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Shortcut Trick

List the prime numbers between 5 and 30: 7, 11, 13, 17, 19, 23, and 29.

Total count of these prime numbers = 7.

Sum = 7 + 11 + 13 + 17 + 19 + 23 + 29 = 119.

Average = Sum ÷ Count = 119 ÷ 7 = 17.

∴ The correct answer is 17.

Alternate Method

Given: Prime numbers lying between 5 and 30.

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

Step-by-step Calculation:

⇒ Prime numbers between 5 and 30 are 7, 11, 13, 17, 19, 23, and 29.

⇒ Sum = 7 + 11 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 119

⇒ Total count of numbers (n) = 7

⇒ Average = 119 ÷ 7

⇒ Average = 17

∴ The correct answer is 17.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself. The smallest prime number is 2, which is also the only even prime.

Average of Numbers

The average (arithmetic mean) is calculated by dividing the sum of all values by the total number of values. For prime numbers, there is no shortcut formula due to their irregular distribution.

Prime Numbers up to 100

There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Shortcut Trick

Remaining work of (A+B+C) for 4 hr (6 − 2) is completed by (A+B) in 7 hr.

Efficiency ratio (A+B+C) : (A+B) = 7 : 4 (Inverse of time ratio 4 : 7).

Efficiency of Pipe C = Efficiency of (A+B+C) − Efficiency of (A+B) = 7 − 4 = 3 units.

Total Work = Efficiency of (A+B+C) × Total time = 7 × 6 = 42 units.

Time taken by C alone = Total Work ÷ Efficiency of C = 42 ÷ 3 = 14 hr.

∴ The correct answer is 14 hr.

Alternate Method

Given:

Time taken by Pipes A, B and C together = 6 hr

Part of tank filled by A, B and C in 2 hr = 2 ÷ 6 = 1/3 part

Remaining part = 1 − 1/3 = 2/3 part

Time taken by (A+B) to fill 2/3 part = 7 hr

Time taken by (A+B) to fill full tank = 7 × (3 ÷ 2) = 10.5 hr

⇒ Efficiency of Pipe C = Efficiency of (A+B+C) − Efficiency of (A+B)

⇒ Efficiency of C = 7 − 4 = 3 units/hr

⇒ Time taken by C alone to fill the tank = Total Work ÷ Efficiency of C

⇒ Time = 42 ÷ 3 = 14 hr

∴ The correct answer is 14 hr.

Additional Information

Pipes & Cisterns Efficiency

The time taken to fill a tank is inversely proportional to the efficiency of the pipe (Time = Work ÷ Efficiency).

Combined Work Formula

If pipe A fills in x hours and pipe B fills in y hours, their combined 1-hour work is (1/x + 1/y).

Part of Work Completed

Part of tank filled in 't' hours = (t ÷ Total time taken to fill the full tank).

Shortcut Trick

To construct a unique quadrilateral, we generally need 5 independent measurements.

Statement (iv) provides 4 sides + 1 diagonal = 5 elements, which is the standard requirement.

Statement (i) involves special quadrilaterals; a Rhombus and Square have fixed properties (diagonals bisect at 90°), allowing construction with only 2 diagonals.

Statements (ii) and (iii) provide only 4 elements each, which is insufficient for a general quadrilateral.

∴ The correct answer is ( i ) and ( iv ).

Alternate Method

Given: Various combinations of sides, angles, and diagonals for constructing a quadrilateral.

Property Used: A unique quadrilateral can be constructed if 5 independent parts are known. Special shapes like the square and rhombus require fewer measurements due to inherent symmetry and properties.

⇒ Condition (i): For a Rhombus, diagonals are perpendicular bisectors. Knowing d₁ and d₂ allows us to plot the vertices uniquely. For a Square, d₁ = d₂, and they are also perpendicular bisectors.

⇒ Condition (ii): 2 sides and 2 angles provide only 4 independent elements. This results in a family of quadrilaterals rather than a unique one.

⇒ Condition (iii): 3 sides and 1 angle provide only 4 independent elements, which is insufficient.

⇒ Condition (iv): 4 sides and 1 diagonal provide 5 independent elements. This allows the construction of two triangles (ABC and ADC) sharing a common base (AC), forming a unique quadrilateral.

∴ The correct answer is ( i ) and ( iv ).

Additional Information

Minimum elements for unique construction

A general quadrilateral requires 5 independent parts: 4 sides + 1 diagonal, 3 sides + 2 diagonals, or 3 sides + 2 included angles.

Properties of a Rhombus

All sides are equal and diagonals bisect each other at 90°. This internal symmetry reduces the number of external measurements needed for construction.

Properties of a Square

A square can be constructed uniquely if only one diagonal is given, because both diagonals are equal and intersect at right angles at their midpoints.

Triangulation Method

Every quadrilateral can be divided into two triangles by a diagonal. Constructing these two triangles using SSS or SAS congruency leads to the unique construction of the quadrilateral.

Shortcut Trick

Convert Discount % to fraction: 25% = 1/4.

If Marked Price is 4 units, Discount is 1 unit, so Net Price is 3 units.

4 units = ₹2,400 ⇒ 1 unit = ₹600.

3 units = 3 × 600 = ₹1,800.

∴ The correct answer is ₹1,800.

