class8-Part 2 chapter 1-Fractions in disguise - solutions

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Class 8 Ganita Prakash · Part 2 · Chapter 1
Fractions in Disguise

Page-wise textbook solutions (pages 1–32, Part 2) · Every step · Figures · Common mistakes · Tips

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🗝️ Key ideas of this chapter

• Per cent means 'out of 100': \(x\% = \dfrac{x}{100}\). Fraction → % : multiply by 100.
• x% of y = \(\dfrac{x}{100} \times y\) = y% of x. Break hard percentages into 50%, 25%, 10%, 5%, 1%.
• Percentages above 100 mean more than the whole: 120% = 1.2 times.
• % change = change ÷ ORIGINAL × 100. Profit % and loss % are on the cost price.
• Successive changes MULTIPLY: +30% then −20% → × 1.3 × 0.8.
• Simple interest: \(p(1 + rt)\). Compound: \(p(1 + r)^t\).

📖 Textbook Page 1

📖 Page 1Example 1

Red paint makes up \(\tfrac{3}{4}\) of Surya's mixture. What percentage is red? What percentage is yellow?

1Make the denominator 100: \(\tfrac{3}{4} = \tfrac{3 \times 25}{4 \times 25} = \tfrac{75}{100}\)

275 out of every 100 → 75%.

3Yellow = the rest = \(\tfrac{1}{4}\) = 100% − 75% = 25%.

Red = 3/4 = 75/100 = 75%; yellow = 1/4 = 25%.

✅ Answer: Red 75%, yellow 25%.

💡 Tip: Fraction → percentage: multiply by 100. \(\tfrac{3}{4} \times 100 = 75\).

📖 Textbook Page 2

📖 Page 2Example 2

Surya saves \(\tfrac{2}{5}\) of his prize money. Express it as a percentage. Complete Method 3 (the boxes).

1Method 1: \(\tfrac{2}{5} = \tfrac{40}{100} = 40\%\)

2Method 2: \(x = \tfrac{2}{5} \times 100 = 40\)

3Method 3: each fifth of the bar is 100 ÷ 5 = 20%, so the boxes are 20%, 40%, 60%, 80%. Savings = 2 fifths = 40%.

Boxes under the bar: 20%, 40%, 60%, 80%.

✅ Answer: 40%

📖 Textbook Page 3

📖 Page 3Example 3

Express 24% as a fraction (and find equivalent forms).

1\(24\% = \tfrac{24}{100}\)

2Divide by 2 again and again: \(\tfrac{12}{50} = \tfrac{6}{25}\)

3Or multiply: \(\tfrac{48}{200}\)

✅ Answer: \(\tfrac{6}{25}\) (simplest form)

📖 Page 3Figure it Out · Q1

Express as percentages: (i) \(\tfrac{3}{5}\) (ii) \(\tfrac{7}{14}\) (iii) \(\tfrac{9}{20}\) (iv) \(\tfrac{72}{150}\) (v) \(\tfrac{1}{3}\) (vi) \(\tfrac{5}{11}\)

1Multiply each fraction by 100.

2(i) \(\tfrac{3}{5} = \tfrac{60}{100}\) = 60%

3(ii) \(\tfrac{7}{14} = \tfrac{1}{2}\) = 50%

4(iii) \(\tfrac{9}{20} = \tfrac{45}{100}\) = 45%

5(iv) \(\tfrac{72}{150} = \tfrac{48}{100}\) = 48%

6(v) \(\tfrac{100}{3} = 33\tfrac{1}{3}\%\) ≈ 33.33%

7(vi) \(\tfrac{500}{11} = 45\tfrac{5}{11}\%\) ≈ 45.45%

✅ Answer: 60%, 50%, 45%, 48%, 33⅓%, 45 5/11 %

📖 Page 3Figure it Out · Q2

Nandini has 25 marbles; 15 are white. What percentage are white? Options: 10%, 15%, 25%, 60%, 40%, none.

1\(\tfrac{15}{25} = \tfrac{60}{100}\)

✅ Answer: (iv) 60%

⚠️ Common mistake: Choosing 15% because there are 15 white marbles — 15 is out of 25, not out of 100.

📖 Textbook Page 4

📖 Page 4Figure it Out · Q3

15 of the 80 students walk to school. What percentage walk?

1\(\tfrac{15}{80} \times 100 = \tfrac{1500}{80} = 18.75\)

✅ Answer: 18.75%

📖 Page 4Figure it Out · Q4

Match each runner A, B, C, D with the approximate percentage of the race completed: 55%, 20%, 38%, 72%, 84%, 93%.

1Compare each position with the full track (Start to Finish = 100%).

2A is about one-third of the way → closest option 38%.

3B is just past halfway → 55%.

4C is about three-quarters → 72%.

5D is almost at the finish → 93%.

620% and 84% are extra options that are not used.

Measure each runner's distance against the whole track (Start → Finish = 100%).

