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Given:

Ratio of upstream speed and speed of current = 7 : 1

Distance covered upstream in 5 hours = 70 km

Concept used:

Speed of Boat = Upstream speed + Speed of current

Downstream speed = Speed of boat + Speed of current

Calculation:

Upstream speed = (70/5) km/hr

⇒ 14 km/hr

Let the upstream speed and speed of current be 7x and x respectively

Now, 7x = 14 km/hr

⇒ x = 2 km/hr

Speed of boat = (14 + 2) km/hr

⇒ 16 km/hr

Downstream speed = (16 + 2) km/hr

⇒ 18 km/hr

Distance covered downstream in 5 hours = (18 × 5) km

⇒ 90 km

∴ Distance covered by boat downstream in 5 hours is 90 km

Shortcut Trick

Simple Interest for 3 years at 12.5% → 3 × 12.5% = 37.5%.

New Principal (P₂) = 8000 + 37.5% of 8000 = 8000 + 3000 = ₹11,000.

Compound Interest for 3 years at 20% → Multiplying factor = (1.2)³ = 1.728.

Final Amount = 11,000 × 1.728 = ₹19,008.

∴ The correct answer is Rs. 19008.

Alternate Method

Given: Initial Principal = Rs. 8000, SI Rate = 12.5%, Time = 3 years, CI Rate = 20%.

Formula Used: SI = (P × R × T)/100 | Amount_CI = P(1 + R/100)^T

⇒ Simple Interest = (8000 × 12.5 × 3) / 100

⇒ Simple Interest = 80 × 37.5 = Rs. 3000

⇒ Total Amount after SI (New Principal) = 8000 + 3000 = Rs. 11,000

⇒ Final Amount (Compound Interest) = 11000 × (1 + 20/100)³

⇒ Final Amount = 11000 × (1.2)³

⇒ Final Amount = 11000 × 1.728 = Rs. 19008

∴ The correct answer is Rs. 19008.

Additional Information

Simple Interest vs Compound Interest

Simple Interest is calculated only on the Principal, while Compound Interest is calculated on the Principal plus the accumulated interest of previous periods.

Effective Rate for 3 years CI

For a rate of 20%, the effective CI rate for 3 years is calculated as 3r.3r²r³ or using the formula (x+y+xy/100), which results in 72.8%.

Fractional Conversion

12.5% = 1/8 and 20% = 1/5. These fractions often simplify multiplication and division in interest-based problems.

Let the average weight of males be x kg

By the alligation method,

Now, 4/(x - 52) = 4/3

⇒ x - 52 = 3

⇒ x = 55

∴ The average weight of males will be 55 kg.

 

Alternate Method

Given:

The average weight of males and females in the town = 52 kg

The ratio of the number of males and females = 4 : 3

The average weight of females = 48 kg

Formula Used:

Average = (Sum of all observations)/(Number of observation)

Calculation:

Let the number of males and female be 4x and 3x respectively.

The average weight of females = 48 kg

⇒ (Sum of the weight of females)/(Total number of females) = 48

⇒ Sum of the weight of females = 48 × (3x)

⇒ Sum of the weight of females = 144x kg

Again, Let the sum of the weight of males be y kg

Now, The average weight of males and females in the town = 52 kg

⇒ (144x + y)/(4x + 3x) = 52

⇒ 144x + y = 52 × 7x

⇒ y = 364x - 144x

⇒ y = 220x

The sum of the weight of males = 220x kg

The average weight of males = (220x)/(4x)

⇒ 55 kg

∴ The average weight of males will be 55 kg.

Shortcut Trick

Remaining Mixture = 50 − 10 = 40 litres.

Ratio 4 : 1 ⇒ Total 5 units = 40 litres ⇒ 1 unit = 8 litres.

Milk = 4 × 8 = 32 litres; Water = 1 × 8 = 8 litres.

Final Milk = 32 + 3 = 35 litres; Final Water = 8 + 5 = 13 litres.

Final Ratio = 35 : 13.

∴ The correct answer is 35 : 13.

Alternate Method

Given: Total mixture = 50 litres, Ratio M:W = 4 : 1.

Formula: Quantity = Total × (Ratio Part / Total Parts).

⇒ Quantity of Milk removed = 10 × (4 / 5) = 8 litres.

⇒ Quantity of Water removed = 10 × (1 / 5) = 2 litres.

⇒ Initial Milk = 50 × (4 / 5) = 40 litres.

⇒ Initial Water = 50 × (1 / 5) = 10 litres.

⇒ New Milk = (40 − 8) + 3 = 35 litres.

⇒ New Water = (10 − 2) + 5 = 13 litres.

⇒ Final Ratio = 35 : 13.

∴ The correct answer is 35 : 13.

Additional Information

Mixture Withdrawal Property

When a portion of a mixture is removed, the ratio of components in the removed part and the remaining part stays the same as the original ratio.

Alligation Rule

Used to find the proportion in which two or more ingredients at a given price/concentration are mixed to produce a mixture of a desired price/concentration.

Replacement Formula

If x units are removed from V volume and replaced with water n times: Final Pure Liquid = Initial Liquid × (1 − x/V)ⁿ.

Shortcut Trick

Apply the Chain Rule formula: M₁D₁ = M₂D₂

30 men × 25 days = M₂ men × 15 days

⇒ M₂ = (30 × 25) ÷ 15 = 2 × 25 = 50 men

∴ The correct answer is 50 men.

Alternate Method

Given:

Initial men (M₁) = 30

Initial days (D₁) = 25

Required days (D₂) = 15

Formula Used:

Total Work = Men × Days

Calculations:

⇒ Total work to dig the pond = 30 × 25 = 750 units

⇒ Number of men required to finish 750 units in 15 days = Total Work ÷ Days

⇒ Required men = 750 ÷ 15

⇒ Required men = 50

∴ The correct answer is 50 men.

Additional Information

Work and Men Relationship

Men and days are inversely proportional. If work is constant, M₁ ÷ M₂ = D₂ ÷ D₁.

General Man-Day Formula

For varying work, use (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where H is hours and W is work.

Calculation:

The series follows the following pattern:

⇒ 6 × 1 – 1 = 5

⇒ 5 × 2 – 1 = 9

⇒ 9 × 3 – 1 = 26

⇒ 26 × 4 – 1 = 103

⇒ 103 × 5 – 1 = 514

∴ The value of ? is 103

Calculation:

The series follows following pattern:

⇒ 3 × 8 = 24

⇒ 24 × 4 = 96

⇒ 96 × 2 = 192

⇒ 192 × 1 = 192

⇒ 192 × 0.5 = 96

∴ The value of ? is None of these

Calculation:

The series follows the following pattern:

⇒ 8 + 12 = 20

⇒ 20 + 16 = 36

⇒ 36 + 20 = 56

⇒ 56 + 24 = 80

⇒ 80 + 28 = 108

∴ The value of ? is 108

Calculation:

The series follows following pattern:

⇒ 4 + 2² = 8

⇒ 8 + 3³ = 35

⇒ 35 + 4² = 51

⇒ 51 + 5³ = 176

⇒ 176 + 6² = 212

∴ The value of ? is 212

Calculation:

The series follows the following pattern:

⇒ 1005 – 5 = 1000

⇒ 1000 – 15 = 985

⇒ 985 – 25 = 960

⇒ 960 – 35 = 925

⇒ 925 – 45 = 880

∴ The value of ? is 880