) If 160 students go.

   Grade 8 students = 40% of 160 = 64 Grade 8 girls = 60% of 64 = 38.4

   Or directly: 24% of 160 = 38.4 ✓

   A note on this answer. 38.4 is not a whole number of people, so the figures given in the question are not quite consistent with each other — you cannot have four-tenths of a student. The mathematics is correct; the total of 160 simply does not divide neatly under these percentages. The intended answer is 38.4, best reported as about 38 students. For the numbers to come out whole, the total would need to be a multiple of 25 (since 24% = 6/25) — 150 would give exactly 36, and 200 would give exactly 48.

   ✦ (i) 24%, since 60% of 40% means 0.6 × 0.4. (ii) 24% of 160 = 38.4, so about 38 students — the figures in the question do not divide into whole students.

9. 93 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

   A shopkeeper sells pencils at a price such that the selling price of 3 pencils equals the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?

   Hint. Set the cost of one pencil to a convenient value and work out the selling price of one pencil.

   Step 1 — Choose a convenient cost. Let the cost of one pencil be ₹1. (Any value works; the percentage is the same.)

   Step 2 — Use the given condition. Selling price of 3 pencils = cost of 5 pencils = ₹5

   So the selling price of one pencil is 5 ÷ 3 = ₹5/3 ≈ ₹1.67

   Step 3 — Compare with the cost. Since ₹1.67 > ₹1, he sells each pencil for more than it cost him — this is a profit.

   Profit per pencil = 5/3 − 1 = ₹2/3

   Step 4 — Find the profit percentage. Profit % = (profit ÷ cost price) × 100 = (2/3 ÷ 1) × 100 = 66.67% (exactly 66⅔%)

   Check with real numbers. Suppose each pencil costs ₹3. Then 5 pencils cost ₹15, and that is what 3 pencils sell for — so each sells at ₹5. Profit per pencil = ₹2 on a cost of ₹3, which is 2/3 = 66.67% ✓

   The quick way to read this kind of question. Whenever the selling price of m items equals the cost of n items:

   $\text{profit \%}=\frac{n-m}{m}\times 100$

   Here n = 5 and m = 3, giving (5 − 3)/3 × 100 = 66.67% ✓

   How to tell profit from loss at a glance: if the number of items whose cost is quoted (5) is larger than the number whose selling price is quoted (3), it is a profit. Reversed — if the SP of 5 equalled the cost of 3 — it would be a loss of (5−3)/5 = 40%.

   ✦ A profit of 66.67% (exactly 66⅔%), since each pencil costing ₹1 sells for ₹5/3.

10. 102 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

    The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase over the last 2 years?

    Hint. The second increase applies to the already-increased fare, so the multipliers combine.

    Step 1 — Turn each increase into a multiplier. • 3% increase → × 1.03 • 4% increase → × 1.04

    Step 2 — Combine them. The second rise applies to the fare after the first rise, so the multipliers are applied in turn:

    combined multiplier = 1.03 × 1.04 = 1.0712

    Step 3 — Read off the overall increase. Since 1.0712 = 1 + 0.0712, the overall rise is 7.12%

    Why it is not simply 7%. Adding the percentages gives 3 + 4 = 7%, which is close but not exact. The extra 0.12% is the 4% charged on the 3% that was already added — a small amount of compounding.

    Working it through on a ₹100 fare: • After the first rise: 100 × 1.03 = ₹103 • After the second: 103 × 1.04 = ₹107.12 The increase is ₹7.12 on ₹100, that is 7.12% ✓

    Had we merely added, we would have expected ₹107 — the missing 12 paise is the 4% of the ₹3 already added.

    The general result. Two successive increases of a% and b% combine to

    $a+b+\frac{ab}{100}\%$

    Here that is 3 + 4 + 12/100 = 7.12% ✓

    When the approximation is safe. For small percentages the extra term is tiny, so adding is a fair estimate. But for large ones it matters a great deal: two successive 50% rises give 50 + 50 + 25 = 125%, not 100%.

    ✦ 7.12%, since 1.03 × 1.04 = 1.0712 — the extra 0.12% above the simple sum is the second year's 4% charged on the first year's 3% rise.

11. 113 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

    If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease?

    Hint. Write the area both before and after. The two must be equal, which pins down the new breadth.

    Step 1 — Write both areas. Let the original length be L and breadth b, so the original area is Lb.