✅ Answer: A ≈ 38%, B ≈ 55%, C ≈ 72%, D ≈ 93%

📖 Page 4Figure it Out · Q5

Use >, < or =: (i) 50% __ 5% (ii) \(\tfrac{5}{10}\) __ 50% (iii) \(\tfrac{3}{11}\) __ 61% (iv) 30% __ \(\tfrac{1}{3}\)

1(i) 50 parts out of 100 is more than 5 parts → >

2(ii) \(\tfrac{5}{10} = \tfrac{1}{2}\) = 50% → =

3(iii) \(\tfrac{3}{11}\) is less than \(\tfrac{3}{9} = \tfrac{1}{3}\), so less than 34%; surely less than 61% → <

4(iv) \(\tfrac{1}{3}\) = 33⅓% → 30% < \(\tfrac{1}{3}\)

✅ Answer: (i) > (ii) = (iii) < (iv) <

📖 Page 4Think

Sugar is \(\tfrac{9}{34}\) of biscuit Variety 1 and \(\tfrac{13}{45}\) of Variety 2. Which is more sugary? Why do we use 100 as the denominator?

1\(\tfrac{9}{34} \times 100 \approx 26.47\%\); \(\tfrac{13}{45} \times 100 \approx 28.89\%\)

2Same denominator (100) makes comparing easy.

3100 fits our base-10 system: 31% = 0.31. It gives enough detail and is easy to picture.

✅ Answer: Variety 2 is more sugary. 100 is round, easy to understand, and works neatly with decimals.

📖 Textbook Page 6

📖 Page 6Example 1

Madhu ate 120 g of biscuits with 25% sugar; Madhav ate 95 g with 35% sugar. Who ate more sugar?

1Madhu: 25 : 100 :: s : 120 → \(s = \tfrac{25}{100} \times 120 = 30\) g

2Madhav: \(\tfrac{35}{100} \times 95 = 33.25\) g

✅ Answer: Madhav (33.25 g against 30 g).

⚠️ Common mistake: Comparing only 25% and 35% — the amounts of biscuit are different.

📖 Textbook Page 7

📖 Page 7Free-hand table

Mentally find 25%, 10%, 20%, 5% (and then 40% and 15%) of 100, 200, 50, 80, 10, 35, 287.

110% → divide by 10. 20% → double the 10%. 5% → half of 10%. 25% → quarter.

240% = 2 × 20%. 15% = 10% + 5%.

✅ Answer:

	100	200	50	80	10	35	287
25%	25	50	12.5	20	2.5	8.75	71.75
10%	10	20	5	8	1	3.5	28.7
20%	20	40	10	16	2	7	57.4
5%	5	10	2.5	4	0.5	1.75	14.35
40%	40	80	20	32	4	14	114.8
15%	15	30	7.5	12	1.5	5.25	43.05

💡 Tip: 20% + 5% = 25% of the same value, because \(\tfrac{20}{100}y + \tfrac{5}{100}y = \tfrac{25}{100}y\).

📖 Textbook Page 8

📖 Page 8Math Talk

How would you mentally find 75%, 90%, 70%, 55% of a value?

175% = 50% + 25% (half + quarter)

290% = 100% − 10%

370% = 50% + 20%

455% = 50% + 5%

✅ Answer: Break the percentage into easy parts: 50%, 25%, 10%, 5%, 1%. Example: 90% of 80 = 80 − 8 = 72.

📖 Page 8FDP table

Complete the Fraction and Decimal for 50%, 100%, 25%, 75%, 10%, 1%, 5%, 43%. Which decimal gives 10%?

1Percentage ÷ 100 = decimal. Simplify the fraction.

✅ Answer:

Per cent	Fraction	Decimal
50%	1/2	0.5
100%	1	1.0
25%	1/4	0.25
75%	3/4	0.75
10%	1/10	0.1
1%	1/100	0.01
5%	1/20	0.05
43%	43/100	0.43

To find 10%, multiply by 0.1.

📖 Page 8Example 3

Maximum marks are 75. Zubin needs 80% for an A grade. What is the minimum score?

1Fraction: \(\tfrac{80}{100} \times 75 = \tfrac{4}{5} \times 75 = 60\)

2Decimal: 0.8 × 75 = 60

3Proportion: 80 : 100 :: ? : 75 → 60

✅ Answer: 60 marks

📖 Textbook Page 9

📖 Page 9Example 4

Millet : water = 2 : 7 for kanji. What percentage is millet? How much millet in 500 mL?

1Millet is 2 parts out of 2 + 7 = 9 → \(\tfrac{2}{9}\).

2Estimate: 20% of 9 = 1.8, 25% of 9 = 2.25 → between 20% and 25%.

3\(\tfrac{2}{9} \times 100 \approx 22.22\%\); water ≈ 77.78%.

4500 mL → \(\tfrac{2}{9} \times 500 \approx 111.1\) mL

✅ Answer: ≈ 22.22% millet; about 111 mL in 500 mL.

⚠️ Common mistake: Using \(\tfrac{2}{7}\) — the whole is 2 + 7 = 9 parts.

📖 Textbook Page 10

📖 Page 10Example 5

A cyclist completes 40% of the Delhi–Agra journey = 92 km. How much is left?

140% = 92 km → 20% = 46 km

2Left = 60% = 3 × 46 = 138 km

3(Total = 92 × 100 ÷ 40 = 230 km; 230 − 92 = 138.)

Each 20% block is 92 ÷ 2 = 46 km. Total 5 × 46 = 230 km.

✅ Answer: 138 km

📖 Textbook Page 11

📖 Page 11Example 6

Kishanlal's daily target is ₹5000. Find the percentage of target for sales of ₹2000, ₹3500, ₹5000, ₹6000, ₹7800 and ₹9550.

1% of target = sales ÷ 5000 × 100

22000 → 40% (60% short); 3500 → 70% (30% short)

35000 → 100%; 6000 → 120% (20% more)

47800 → 156%; 9550 → 191%

Sales above the target give a percentage greater than 100.

