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Concept used:

Follow BODMAS rule to solve this question, as per the order given below,

Calculation:

[(-251) × 21 × (-12)] ÷ ? = 63

⇒ [(-251) × 21 × (-12)]/63 = ?

⇒ (251 × 12)/3 = ?

⇒ 251 × 4 = ?

⇒ ? = 1004

∴ The value of ? is 1004

Concept used:

Calculation:

(36 + ? + 121)1/2​ + 26 = 39

⇒ (36 + ? + 121)1/2​ = 39 - 26

⇒ (36 + ? + 121) = 13²

⇒ (? + 157) = 169

⇒ ? = 169 - 157 = 12

Calculation:

[(130)2 ÷ 25 × 15] ÷ 30

⇒ $(\frac{130 × 130}{25} × 15)÷30$ = (130 × 26 × 3) ÷ 30

⇒ 13 × 26 = 338

New income = 10,000 + 20% of 10,000

⇒ Rs. 12,000

Expenses = Rs.1,000

New saving = Rs. (12,000 – 1,000)

⇒ Rs. 11,000

Expenses = Rs. 6,000

Saving = Rs. 3,000

Required percent = [(6,000/3,000)] × 100

⇒ 200%

Saving = Rs. 6000

Income = Rs. 12,000

Required percentage = [(6,000/12,000)] × 100

⇒ 50%

Shortcut Trick

Ratio of Amount (A) to Principal (P): 51840 / 25000 = 5184 / 2500 = 1296 / 625.

Rate (R) = 20% → Multiplying factor = (1 + 20/100) = 6/5.

Check power: (6/5)ⁿ = 1296/625 ⇒ (6/5)⁴ = 1296/625.

∴ The correct answer is 4 years.

Alternate Method

Given: Principal (P) = ₹25000, Amount (A) = ₹51840, Rate (R) = 20% p.a.

Formula Used: A = P(1 + R/100)ⁿ

⇒ 51840 = 25000(1 + 20/100)ⁿ

⇒ 51840 / 25000 = (120/100)ⁿ

⇒ 5184 / 2500 = (6/5)ⁿ

⇒ 1296 / 625 = (6/5)ⁿ

⇒ (6/5)⁴ = (6/5)ⁿ

⇒ n = 4

∴ The correct answer is 4 years.

Additional Information

Compound Interest Formula

The amount A is calculated as A = P(1 + r/n)^nt, where n is the number of times interest is compounded per year.

Effective Rate of Interest

For half-yearly compounding, the rate becomes R/2 and the time becomes 2n.

Rule of 72

A simplified way to determine how long an investment will take to double, given a fixed annual rate of interest: Time ≈ 72 / Rate.

Shortcut Trick

Ratio of Boat speed (B) to Current speed (C) = 9 : 2.

Upstream speed (U) = 9 − 2 = 7 units; Downstream speed (D) = 9 + 2 = 11 units.

Difference = (5 × 11) − (2 × 7) = 55 − 14 = 41 units.

41 units = 123 km ⇒ 1 unit = 3 kmph.

Boat speed = 9 units = 9 × 3 = 27 kmph.

∴ The correct answer is 27 kmph.

Alternate Method

Given:

Ratio of Still Water Speed (u) and Current Speed (v) = 9x : 2x.

Formula Used:

Downstream Speed = u + v | Upstream Speed = u − v | Distance = Speed × Time.

Calculations:

⇒ Downstream Speed = 9x + 2x = 11x

⇒ Upstream Speed = 9x − 2x = 7x

⇒ Distance Downstream (5 hrs) = 11x × 5 = 55x

⇒ Distance Upstream (2 hrs) = 7x × 2 = 14x

⇒ Difference: 55x − 14x = 123

⇒ 41x = 123

⇒ x = 3

⇒ Speed of boat in still water = 9x = 9 × 3 = 27 kmph

∴ The correct answer is 27 kmph.

Additional Information

Relative Speed in Streams

Downstream speed (D) is the sum of boat speed (u) and stream speed (v). Upstream speed (U) is the difference (u − v).

Calculating Component Speeds

Boat speed in still water u = (D + U) ÷ 2 and Stream speed v = (D − U) ÷ 2.

Average Speed

Average speed for a round trip (upstream and downstream) = (D × U) ÷ u = (u² − v²) ÷ u.

Shortcut Trick Loss = CP of 5 bananas Loss occurs when CP > SP. For 20 bananas: Loss = CP of 20 − SP of 20 Equating: CP of 5 = CP of 20 − SP of 20 ⇒ SP of 20 = CP of 15 Given SP of 20 = Rs.600, therefore CP of 15 = Rs.600 CP of 1 banana = 600 ÷ 15 = Rs.40 ∴ The correct answer is Rs.40.

Alternate Method Given: Number of bananas = 20 | Selling Price (SP) = Rs.600 | Loss = CP of 5 bananas. Formulas: Loss = CP − SP | Cost Price (CP) = Total Cost ÷ Quantity.

⇒ Let the CP of 1 banana be x. ⇒ CP of 20 bananas = 20x. ⇒ Loss = CP of 5 bananas = 5x. ⇒ SP of 20 bananas = CP − Loss = 20x − 5x = 15x. ⇒ According to the question, 15x = 600. ⇒ x = 600 ÷ 15 = 40. ∴ The correct answer is Rs.40.

Additional Information Loss in Terms of CP When loss is expressed as the Cost Price of 'n' articles, simply subtract 'n' from the total quantity to find the equivalent CP representing the SP. Profit in Terms of CP If there were a profit equal to CP of 'n' articles, SP = CP × (Total + n).

Shortcut Trick

Distance covered by Bus in (20 − 16) minutes = Distance covered by Man in 16 minutes.

Ratio of Time taken for same distance (Bus : Man) = 4 : 16 = 1 : 4.

Ratio of Speed (Bus : Man) = 4 : 1.

Since Bus speed is 40 km/hr (4 units), Man speed (1 unit) = 40 ÷ 4 = 10 km/hr.

∴ The correct answer is 10 km/hr.

Alternate Method

Given: Speed of Bus (S_b) = 40 km/hr, Time interval of buses (T₁) = 20 min, Meeting interval (T₂) = 16 min.

Formula Used: S_b × (T₁ − T₂) = S_m × T₂

⇒ Distance between two consecutive buses = Speed × Time = 40 × (20/60) km.

⇒ When man moves towards terminal, Relative Speed = (S_b + S_m).

⇒ Distance = Relative Speed × Meeting Time

⇒ 40 × 20 = (40 + S_m) × 16

⇒ 800 = 640 + 16 × S_m

⇒ 16 × S_m = 160

⇒ S_m = 10 km/hr.

∴ The correct answer is 10 km/hr.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂).

Constant Distance Rule

For a constant distance, Speed is inversely proportional to Time. If time ratio is a:b, then speed ratio is b:a.

Bus and Man logic

If the man moves away from the terminal, he would meet the bus at intervals longer than the original frequency (T₁ + Δt).