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Given:

The percent of tigers in Z1 is 15%.

The percent of tigers in Z3 is 30%.

Calculation:

We know that,

100% ≡ 360º ( central angle of the pie chart is 360º)

Now, the central angle of the sectors for Z1 = 15% × 360º

The central angle of the sectors for Z3 = 30% × 360º

According to the question,

Z3 - Z1 =  360º × (30% -  15%)

⇒ 360º × (15/100) = 54º

∴ The difference in the central angle of the sectors for Z1 and Z3 is 54º.

Shortcut Trick

Initial average of 5 numbers = 38

New average of 4 numbers = 30 (Decrease of 8 per number)

Removed value = Old Average + (Change in Average × Number of remaining items)

Removed value = 38 + (8 × 4) = 38 + 32 = 70

∴ The correct answer is 70.

Alternate Method

Given:

Average of 5 numbers = 38

Average of 4 numbers = 30

Formula used:

Sum = Average × Number of observations

Calculations:

⇒ Sum of 5 numbers = 5 × 38 = 190

⇒ Sum of 4 numbers = 4 × 30 = 120

⇒ Removed number = Sum of 5 numbers − Sum of 4 numbers

⇒ Removed number = 190 − 120

⇒ Removed number = 70

∴ The correct answer is 70.

Additional Information

Average Concept

The average (arithmetic mean) is the sum of all values divided by the total number of values: Average = Total Sum ÷ Count.

Effect of removing a number

If the removed number is greater than the average, the new average will decrease. If the removed number is less than the average, the new average will increase.

Weighted Average

When combining two groups, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Given:

Radius of cylinder = R

Height of cylinder = 2R

Radius of hemisphere = R

Formula Used:

Volume of cylinder = π × radius² × height

Volume of hemisphere = (2/3) × π × radius³

Calculation:

⇒ Volume of cylinder = π × R² × 2R = 2πR³

⇒ Volume of hemisphere = (2/3) × π × R³ = (2πR³) ÷ 3

⇒ Ratio of volumes = Volume of cylinder ÷ Volume of hemisphere

⇒ Ratio = (2πR³) ÷ ((2πR³) ÷ 3)

⇒ Ratio = 3 : 1

∴ The correct answer is 3 ∶ 1.

Shortcut Trick

Total Area = Area of Rectangle + Area of 1 Full Circle (since 2 semicircles = 1 circle)

Area = (Length × Width) + πr²

Area = (14 × 7) + (3.14159 × 7²)

Area = 98 + 153.9379

Area ≈ 251.94 cm²

∴ The correct answer is 251.94 cm².

Alternate Method

Given:

Length of the rectangle (l) = 14 cm

Width of the rectangle (w) = 7 cm

Radius of each semicircle (r) = 7 cm

Formula Used:

Area of Rectangle = Length × Width

Area of Semicircle = (1/2) × πr²

⇒ Area of the rectangle = 14 × 7 = 98 cm²

⇒ Area of one semicircle = (1/2) × π × (7)²

⇒ Area of two semicircles = 2 × [(1/2) × π × 49]

⇒ Area of two semicircles = π × 49 ≈ 3.14159 × 49 ≈ 153.94 cm²

⇒ Total area of the figure = Area of rectangle + Area of two semicircles

⇒ Total Area = 98 + 153.94 = 251.94 cm²

∴ The correct answer is 251.94 cm².

Additional Information

Area of a Circle

The area of a full circle is calculated using the formula A = πr², where r is the radius.

Perimeter of the Combined Figure

The perimeter of this specific figure includes only the two widths of the rectangle and the two curved arcs: P = 2w + 2πr.

Properties of Semicircles

A semicircle is exactly half of a circle. Its area is (1/2)πr² and its arc length is πr.

Shortcut Trick

The train covers its own length in 12 seconds and (Length + 180 m) in 20 seconds.

The extra distance of 180 m is covered in the extra time: 20 − 12 = 8 seconds.

Speed of the train = 180 ÷ 8 = 22.5 m/s.

Length of the train = Speed × Time to cross the man = 22.5 × 12 = 270 m.

∴ The correct answer is 270 m.

Alternate Method

Given:

Length of platform = 180 m

Time taken to cross platform = 20 s

Time taken to cross man = 12 s

Formula Used:

Speed = Distance ÷ Time

⇒ Let the length of the train be L meters.

⇒ Speed of the train while crossing the man = L ÷ 12

⇒ Speed of the train while crossing the platform = (L + 180) ÷ 20

⇒ Equating the speeds: L ÷ 12 = (L + 180) ÷ 20

⇒ 20 × L = 12 × (L + 180)

⇒ 20L = 12L + 2160

⇒ 20L − 12L = 2160

⇒ 8L = 2160

⇒ L = 2160 ÷ 8

⇒ L = 270 m

∴ The correct answer is 270 m.

Additional Information

Crossing a Stationary Point

When a train crosses a point object like a pole or a man, the distance covered is exactly the length of the train.

Crossing a Stationary Object with Length

When a train crosses a platform or a bridge, the total distance covered is the sum of the train length and the object length.

Relative Speed

Relative speed in the same direction is S₁ − S₂, and in the opposite direction, it is S₁ + S₂.

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

The correct answer is Profit of 2.5%

Shortcut Trick

When the Profit on one price equals the Loss on another, the Cost Price (CP) is the average of the two selling prices.