✅ Answer: 40%, 70%, 100%, 120%, 156%, 191%

📖 Textbook Page 12

📖 Page 12Think

On Day 7 he achieved 150% and on Day 8, 210% of his target. Find the sales.

1150% of 5000 = 1.5 × 5000

2210% of 5000 = 2.1 × 5000

✅ Answer: Day 7: ₹7,500; Day 8: ₹10,500

📖 Page 12Table

Write each as a fraction and a decimal and mark it on a number line: 90%, 110%, 200%, 250%, 15%, 173%, 358%, 28.9%, 305%.

1Divide by 100 for the decimal; simplify the fraction.

Approximate positions: 100% = 1 whole, 200% = 2 wholes, and so on.

✅ Answer:

Per cent	Fraction	Decimal
90%	9/10	0.9
110%	11/10	1.1
200%	2	2.0
250%	5/2	2.5
15%	3/20	0.15
173%	173/100	1.73
358%	179/50	3.58
28.9%	289/1000	0.289
305%	61/20	3.05

📖 Page 12Example 7

Last year 260 kg of wheat; this year 650 kg. This year is what percentage of last year?

1\(\tfrac{650}{260} \times 100 = 250\)

✅ Answer: 250% (2.5 times)

📖 Page 12Figure it Out · Q1

Find the missing numbers in the bar models. (ii) 10 equal parts, whole = 90, ? covers 6 parts. (iii) 4 equal parts, whole = 140, ? covers 3 parts.

1(i) worked: 5 parts → each 20%; 60 is 4 parts of 75 = 80%.

2(ii) 10 parts → each part 10%. Left: ? = 10%. Right: 6 parts = 60% of 90 = 54.

3(iii) 4 parts → each 25%. Left: ? = 25%. Right: 3 parts = 75% of 140 = 105.

(ii) Each part is 9; six parts = 54.

✅ Answer: (ii) 10% and 54 (iii) 25% and 105

📖 Textbook Page 13

📖 Page 13Figure it Out · Q2

Find and draw bar models: (i) 25% of 160 (ii) 16% of 250 (iii) 62% of 360 (iv) 140% of 40 (v) 1% of 1 hour (vi) 7% of 10 kg

1(i) ¼ × 160 = 40

2(ii) 0.16 × 250 = 40

3(iii) 0.62 × 360 = 223.2

4(iv) 1.4 × 40 = 56 (bar longer than the whole!)

5(v) 1% of 60 min = 0.6 min = 36 seconds

6(vi) 0.07 × 10 kg = 0.7 kg = 700 g

(i) 160 split into 4 parts of 40.

✅ Answer: 40, 40, 223.2, 56, 36 s, 700 g

💡 Tip: 16% of 250 = 250% of 16 = 40! (See Q7 on page 29.)

📖 Page 13Figure it Out · Q3

Surya made 60 mL of paint; red is \(\tfrac{3}{4}\) of it. How much red?

1\(\tfrac{3}{4} \times 60 = 45\)

✅ Answer: 45 mL

📖 Page 13Figure it Out · Q4

Use >, <, =: (i) 50% of 510 __ 50% of 515 (ii) 37% of 148 __ 73% of 148 (iii) 29% of 43 __ 92% of 110 (iv) 30% of 40 __ 40% of 50 (v) 45% of 200 __ 10% of 490 (vi) 30% of 80 __ 24% of 64

1(i) same %, smaller number → <

2(ii) same number, smaller % → <

3(iii) smaller % of a smaller number → < (12.47 vs 101.2)

4(iv) 12 vs 20 → <

5(v) 90 vs 49 → >

6(vi) 24 vs 15.36 → >

✅ Answer: (i) < (ii) < (iii) < (iv) < (v) > (vi) >

📖 Page 13Figure it Out · Q5

(i) 30% of k is 70: find 60%, 90%, 120% of k. (ii) 100% of m is 215: find 10%, 1%, 6%. (iii) 90% of n is 270: find 9%, 18%, 100%.

1(i) 60% = 2 × 30% = 140; 90% = 3 × 30% = 210; 120% = 4 × 30% = 280

2(ii) 10% = 21.5; 1% = 2.15; 6% = 6 × 2.15 = 12.9

3(iii) 9% = 270 ÷ 10 = 27; 18% = 54; 10% = 30 so 100% = 300

4(iv) Example: 25% of p is 40 → 50% is 80, 75% is 120, 100% is 160.

✅ Answer: (i) 140, 210, 280 (ii) 21.5, 2.15, 12.9 (iii) 27, 54, 300

📖 Page 13Figure it Out · Q6

(i) 3 is __% of 300 (ii) __ is 40% of 4 (iii) 40 is 80% of __

1(i) \(\tfrac{3}{300} \times 100 = 1\)

2(ii) 0.4 × 4 = 1.6

3(iii) 80% → 40, so 10% → 5, 100% → 50

✅ Answer: (i) 1% (ii) 1.6 (iii) 50

📖 Page 13Figure it Out · Q7

Is 10% of a day longer than 1% of a week?

110% of 24 h = 2.4 h = 144 min

21 week = 168 h; 1% = 1.68 h = 100.8 min

✅ Answer: Yes, 10% of a day (144 min) is longer than 1% of a week (about 101 min).

📖 Page 13Figure it Out · Q8

Each day the bull is given n + 1 units and eats n units (day 1: 1 of 2 … day 99: 99 of 100). Write as percentages. What do you notice?