    After the change: • new length = 1.1L • new breadth = b′ (unknown) • new area = 1.1L × b′

    Step 2 — Set the areas equal. 1.1L × b′ = Lb

    The L cancels from both sides: 1.1 b′ = b b′ = b ÷ 1.1 = b × 10/11

    Step 3 — Find the percentage decrease. The breadth has fallen from b to (10/11)b, so the drop is b − (10/11)b = (1/11)b

    As a percentage of the original breadth: (1/11) × 100 = 9.0909…% = 9 1/11 %

    Why the answer is not 10%. The instinct is that a 10% rise must be undone by a 10% fall — but a 10% fall would give 0.9b, and 1.1 × 0.9 = 0.99, not 1. That leaves the area 1% smaller, not unchanged. The required decrease is slightly less than 10%, because it is measured against the original breadth while acting on a length that has already grown.

    Check: 1.1 × (10/11) = 11/10 × 10/11 = 1 exactly ✓ so the area is genuinely unchanged.

    The general rule. To keep a product constant when one factor rises by r%, the other must fall by

    $\frac{r}{100+r}\times 100\%$

    Here that is (10/110) × 100 = 9.09% ✓

    This is why the question says exactly — 9.09% is a rounded value, and the exact answer is the fraction 9 1/11 %.

    ✦ The breadth decreases by exactly 1/11, that is 9 1/11 % ≈ 9.09% — not 10%, since a 10% fall would leave the area 1% smaller.

12. 122 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

    The percentage of ingredients in a 65 g chips packet is shown in the picture. Find the weight each ingredient makes up in this packet.

    Hint. Find 1% of 65 g once, then multiply by each ingredient's percentage.

    Note: the individual percentages are printed on the packet label in the figure on page 29. The method below is what earns the marks, and it works for whatever percentages your copy shows.

    The efficient method.

    Find 1% of the packet once, then every ingredient is a single multiplication: 1% of 65 g = 65 ÷ 100 = 0.65 g

    So an ingredient making up p% of the packet weighs p × 0.65 g.

    A worked illustration. Suppose the label shows Potato 60%, Oil 25%, Salt 10% and Spices 5%:

    Ingredient	Percentage	Weight
    Potato	60%	60 × 0.65 = 39 g
    Oil	25%	25 × 0.65 = 16.25 g
    Salt	10%	10 × 0.65 = 6.5 g
    Spices	5%	5 × 0.65 = 3.25 g
    Total	100%	65 g ✓

    Two checks that must both hold.

    1. The percentages must add to 100%.
    2. The weights must add to 65 g. If either fails, an ingredient has been misread or miscalculated.

    A useful shortcut for common values: • 50% = half of 65 = 32.5 g • 25% = a quarter = 16.25 g • 10% = 6.5 g • 1% = 0.65 g Most labels use round percentages, so these four cover almost everything.

    Why food labels are given as percentages at all. Percentages let you compare two packets of different sizes directly — a 65 g packet and a 200 g packet with the same percentage of oil have the same proportion of oil, even though the actual weights differ.

    ✦ Find 1% of the packet first — 65 ÷ 100 = 0.65 g — then each ingredient of p% weighs p × 0.65 g. The weights must total exactly 65 g.

13. 134 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 29

    Three shops sell the same items at the same price. Shop A: "Buy 1 get 1 free". Shop B: "Buy 2 get 1 free". Shop C: "Buy 3 get 1 free". (i) If one item costs ₹100, what is the effective price per item in each shop? Arrange from cheapest to costliest. (ii) Calculate the percentage discount at each shop. (iii) If you need 4 items, which shop would you choose and why?

    Hint. For each deal, work out how much you pay and how many items you take home.

    (i) Effective price per item.

    The effective price is what you pay divided by how many items you actually receive.

    Shop	Deal	You pay	You get	Effective price
    A	Buy 1 get 1 free	₹100	2 items	100 ÷ 2 = ₹50
    B	Buy 2 get 1 free	₹200	3 items	200 ÷ 3 = ₹66.67
    C	Buy 3 get 1 free	₹300	4 items	300 ÷ 4 = ₹75

    Cheapest to costliest: A, then B, then C.

    (ii) Percentage discount.

    As the hint suggests, compare the free items with the total items received:

    Shop	Free ÷ total received	Discount
    A	1 out of 2	50%
    B	1 out of 3	33.33%
    C	1 out of 4	25%

    Check against part (i): a 50% discount on ₹100 gives ₹50 ✓, a 33.33% discount gives ₹66.67 ✓, and a 25% discount gives ₹75 ✓

    The trap: "Buy 1 get 1 free" is a 50% discount, not 100%. You still pay for one of the two items. Comparing the free item with the total received — not with the number bought — is what gets this right.

    (iii) Buying exactly 4 items.