CP = (2200 + 1800) ÷ 2 = ₹2000

New Selling Price (SP) = ₹2050

Profit = 2050 − 2000 = ₹50

Profit % = (50 ÷ 2000) × 100 = 2.5%

∴ The correct answer is Profit of 2.5%.

Alternate Method

Given:

SP₁ = ₹2200 | SP₂ = ₹1800 | New SP = ₹2050

Formula Used:

Profit = SP − CP | Loss = CP − SP | Profit % = (Profit ÷ CP) × 100

⇒ Let the Cost Price be x.

⇒ Profit = 2200 − x

⇒ Loss = x − 1800

⇒ 2200 − x = x − 1800

⇒ 2x = 4000 ⇒ x = 2000

⇒ New Profit = 2050 − 2000 = ₹50

⇒ Profit % = (50 ÷ 2000) × 100 = 2.5%

∴ The correct answer is Profit of 2.5%.

Additional Information

Marked Price (MP)

The price printed on the label of an item. Discount is always calculated on the Marked Price.

Successive Discount

If two discounts d₁% and d₂% are given, the Effective Discount = [d₁ + d₂ − (d₁ × d₂) ÷ 100]%.

Dishonest Dealer

Profit % = [Error ÷ (True Value − Error)] × 100, where error is the weight difference.

Shortcut Trick

Relative speed = 100 − 80 = 20 km/h.

Gap reduced in 90 seconds = 20 × (5/18) × 90 = 500 m.

Initial gap = 1 km = 1000 m.

Distance after 90 seconds = 1000 − 500 = 500 m.

∴ The correct answer is 500 m.

Alternate Method

Given: Initial distance = 1 km = 1000 m, Police speed = 100 km/h, Stolen car speed = 80 km/h, Time = 90 seconds.

Formula Used: Relative Speed (same direction) = u − v; Distance = Speed × Time.

⇒ Relative Speed = 100 − 80 = 20 km/h

⇒ Speed in m/s = 20 × (5/18) = 50/9 m/s

⇒ Distance covered in 90 seconds = (50/9) × 90 = 500 m

⇒ Final distance between them = 1000 − 500 = 500 m

∴ The correct answer is 500 m.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is the difference (u − v). In opposite directions, it is the sum (u + v).

Unit Conversion

To convert speed from km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Catching Time

The time taken to catch an object is calculated as: Time = Initial Distance ÷ Relative Speed.

Shortcut Trick

Profit Ratio = (Capital × Time) ratio

X : Y = (75,000 × 12) : (1,00,000 × 6 + 60,000 × 6)

X : Y = 9,00,000 : 9,60,000 = 15 : 16

Total units = 15 + 16 = 31 units

X's share = (15 / 31) × 1,05,000 ≈ 50,806.45

∴ The correct answer is ₹50,806.45.

Alternate Method

Given:

Investment of X = ₹75,000 for 12 months

Investment of Y = ₹1,00,000 for 6 months, then ₹60,000 for 6 months

Total Profit = ₹1,05,000

Formula Used:

Profit Share Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ X's total investment units = 75,000 × 12 = 9,00,000

⇒ Y's total investment units = (1,00,000 × 6) + (60,000 × 6)

⇒ Y's total investment units = 6,00,000 + 3,60,000 = 9,60,000

⇒ Ratio of Profit (X : Y) = 9,00,000 : 9,60,000

⇒ Ratio X : Y = 90 : 96 = 15 : 16

⇒ Total Profit units = 15 + 16 = 31 units

⇒ 31 units = ₹1,05,000

⇒ 1 unit = 1,05,000 ÷ 31 ≈ 3387.0967

⇒ X's share = 15 × 3387.0967 = 50,806.45

∴ The correct answer is ₹50,806.45.

Additional Information

Compound Partnership

When partners invest different amounts for different time periods, the profit is shared in the ratio of the product of their capital and time period.

Working vs Sleeping Partner

A working partner manages the business and may receive a salary or a fixed percentage of profit before the remaining profit is distributed in the investment ratio.

Ratio of Capitals

If the time period is the same for all partners, the profit is shared simply in the ratio of their initial investments (Capitals).

Calculation:

The total number of Goals Score by F5 in both the Matches = 12 + 18 = 30

The total number of Goals Score Made by F1 in both the  Matches = 24 + 18 = 42 

The ratio = (12 + 18)  : (24 + 18) = 5 : 7

Shortcut Trick

7% of the number = 21

1% of the number = 21 ÷ 7 = 3

Number (100%) = 3 × 100 = 300

Required value = Number − 25% of Number = 75% of Number

Required value = 75 × 3 = 225

∴ The correct answer is 225.

Alternate Method

Given: 7% of a number is 21.

Formula Used: Value = (Percentage ÷ 100) × Base Number

⇒ Let the number be x.

⇒ 7% of x = 21

⇒ (7 ÷ 100) × x = 21

⇒ x = (21 × 100) ÷ 7

⇒ x = 3 × 100 = 300

⇒ 25% of the number = (25 ÷ 100) × 300 = 75

⇒ Value = Number − 25% of Number

⇒ Value = 300 − 75 = 225

∴ The correct answer is 225.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, use standard fractions: 25% = 1/4, 50% = 1/2, 75% = 3/4, and 10% = 1/10.

Finding the Whole from a Part

If P% of a number is V, then the number is calculated as (V ÷ P) × 100.

Percentage Change

A number decreased by k% is equivalent to (100 − k)% of the original number.