1Day 1: \(\tfrac{1}{2}\) = 50%

2Day 2: \(\tfrac{2}{3}\) ≈ 66.67%

3Day 3: \(\tfrac{3}{4}\) = 75%; Day 4: 80%; Day 9: 90%

4Day 99: \(\tfrac{99}{100}\) = 99%

✅ Answer: The percentage keeps increasing and gets closer and closer to 100%, but never reaches it.

📖 Textbook Page 14

📖 Page 14Figure it Out · Q9

Workers pick 20% of a plantation in 18 days. How long for the whole plantation?

120% → 18 days; 100% is 5 times 20%.

25 × 18 = 90 days

✅ Answer: 90 days — assuming the same number of workers work at the same speed (otherwise the time would change).

📖 Page 14Figure it Out · Q10

Warm-up : play : cool-down = 10% : 80% : 10% of a 90-minute session. Find each time.

110% of 90 = 9

280% of 90 = 72

✅ Answer: Warm-up 9 min, play 72 min, cool-down 9 min

📖 Page 14Figure it Out · Q11

About 90% of the world's population lives in the Northern Hemisphere. Estimate the number using this year's population.

1The world population is a little over 8 billion (about 8.2–8.3 billion; the textbook uses 8.2 billion for 2025).

290% of 8.2 billion = 0.9 × 8.2 ≈ 7.4 billion

✅ Answer: About 7.4 billion people (check this year's figure and recalculate).

📖 Page 14Figure it Out · Q12

Halwa for 4: rava 40%, sugar 40%, ghee 20%. (i) Proportions for 8 people? (ii) Amounts in 2 kg?

1(i) Proportions don't change with the number of people — only amounts change.

2(ii) Rava 40% of 2000 g = 800 g; sugar 800 g; ghee 20% = 400 g.

✅ Answer: (i) Still 40% : 40% : 20% (ii) Rava 800 g, sugar 800 g, ghee 400 g

⚠️ Common mistake: Doubling the percentages for 8 people. Percentages stay the same; the quantities double.

📖 Page 14Example 1

Eesha scored 42/50 in English and 70/80 in Science. Where did she do better?

1English \(\tfrac{42}{50} \times 100 = 84\%\)

2Science \(\tfrac{70}{80} \times 100 = 87.5\%\)

✅ Answer: Science (87.5% > 84%).

💡 Tip: Different maximum marks? Convert both to percentages.

📖 Textbook Page 15

📖 Page 15Example 2 (KYC)

DEF (150 g): sugar 99 g, milk solids 30 g, badam 12 g, chemicals 9 g. Zacni (400 g): sugar 272 g, milk solids 64 g, badam 40 g, chemicals 24 g. Which has more badam? Fewer chemicals?

1Divide by the total weight and × 100.

2DEF: 66%, 20%, 8%, 6% (total 100%)

3Zacni: 68%, 16%, 10%, 6% (total 100%)

✅ Answer:

	Sugar	Milk solids	Badam	Food chemicals
DEF	66%	20%	8%	6%
Zacni	68%	16%	10%	6%

Zacni has a larger share of badam (10%). Both have the same share of food chemicals (6%). Both are mostly sugar!

📖 Page 15Increase / decrease

Tomatoes went from ₹30 to ₹42; theatre footfall fell from 160 to 100. Find the percentage change.

1% change = change ÷ ORIGINAL × 100

2Tomatoes: \(\tfrac{12}{30} \times 100 = 40\%\) increase

3Footfall: \(\tfrac{60}{160} \times 100 = 37.5\%\) decrease

✅ Answer: 40% increase; 37.5% decrease

⚠️ Common mistake: Dividing by the new value instead of the original value.

📖 Textbook Page 16

📖 Page 16Example 3

Do these mean the same? (i) 1991 population is 165% of 1961. (ii) It increased by 65% from 1961 to 1991.

1(i) q = 1.65p

2(ii) q = p + 0.65p = 1.65p

✅ Answer: Yes. Both say the 1991 population is 1.65 times that of 1961.

📖 Textbook Page 17

📖 Page 17Example 4

Kishanlal buys a sweater at ₹300 and sells at ₹430. Find his profit %. Also find the profit % of the wholesaler (CP ₹253, SP ₹300) and manufacturer (CP ₹230, SP ₹253).

1Profit % = profit ÷ CP × 100

2Kishanlal: \(\tfrac{130}{300} \times 100 \approx 43.3\%\)

3Wholesaler: \(\tfrac{47}{253} \times 100 \approx 18.6\%\)

4Manufacturer: \(\tfrac{23}{230} \times 100 = 10\%\)

One seller's selling price becomes the next seller's cost price.

✅ Answer: 43.3%, 18.6%, 10%

📖 Page 17Think

Shambhavi buys notebooks at ₹36 and adds a 20% margin. Selling price? She sells crayons at ₹50 with a 25% margin. Cost price?

1Notebook SP = 120% of 36 = 1.2 × 36 = 43.20

2Crayons: 125% of CP = 50 → CP = 50 ÷ 1.25 = 40

✅ Answer: Notebook ₹43.20; crayon box cost ₹40

⚠️ Common mistake: Taking 25% of ₹50 off (₹37.50). The 25% is of the cost price, not of ₹50.

📖 Page 17Example 5

Raghu bought rice at ₹35/kg and sold 10 kg for ₹300. Loss %? Would the loss % per kg be the same?

