Case-Based MCQs (Grade 8): Exponents

Grade 8 • Case-Based MCQs Exponents & Counting

Power Play: Exponential Growth, Counting & Estimation

Six real-life passages with MCQs testing understanding of doubling, powers, geometric growth, counting by products, and estimation. Includes collapsible answers with explanations.

Contents:

• Case 1 — The Magical Folding Paper
• Case 2 — The King’s Diamonds
• Case 3 — The Magical Lotus Pond
• Case 4 — Dresses, Caps & Shoes
• Case 5 — Safe Passwords
• Case 6 — Tulābhāra (Coins & Weight)

Case 1 — The Magical Folding Paper

A sheet of paper has an initial thickness of 0.001 cm. When you fold it once, its thickness doubles. This doubling continues with every fold (exponential growth).

After 10 folds: 1.024 cm (≈ 1 cm)

After 17 folds: ≈ 131 cm (≈ 1.31 m)

After 26 folds: ≈ 670 m

After 30 folds: ≈ 10.7 km (typical airplane altitude)

After 46 folds: > 700,000 km (Earth–Moon scale)

General rule: after n folds, thickness t(n) = 0.001 × 2n cm.

Q1. The thickness after 1 fold is

1. 0.001 cm
2. 0.002 cm
3. 0.01 cm
4. 0.1 cm

Show Answer & Explanation

Answer: (b) 0.002 cm. One fold doubles the thickness.

Q2. After 10 folds, the thickness is about

1. 1 cm
2. 10 cm
3. 100 cm
4. 0.1 cm

Show Answer & Explanation

Answer: (a) 1 cm. Exactly 1.024 cm since 0.001 × 2¹⁰ = 1.024 cm.

Q3. 131 cm (after 17 folds) is closest to

1. length of a pencil
2. height of a chair
3. height of a person
4. height of a door

Show Answer & Explanation

Answer: (c) height of a person (~1.3 m).

Q4. After 26 folds ≈ 670 m. Which is closest?

1. Eiffel Tower (330 m)
2. Burj Khalifa (830 m)
3. Qutub Minar (73 m)
4. Empire State Building (443 m)

Show Answer & Explanation

Answer: (b) Burj Khalifa (830 m) is closest among options.

Q5. 10.7 km (after 30 folds) is comparable to

1. Mariana Trench depth
2. Height at which airplanes fly
3. Height of Mount Everest
4. Depth of a swimming pool

Show Answer & Explanation

Answer: (b) airplane cruising altitude (~10–12 km).

Q6. After 20 folds, thickness is approximately

1. 1 m
2. 10 m
3. 100 m
4. 1 km

Show Answer & Explanation

Answer: (b) 10 m. 0.001 × 2²⁰ = 1048.576 cm ≈ 10.49 m.

Q7. The thickness increases after every 3 folds by

1. 2 times
2. 4 times
3. 6 times
4. 8 times

Show Answer & Explanation

Answer: (d) 8 times, since 2³ = 8.

Q8. Exponential growth here means

1. increase by a fixed number
2. double every step
3. decrease every step
4. remain constant

Show Answer & Explanation

Answer: (b) doubles every fold.

Q9. After 46 folds the thickness reaches about 700,000 km. This can reach

1. the Sun
2. the Moon
3. the Mariana Trench
4. Mount Everest

Show Answer & Explanation

Answer: (b) the Moon (Earth–Moon distance scale).

Q10. This growth pattern follows

1. Arithmetic progression
2. Linear growth
3. Geometric progression
4. Subtraction pattern

Show Answer & Explanation

Answer: (c) Geometric progression (common ratio 2).

Case 2 — The King’s Diamonds

A king has 3 daughters. Each daughter has 3 baskets. Each basket has 3 keys. Each key opens 3 rooms. Continuing the multiplicative pattern yields seven factors of 3 overall.

Total diamonds = 3⁷ = 2187 (since 3⁴ = 81 rooms and 3 more layers: 81 × 27 = 2187).

Q1. Total baskets

1. 3
2. 6
3. 9
4. 27

Show Answer & Explanation

Answer: (c) 9, since 3 daughters × 3 baskets.