    Shop	How to get 4 items	Total cost
    A	Buy 2, get 2 free	₹200
    B	Buy 2 get 1 free (3 items), then buy 1 more	₹300
    C	Buy 3, get 1 free	₹300

    Choose Shop A, paying ₹200 — a saving of ₹100 over either of the others.

    Note that Shop B's deal does not divide neatly into 4: you take the offer once for 3 items, then pay full price for the fourth, so its effective price for exactly 4 items rises to ₹75 — the same as Shop C, despite B being cheaper in general.

    The wider lesson. A deal's advertised generosity is not the same as its value. Always convert to an effective price per item, and then check whether the number you actually need fits the deal cleanly.

    ✦ (i) A ₹50, B ₹66.67, C ₹75 — cheapest to costliest A, B, C. (ii) 50%, 33.33%, 25% respectively. (iii) Shop A, at ₹200 for 4 items against ₹300 elsewhere.

14. 144 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 30

    In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?

    Hint. Only left-handed people leave, so the number of right-handed people never changes. Work with them instead.

    Step 1 — Set out the starting position. Of the 100 people: • left-handed = 99% of 100 = 99 • right-handed = 1

    Step 2 — Notice what stays fixed. Only left-handed people leave, so the single right-handed person remains throughout. This is the key that unlocks the problem — tracking the unchanging quantity is far easier than tracking the changing one.

    Step 3 — Use the right-handed person as the anchor. When the room is 98% left-handed, it is 2% right-handed. That 2% is still just the one person, so

    1 = 2% of the new total new total = 1 ÷ 0.02 = 50 people

    Step 4 — Work out how many left. New total = 50, of whom 1 is right-handed, so 49 are left-handed.

    Left-handed people who left = 99 − 49 = 50

    Step 5 — Check. Room now holds 49 left-handed + 1 right-handed = 50 people. 49 ÷ 50 = 0.98 = 98% ✓

    Why the answer is so startling. Dropping the percentage by a single point — from 99% to 98% — requires half the room to leave. The reason is that the percentage is controlled by the one right-handed person: for that single person to represent 2% instead of 1%, the room must shrink to half its size.

    The general principle. When one small group is fixed in number, the percentage is governed by the total. Doubling a fixed group's percentage share means halving the total.

    The wrong approach to avoid: trying to find a number x such that (99 − x)/(100 − x) = 0.98 also works, but it is far more error-prone. Solving it gives 99 − x = 98 − 0.98x, hence 0.02x = 1 and x = 50 ✓ — the same answer, reached with more effort.

    ✦ 50 left-handed people must leave. The one right-handed person is unchanged, so for them to be 2% of the room the total must fall to 50, leaving 49 left-handed people.

15. 152 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 30

    Look at the graph showing ability to use computers by age and gender (2023), with bars for children, teenagers, twenties, thirties, forties, fifties and seniors. Interpret what it shows.

    Hint. Read the axis labels first, then compare the bars within each age group and across age groups.

    Note: the exact percentages are printed on the bar graph on page 30. The readable values in the text include 4%, 24%, 26% and 29%, and the graph's own caption states that ability is highest among those in their twenties and teenagers. The method below is what the question tests.

    How to read a grouped bar graph — in order.

    1. Read the axes and the key first. The horizontal axis shows percentages starting at 0%; the vertical axis lists seven age groups. The key shows two bars per group, one for female and one for male. Never compare bars without checking which is which.

    2. Compare within each age group. For each age band, look at the two bars side by side. This answers questions about the gender gap at a particular age — and whether it is wide, narrow, or reversed.

    3. Compare across age groups. Read down the chart to see how ability changes with age. The caption tells us it peaks in the twenties and teenage years and falls away on either side — lower among children, who have had less exposure, and declining steadily through the thirties, forties, fifties and into the senior group.

    4. Watch for the trap. A bar showing 24% and another showing 29% differ by 5 percentage points, but that is a relative increase of 5/24 ≈ 21%. "Percentage points" and "per cent" are not the same thing, and conflating them is the standard error in graph questions.

    Typical conclusions such a graph supports. • Computer ability rises through childhood, peaks in the teens and twenties, and then declines with age. • The decline is gradual through the middle years and steepest for the senior group. • Any gender gap should be quoted with its size, not merely noted as existing.

    A check before writing: every figure you quote should be traceable to a specific bar. If you cannot point to the bar, do not claim the number.

    ✦ Read the axes and key first, compare the two bars within each age band for the gender gap, then read across bands for the age trend — ability peaks in the teens and twenties and declines with age. Quote differences as percentage points, not per cent.

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.