1CP = 350, loss = 50 → \(\tfrac{50}{350} \times 100 \approx 14.29\%\)

2Per kg: CP 35, SP 30, loss 5 → \(\tfrac{5}{35} \times 100\) — the same.

✅ Answer: ≈ 14.29% loss; yes, the same per kg (a percentage does not depend on the quantity).

📖 Textbook Page 18

📖 Page 18Example 6

A vase costs ₹2650; it is sold at an 18% loss. Selling price?

1SP = 82% of 2650 = 0.82 × 2650 = 2173

2(or loss = 0.18 × 2650 = 477; 2650 − 477)

✅ Answer: ₹2,173

📖 Page 18Think

Snehal sells strawberries at ₹80/kg with a 12% loss. Cost price?

1SP = 88% of CP

2CP = 80 ÷ 0.88 ≈ 90.91

✅ Answer: About ₹90.91 per kg

⚠️ Common mistake: Adding 12% of 80 (= ₹89.60). The 12% is of the cost price.

📖 Page 18Think

A cooker (MRP ₹1800) has a 35% discount. SP? If CP is ₹900, profit %?

1SP = 65% of 1800 = 1170

2Profit = 270 → \(\tfrac{270}{900} \times 100 = 30\%\)

✅ Answer: SP ₹1,170; profit 30%

📖 Page 18Gross & net profit

Manisha buys fertiliser at ₹500 and sells at ₹750. Profit % on cost and on sales? Monthly: sales ₹1,50,000, cost ₹1,00,000, expenses ₹5,000 — net profit %?

1On cost: \(\tfrac{250}{500} \times 100 = 50\%\)

2On sales: \(\tfrac{250}{750} \times 100 \approx 33.33\%\)

3Net profit = 50000 − 5000 = 45000 → \(\tfrac{45000}{150000} \times 100 = 30\%\)

✅ Answer: 50% on cost, 33.33% on sales; net profit 30% of revenue

📖 Textbook Page 19

📖 Page 19Taxes

Check the bill: 3 CFL bulbs at ₹150, CGST 9%, SGST 9%, total ₹531.

1Subtotal 3 × 150 = 450

2CGST 9% of 450 = 40.50; SGST 9% = 40.50

3Total = 450 + 81 = 531 (= 118% of 450)

✅ Answer: The bill is correct.

📖 Page 19Figure it Out · Q1

A geometry box is bought for ₹75 and sold for ₹110. Profit margin on cost?

1Profit = 35

2\(\tfrac{35}{75} \times 100 \approx 46.67\)

✅ Answer: ≈ 46.67%

📖 Page 19Figure it Out · Q2

Materials for a chair cost ₹475; I want a 50% margin. Selling price?

1150% of 475 = 1.5 × 475

✅ Answer: ₹712.50

📖 Page 19Figure it Out · Q3

Revenue ₹2.5 crore, profit margin 25%. Total expenditure?

1A profit margin of a company is usually taken on revenue: profit = 25% of 2.5 crore = ₹0.625 crore.

2Expenditure = 2.5 − 0.625 = ₹1.875 crore.

3(If the 25% were on cost: cost × 1.25 = 2.5 → ₹2 crore.)

✅ Answer: ₹1.875 crore (₹1,87,50,000)

📖 Page 19Figure it Out · Q4

25% discount on a ₹300 shirt. What does Anwar pay?

175% of 300

✅ Answer: ₹225

📖 Page 19Figure it Out · Q5

Petrol: ₹60 (2015) → ₹100 (2025). Percentage increase?

1Increase 40 on the original 60

2\(\tfrac{40}{60} \times 100 \approx 66.67\%\)

✅ Answer: (iv) 66.66%

⚠️ Common mistake: Choosing 40% — that divides by 100, not by the original 60.

📖 Textbook Page 20

📖 Page 20Figure it Out · Q3 (p20)

Samson paid ₹4,40,000 after a 15% discount. Original price?

185% of original = 440000

2Original = 440000 ÷ 0.85 ≈ 517647

✅ Answer: About ₹5,17,647

📖 Page 20Figure it Out · Q4 (p20)

1600 people voted; the winner got 500. What percent? Minimum number of candidates?

1\(\tfrac{500}{1600} \times 100 = 31.25\%\)

2The other 1100 votes went to others, and each must have fewer than 500.

32 others can have at most 499 + 499 = 998 < 1100. 3 others can (e.g. 400 + 400 + 300).

✅ Answer: 31.25%; at least 4 candidates.

📖 Page 20Figure it Out · Q5 (p20)

Rice: ₹38 (2024) → ₹42 (2025). Inflation rate?

1\(\tfrac{4}{38} \times 100 \approx 10.53\)

✅ Answer: ≈ 10.5%

📖 Page 20Figure it Out · Q6 (p20)

A number increased by 20% becomes 90. Find it.

1120% of x = 90

2x = 90 ÷ 1.2 = 75

✅ Answer: 75

⚠️ Common mistake: Taking 20% off 90 (= 72). Check: 72 + 20% = 86.4, not 90.

📖 Page 20Figure it Out · Q7 (p20)

Two buffaloes are sold at ₹80,000 each: 5% profit on one, 10% loss on the other. Overall?