Q2. Total keys

1. 9
2. 18
3. 27
4. 81

Show Answer & Explanation

Answer: (c) 27, because 9 baskets × 3 keys each.

Q3. Total rooms

1. 27
2. 34
3. 81
4. 243

Show Answer & Explanation

Answer: (c) 81 = 3⁴.

Q4. Total number of diamonds

1. 243
2. 729
3. 2187
4. 6561

Show Answer & Explanation

Answer: (c) 2187 = 3⁷. Seven successive ×3 stages.

Q5. Diamonds can be expressed as

1. 3⁴
2. 3⁵
3. 3⁶
4. 3⁷

Show Answer & Explanation

Answer: (d) 3⁷.

Q6. If 3⁴ = 81 (rooms), to get 3⁷ multiply 81 by

1. 3
2. 9
3. 27
4. 243

Show Answer & Explanation

Answer: (c) 27 = 3³.

Q7. The concept illustrated is

1. Arithmetic progression
2. Powers & exponents
3. Fractions
4. Decimals

Show Answer & Explanation

Answer: (b) Powers & exponents.

Case 3 — The Magical Lotus Pond

The number of lotuses doubles daily. The pond is fully covered on day 30.

Fully covered = 2³⁰ lotuses; half covered = 2²⁹. Doubling means day 29 must be half of day 30.

Damayanti moves flowers: after 4 days of doubling then 4 days of tripling: 1 × 2⁴ × 3⁴ = 1296. Reversing the order gives the same (commutative property).

Q1. If full on day 30, half-full on

1. 28th
2. 29th
3. 30th
4. 31st

Show Answer & Explanation

Answer: (b) 29th day.

Q2. If full = 2³⁰, then day 29 lotuses =

1. 2²⁸
2. 2²⁹
3. 2³⁰
4. 2³¹

Show Answer & Explanation

Answer: (b) 2²⁹.

Q3. Day 29 count equals

1. (2³⁰)/2
2. 2 × 2³⁰
3. 2³⁰ + 2
4. (2³⁰)/4

Show Answer & Explanation

Answer: (a) (2³⁰)/2 = 2²⁹.

Q4. After 4 days in doubling pond, lotuses

1. 2² = 4
2. 2³ = 8
3. 2⁴ = 16
4. 2⁵ = 32

Show Answer & Explanation

Answer: (c) 16.

Q5. After moving to tripling pond for 4 more days

1. 16 × 3²
2. 16 × 3³
3. 16 × 3⁴
4. 16 × 3⁵

Show Answer & Explanation

Answer: (c) 16 × 3⁴.

Q6. Final lotuses after both transfers

1. 648
2. 1296
3. 2187
4. 4096

Show Answer & Explanation

Answer: (b) 1296 = 16 × 81.

Q7. Reversing the order (tripling then doubling) yields

1. Different
2. Same
3. Half
4. Double

Show Answer & Explanation

Answer: (b) Same. 1 × 3⁴ × 2⁴ = 1 × 2⁴ × 3⁴.

Case 4 — Dresses, Caps & Shoes

Estu has 4 dresses and 3 caps. Total outfits = 4 × 3 = 12. Roxie has 7 dresses, 2 hats, 3 pairs of shoes. Total outfits = 7 × 2 × 3 = 42.

Q1. Estu’s outfits

1. 7
2. 12
3. 16
4. 24

Show Answer & Explanation

Answer: (b) 12 = 4 × 3.

Q2. Expression for Estu’s combinations

1. 4 + 3
2. 4 × 3
3. 4³
4. 3⁴

Show Answer & Explanation

Answer: (b) 4 × 3 (multiplication principle).

Q3. Roxie’s outfits

1. 12
2. 24
3. 42
4. 84

Show Answer & Explanation

Answer: (c) 42 = 7 × 2 × 3.

Q4. Correct expression for Roxie

1. 7 + 2 + 3
2. 7 × 2 × 3
3. 7² × 3
4. 7 × 2³

Show Answer & Explanation

Answer: (b) 7 × 2 × 3.