1CP₁ = 80000 ÷ 1.05 ≈ 76,190.48

2CP₂ = 80000 ÷ 0.90 ≈ 88,888.89

3Total CP ≈ 1,65,079.37; total SP = 1,60,000

4Loss ≈ ₹5,079.37 → \(\tfrac{5079.37}{165079.37} \times 100 \approx 3.08\%\)

✅ Answer: Overall loss of about ₹5,079 (≈ 3.08%).

⚠️ Common mistake: Saying 10% − 5% = 5% loss. The two cost prices are different.

📖 Page 20Figure it Out · Q8 (p20)

Elephants increased by 5% from p. The population now is?

1p + 5% of p = p × 1.05

✅ Answer: (iv) p × 1.05

📖 Page 20Figure it Out · Q9 (p20)

'Demand has fallen by 85%.' Which statements mean the same?

1Now = old − 85% of old = 15% of old.

✅ Answer: (iii) The demand now is 15% of the demand a decade ago.

📖 Textbook Page 21

📖 Page 21Interest

₹6000 at 10% p.a. What is the amount after 1 year? After 3 years with interest paid out (no compounding) and with compounding?

11 year: 6000 × 1.1 = 6600

2No compounding: ₹600 each year → 6000 + 3 × 600 = 7800

3Compounding: 6000 → 6600 → 7260 → 7986

✅ Answer: ₹6600; ₹7800 without compounding; ₹7986 with compounding.

Compounding	Interest added	Amount
Year 1	600	6600
Year 2	660	7260
Year 3	726	7986

📖 Textbook Page 22

📖 Page 22Example 8

What percent of the deposit is received in each option?

17800 ÷ 6000 × 100 = 130% (= 6000 × 1.3)

27986 ÷ 6000 × 100 = 133.1% (= 6000 × 1.1³)

✅ Answer: 130% (gain 30%) and 133.1% (gain 33.1%).

📖 Page 22Figure it Out · Q1

₹20,000 at 10% p.a. for 2 years — with and without compounding.

1Without: 20000 + 2 × 2000 = 24000

2With: 20000 × 1.1 × 1.1 = 24200

✅ Answer: ₹24,000 vs ₹24,200 (₹200 more)

📖 Textbook Page 23

📖 Page 23Figure it Out · Q2

₹20,000 at 5% p.a. for 4 years — with and without compounding.

1Without: 20000 + 4 × 1000 = 24000

2With: \(20000 \times 1.05^4 = 20000 \times 1.21550625 \approx 24310.13\)

✅ Answer: ₹24,000 vs about ₹24,310

📖 Page 23Figure it Out · Q3

What do you notice in Q1 and Q2?

1Without compounding both give ₹24,000 (10% × 2 = 5% × 4 = 20%).

2With compounding the 4-year, 5% deposit gives more (₹24,310 > ₹24,200): more terms → more interest on interest.

✅ Answer: Same simple interest, but compounding over more terms grows more.

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📖 Page 24Math Talk

Formula for the total INTEREST earned (principal p, rate r, t terms)?

1No compounding: interest = prt

2Compounding: interest = \(p(1+r)^t - p\)

✅ Answer: prt and \(p(1+r)^t - p\)

📖 Page 24Figure it Out · Q4

Jasmine invests p for 4 years at 6% p.a., no compounding. Which expression gives the total amount?

1Interest each year = p × 0.06; for 4 years = p × 0.06 × 4.

2Total = p + (p × 0.06 × 4) = 1.24p.

3(i)–(vi) are not equal to 1.24p (e.g. p × 1.06 × 4 = 4.24p).

✅ Answer: Only (vii) p + (p × 0.06 × 4)

📖 Page 24Figure it Out · Q5

₹50,000 at 7% p.a. for 3 years. Interest without compounding? How much more with compounding?

1Simple: 50000 × 0.07 × 3 = 10500

2Compound amount: \(50000 \times 1.07^3 = 50000 \times 1.225043 \approx 61252.15\) → interest ≈ 11252.15

3Difference ≈ 752.15

✅ Answer: ₹10,500; about ₹752 more with compounding

📖 Page 24Figure it Out · Q6

Giridhar: ₹12,500 at 12% for 3 years, simple. Raghava: ₹12,500 at 10% for 3 years, compounded. Who pays more interest?

1Giridhar: 12500 × 0.12 × 3 = 4500

2Raghava: 12500 × 1.331 − 12500 = 4137.50

✅ Answer: Giridhar pays ₹362.50 more.

📖 Page 24Figure it Out · Q7

₹1000 at 10% p.a. — how long to double with and without compounding?

1Without: ₹100 every year → ₹2000 after 10 years.

2With: 1.1⁷ ≈ 1.95, 1.1⁸ ≈ 2.14 → it doubles during the 8th year (about 7.3 years).

₹1000 at 10% p.a.: compounding crosses ₹2000 during year 8; simple interest reaches it at year 10.

✅ Answer: About 7–8 years with compounding vs 10 years without. Yes — compounding grows exponentially (multiplying), simple interest grows linearly (adding the same amount).

📖 Page 24Figure it Out · Q8

A city of 1.5 crore grows 3% a year. Population after 3 years?

1\(1.5 \times 1.03^3 = 1.5 \times 1.092727\)

✅ Answer: ≈ 1.639 crore (about 1,63,90,900)

📖 Page 24Figure it Out · Q9

Bacteria: 5,06,000 growing 2.5% per hour. Count after 2 hours?

1\(506000 \times 1.025^2 = 506000 \times 1.050625\)

✅ Answer: ≈ 5,31,616

📖 Textbook Page 25

📖 Page 25Example 10

A TV bought for ₹21,000 depreciates 5% in a year. Value after 1 year?

195% of 21000 = 0.95 × 21000

✅ Answer: ₹19,950

📖 Page 25Example 11

A village of 1250 falls 10% every decade. Population after 3 decades?

1\(1250 \times 0.9^3 = 1250 \times 0.729 = 911.25\)

✅ Answer: About 910 people.

⚠️ Common mistake: Subtracting 30% at once (= 875). Each decade's 10% is of a smaller number.

📖 Page 25Would you rather?

Option A: deposit ₹100, get ₹300. Option B: deposit ₹1000, get ₹1500. Percentage gain? Which would you choose?

1A: gain 200 → 200%

2B: gain 500 → 50%

✅ Answer: A has the bigger percentage (200%) but B gives more money (₹500). If you can afford ₹1000, B is better — compare amounts, not just percentages.

📖 Textbook Page 26

📖 Page 26Think

20% discount or ₹50 off (above ₹150). Which is better for ₹180, ₹225, ₹300?

120% of 180 = 36 → ₹50 off is better

220% of 225 = 45 → ₹50 off is better

320% of 300 = 60 → 20% is better

4They are equal at ₹250.

✅ Answer: ₹180: ₹50 off; ₹225: ₹50 off; ₹300: 20% off

📖 Page 26Example 12

Cakely: 30% + 20% off; Cakify: 50% off. Which is cheaper for a ₹200 cake?

1Cakely: 200 → 140 (30% off) → 112 (20% off 140)

2Cakify: 50% off → 100

330% + 20% = 0.7 × 0.8 = 0.56 → only 44% off.

✅ Answer: Cakify (₹100 against ₹112).

⚠️ Common mistake: Adding the discounts: 30% + 20% is NOT 50% off when applied one after another.

📖 Textbook Page 27

📖 Page 27Example 13

Surbhi adds a 50% margin, then gives 50% off. (i) Any loss? (ii) If she sold goods for ₹12,000, the loss and loss %? (iii) What discount would give no loss?

1Cost x → marked 1.5x → after 50% off 0.75x.

2(i) Yes — 25% loss.

3(ii) 0.75x = 12000 → x = 16000 → loss ₹4000 (25%).

4(iii) 1.5x(1 − d) = x → \(d = \tfrac{1}{3}\) = 33⅓%

50% up and then 50% down does NOT bring you back: 1.5 × 0.5 = 0.75 (a 25% loss).

✅ Answer: (i) 25% loss (ii) ₹4,000, 25% (iii) 33⅓% discount

📖 Page 27Try This

Ariba has 120% of Arun's marbles. What can Arun say?

1Ariba = 1.2 × Arun → Arun = Ariba ÷ 1.2 = \(\tfrac{5}{6}\) of Ariba

2\(\tfrac{5}{6}\) ≈ 83.33%

✅ Answer: "I have about 83.33% of Ariba's marbles" (16⅔% fewer — not 20% fewer).

📖 Textbook Page 28

📖 Page 28Figure it Out · Q1

Bengaluru 2025 is 250% of 2000 (50 lakh). Population in 2025?

12.5 × 50 lakh

✅ Answer: 125 lakh = 1.25 crore

📖 Page 28Figure it Out · Q2

World 8.2 billion. Match: Germany 83 million, India 1.46 billion, Bangladesh 175 million, USA 347 million with 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1%.

11% of 8.2 billion = 82 million.

2Germany 83 million ≈ 1%

3India 1460 ÷ 8200 ≈ 17.8% → 18%

4Bangladesh 175 ÷ 8200 ≈ 2.1% → 2%

5USA 347 ÷ 8200 ≈ 4.2% → about 4%

✅ Answer: Germany 1%, India 18%, Bangladesh 2%, USA ≈ 4%. (4% is not printed among the options; the nearest boxes are 2% and 8%. 4.2% is the correct value.)

📖 Page 28Figure it Out · Q3

Phone ₹8,250 + 18% GST. Which give the final price?

1Final = 8250 + 18% of 8250 = 8250 × 1.18 = 9735

✅ Answer: (v) 8250 × 1.18 and (vi) 8250 + 8250 × 0.18 (= ₹9,735)

📖 Page 28Figure it Out · Q4

Mice: +5%, −2%, −3% in three months from p. Which statements are true?

1Each month multiply: p × 1.05 × 0.98 × 0.97

21.05 × 0.98 × 0.97 = 0.99813 → slightly less than p

✅ Answer: (ii) and (vi) are true.

⚠️ Common mistake: Thinking +5 − 2 − 3 = 0 so the population is back to p.

📖 Page 28Figure it Out · Q5

35% margin, then 30% discount on the selling price. Profit or loss?

1Cost x → 1.35x → 0.7 × 1.35x = 0.945x

✅ Answer: Loss of 5.5% (he gets 94.5% of his cost).

📖 Page 28Figure it Out · Q6

What percentage of the area is region E?

1Big square 8 × 8 = 64 small squares.

2E is half of a 4 × 4 square = 8.

3\(\tfrac{8}{64} \times 100 = 12.5\)

E is half of a 4 × 4 square.

✅ Answer: 12.5%

📖 Textbook Page 29

📖 Page 29Figure it Out · Q7

5% of 40 and 40% of 5; 25% of 12 and 12% of 25; 15% of 60 and 60% of 15. What do you notice? Prove it.

12 and 2; 3 and 3; 9 and 9.