Q5. If Estu had 5 dresses and 4 caps, total outfits

1. 5 × 4 = 20
2. 5 + 4 = 9
3. 5⁴
4. 4⁵

Show Answer & Explanation

Answer: (a) 20 (product of choices).

Q6. The concept used is

1. Addition principle
2. Multiplication principle
3. Division
4. Subtraction

Show Answer & Explanation

Answer: (b) Multiplication principle (independent choices multiply).

Case 5 — Safe Passwords

Digit locks: A 2-digit lock has 10 × 10 = 10² = 100 combinations. A 3-digit lock has 10³ = 1000. A 5-digit lock has 10⁵ = 100000 combinations.

Letter lock: With 6 slots using A–Z (26 letters), combinations = 26⁶.

Q1. 2-digit lock combinations

1. 10
2. 20
3. 10² = 100
4. 200

Show Answer & Explanation

Answer: (c) 10² = 100.

Q2. 3-digit lock combinations

1. 10² = 100
2. 10³ = 1000
3. 999
4. 900

Show Answer & Explanation

Answer: (b) 10³ = 1000.

Q3. 5-digit lock combinations (all tried)

1. 9999
2. 50000
3. 10⁵ = 100000
4. 5¹⁰

Show Answer & Explanation

Answer: (c) 10⁵ = 100000.

Q4. 6-slot letter lock (A–Z) combinations

1. 6⁶
2. 26⁶
3. 10⁶
4. 26 × 6

Show Answer & Explanation

Answer: (b) 26⁶.

Q5. Which are combinatorial codes?

1. Only PIN codes
2. Only mobile numbers
3. (i) and (ii)
4. (i), (ii), and (iii)

Items: (i) PIN codes in India, (ii) Mobile numbers, (iii) Vehicle registration numbers.

Show Answer & Explanation

Answer: (d) All three use structured digit/letter choices.

Case 6 — Tulābhāra (Coins & Weight)

Nanjundappa donates jaggery equal to Roxie’s weight and wheat equal to Estu’s weight.

• Worth of jaggery (₹) = Roxie’s weight (kg) × cost per kg
• Worth of wheat (₹) = Estu’s weight (kg) × cost per kg

Assume: Roxie = 45 kg, jaggery = ₹70/kg → 45 × 70 = ₹3150. Estu = 50 kg, wheat = ₹50/kg → 50 × 50 = ₹2500.

Tulābhāra is an old practice (symbol of bhakti and community support).

Q1. Worth of jaggery

1. ₹2500
2. ₹3000
3. ₹3150
4. ₹3500

Show Answer & Explanation

Answer: (c) ₹3150 = 45 × 70.

Q2. Worth of wheat

1. ₹2000
2. ₹2500
3. ₹2700
4. ₹3000

Show Answer & Explanation

Answer: (b) ₹2500 = 50 × 50.

Q3. If a ₹1 coin weighs 5 g, coins equal to Roxie’s 45 kg

1. 450
2. 4500
3. 9000
4. 45 × 200

Show Answer & Explanation

Answer: (c) 9000. 45 kg = 45000 g; 45000 ÷ 5 = 9000.

Q4. Tulābhāra symbolizes

1. Wealth accumulation
2. Bhakti (surrender & gratitude)
3. Business
4. Entertainment

Show Answer & Explanation

Answer: (b) Bhakti (and community support).

Q5. Estu 50 kg, ₹2 coins of 5 g each — number of coins

1. 5000
2. 10000
3. 25000
4. 20000

Show Answer & Explanation

Answer: (b) 10000 coins. 50 kg = 50000 g; 50000 ÷ 5 = 10000 coins (value would be ₹20000).

Teacher Tips

• Estimate vs exact: Compute a few exact powers, then compare with real-world scales (m, km, currency).
• Structure trees: Draw branching trees (e.g., King’s Diamonds) to “count layers” of multiplication.
• Multiply choices: Reinforce that independent choices multiply (outfits, passwords).
• Vocabulary: Exponential growth, geometric progression, multiplication principle, estimation.