2x% of y = \(\tfrac{x}{100} \times y = \tfrac{xy}{100}\) = \(\tfrac{y}{100} \times x\) = y% of x

✅ Answer: x% of y always equals y% of x. (Useful trick: 8% of 50 = 50% of 8 = 4.)

📖 Page 29Figure it Out · Q8

40% of students are Grade 8; 60% of them are girls. (i) % Grade 8 girls? (ii) If 160 students, how many Grade 8 girls?

1(i) 60% of 40% = 0.6 × 0.4 = 0.24 = 24%

2(ii) 24% of 160 = 38.4

60% of 40% = 0.6 × 0.4 = 0.24 = 24% of all students.

✅ Answer: (i) 24% (ii) 38.4 — about 38. (The book's numbers do not give a whole number; with exact counts it would be 38 or 39.)

📖 Page 29Figure it Out · Q9

The selling price of 3 pencils = the cost of 5 pencils. Profit or loss %?

1Let CP of 1 pencil = ₹1. SP of 3 = ₹5 → SP of 1 = \(\tfrac{5}{3}\)

2Profit = \(\tfrac{2}{3}\) on 1 → \(\tfrac{2}{3} \times 100 \approx 66.67\%\)

✅ Answer: Profit of 66⅔%

📖 Page 29Figure it Out · Q10

Bus fares rose 3% and then 4%. Overall increase?

11.03 × 1.04 = 1.0712

✅ Answer: 7.12% (not 7%)

📖 Page 29Figure it Out · Q11

Length increases by 10% and the area stays the same. By what % does the breadth decrease?

1New length = 1.1L; area L × B = 1.1L × B′ → B′ = \(\tfrac{B}{1.1} = \tfrac{10}{11}B\)

2Decrease = \(\tfrac{1}{11}\) = 9.09%

✅ Answer: \(9\tfrac{1}{11}\%\) ≈ 9.09%

⚠️ Common mistake: Saying 10%. Check: 1.1 × 0.9 = 0.99, so area would drop.

📖 Page 29Figure it Out · Q12

A 65 g chips packet: potato 70%, oil 24%, salt 3%, spices 3%. Find each weight.

170% of 65 = 45.5

224% of 65 = 15.6

33% of 65 = 1.95 (salt and spices each)

✅ Answer: Potato 45.5 g, oil 15.6 g, salt 1.95 g, spices 1.95 g (total 65 g)

📖 Page 29Figure it Out · Q13

Shop A: buy 1 get 1; Shop B: buy 2 get 1; Shop C: buy 3 get 1. Item ₹100. (i) Effective price (ii) % discount (iii) Best for 4 items?

1(i) A: ₹200 for 2 → ₹50; B: ₹200 for 3 → ₹66.67; C: ₹300 for 4 → ₹75. Cheapest: A, B, C.

2(ii) Free ÷ total: A \(\tfrac{1}{2}\) = 50%; B \(\tfrac{1}{3}\) ≈ 33.3%; C \(\tfrac{1}{4}\) = 25%

3(iii) 4 items: A → pay for 2 = ₹200; B → 3 for ₹200 + 1 for ₹100 = ₹300; C → pay for 3 = ₹300.

✅ Answer: Shop A (₹200 for 4 items).

📖 Textbook Page 30

📖 Page 30Figure it Out · Q14

100 people, 99% left-handed. How many left-handed people must leave to make it 98%?

1Right-handed = 1 person, and stays.

2At 98% left-handed, the 1 right-handed person is 2% of the room → room = 50 people.

3100 − 50 = 50 must leave.

✅ Answer: 50 people! (Check: 49 out of 50 = 98%.)

⚠️ Common mistake: Guessing 1 — one leaving gives 98/99 ≈ 98.99%.

📖 Page 30Figure it Out · Q15

From the graph (computer use by age and gender), which statements are valid?

Redrawn from the textbook graph (ability to use a computer, 2023). For Children the book shows one value, 4%.

1(i) Twenties have the highest bars (26%, 37%) → valid

2(ii) Female bar is shorter than male in every age group → valid

3(iii) The graph shows percentages, not how many people → NOT valid

4(iv) Thirties: 14% and 25% — neither is more than 25% → NOT valid

5(v) Seniors: 2% and 4% — less than 10% → valid

6(vi) Twenties: 26% and 37%, not 50% → NOT valid

✅ Answer: Valid: (i), (ii), (v).

📖 Textbook Page 32

📖 Page 32Puzzle: Peaceful Knights

Place 8 knights on a chessboard so that no knight attacks another.

1A knight moves 2 squares one way and 1 square the other way.

2So a knight never attacks a square in its own row (it always changes row).

3Put all 8 knights in one row — done!

4Another idea: a knight always jumps to a square of the OTHER colour, so knights placed only on white squares never attack each other (you could even fit 32!).

One answer: all 8 knights on one row. A knight always jumps 2 rows or 1 row, so it can never land on a square in its own row.

✅ Answer: Put all 8 on one row (or all on squares of the same colour).

🌍 Where do we use this?

• Marks, discounts, GST bills, food labels, bank interest, loans, population growth, depreciation.

🚀 Link to higher classes

• Class 9–10: compound interest, growth and decay. Class 11: exponential functions, statistics.

⬆ Top 🏠 Class 8 Hub

🔑 keytoenjoylearningmaths.blogspot.com · Solutions written in our own words, based on NCERT Ganita Prakash Class 8 (Part 2)