Alternate Method

Given:

Marked Price (MP) = ₹2,400

Discount Percentage = 25%

Formula Used:

Net Price = Marked Price × (100 − Discount%)/100

Calculations:

⇒ Discount Amount = 25% of 2,400

⇒ Discount Amount = (25/100) × 2,400 = ₹600

⇒ Net Price = MP − Discount Amount

⇒ Net Price = 2,400 − 600

⇒ Net Price = ₹1,800

∴ The correct answer is ₹1,800.

Additional Information

Marked Price (MP)

The price at which a product is listed or labeled by the manufacturer or wholesaler before any discount.

Selling Price (SP) / Net Price

The actual price the buyer pays after deducting the discount from the marked price: SP = MP − Discount.

Successive Discounts

When two discounts d₁% and d₂% are given, the effective discount is [d₁ + d₂ − (d₁ × d₂)/100]%.

Shortcut Trick

Use the ratio method: Effective SP/CP Ratio = (Faulty Weight Reading ÷ Actual Weight) × (Stated SP ÷ Actual CP)

Ratio = (1000 ÷ 900) × (150 ÷ 120)

Ratio = (10 ÷ 9) × (5 ÷ 4) = 50 ÷ 36 = 25/18

Actual Profit % = (25 − 18) ÷ 18 × 100 = 7/18 × 100 = 38(8/9)%

∴ The correct answer is 38(8)/(9)%.

Alternate Method

Given:

Actual Cost Price (CP) of 1000 g = ₹120

Faulty Selling Price (SP) of 1000 g = ₹150

Actual weight delivered = 900 g for every 1000 g reading.

Calculations:

⇒ Actual CP of the 900 g fruit sold = (120 ÷ 1000) × 900 = ₹108

⇒ Actual SP received (for the 1 kg reading) = ₹150

⇒ Profit Amount = SP − CP = 150 − 108 = ₹42

⇒ Actual Profit Percentage = (Profit ÷ Actual CP) × 100

⇒ Profit % = (42 ÷ 108) × 100

⇒ Profit % = (7 ÷ 18) × 100 = 700/18 = 350/9 = 38(8/9)%

∴ The correct answer is 38(8)/(9)%.

Additional Information

Profit Percentage Formula

Profit % = [(SP − CP) ÷ CP] × 100. Always ensure both CP and SP are calculated for the same quantity of goods.

Faulty Weight Gain

When a seller uses a weight of 'w' grams instead of 'W' grams, the weight gain factor is (W ÷ w). This acts as a successive multiplier to the price-based profit.

Successive Percentage Change

The net profit when there is both a markup (m%) and a weight error (e%) can be calculated using the combined ratio of SP/CP = (1 + m/100) × (True Weight ÷ Faulty Weight).

Shortcut Trick

Relative Speed in opposite direction = 40 + 60 = 100 km/h

Convert to m/s: 100 × 5/18 = 250/9 m/s

Total Distance = Sum of lengths = 100 + 150 = 250 m

Time = 250 ÷ (250/9) = 9 sec

∴ The correct answer is 9 sec.

Alternate Method

Given:

Length of first train (L₁) = 100 m

Length of second train (L₂) = 150 m

Speed of first train (S₁) = 40 km/h

Speed of second train (S₂) = 60 km/h

Formula Used:

Time = (Total Distance) ÷ (Relative Speed)

Calculations:

⇒ Total Distance = L₁ + L₂ = 100 + 150 = 250 m

⇒ Relative Speed = S₁ + S₂ = 40 + 60 = 100 km/h

⇒ Relative Speed in m/s = 100 × (5/18) m/s

⇒ Relative Speed = 500/18 = 250/9 m/s

⇒ Time Taken = Total Distance ÷ Relative Speed

⇒ Time Taken = 250 ÷ (250/9)

⇒ Time Taken = 250 × (9/250)

⇒ Time Taken = 9 sec

∴ The correct answer is 9 sec.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the difference of their individual speeds: |S₁ − S₂|.

Unit Conversion Shortcut

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

Crossing a Stationary Object

When a train crosses a pole or a man, the distance covered is equal to the length of the train. When it crosses a platform or bridge, the distance is the sum of their lengths.

Shortcut Trick

Use the successive percentage change formula: Net Change = [x + y + (xy ÷ 100)]%.

Here, x = −30 (decrease) and y = +50 (increase).

Net Change = −30 + 50 + [(−30 × 50) ÷ 100].

Net Change = 20 − 15 = 5%.

A positive result indicates an overall increase.

∴ The correct answer is 5% increase.

Alternate Method

Given:

Initial decrease = 30%

Subsequent increase = 50%

Assume the initial portfolio value is 100 units.

Step 1: Calculate value after the first quarter (30% decrease).

⇒ New Value = 100 − (30% of 100)

⇒ New Value = 100 − 30 = 70 units.

Step 2: Calculate value after the next quarter (50% increase).

⇒ Final Value = 70 + (50% of 70)

⇒ Final Value = 70 + 35 = 105 units.

Step 3: Calculate overall percentage change.

⇒ Net % Change = [(Final Value − Initial Value) ÷ Initial Value] × 100

⇒ Net % Change = [(105 − 100) ÷ 100] × 100 = 5%.

∴ The correct answer is 5% increase.

Additional Information

Successive Percentage Change Rule

The net effect of two successive changes of x% and y% is always [x + y + (xy ÷ 100)]%. This is widely used in compound interest and profit/loss calculations.

Multiplier Method

For a series of changes, the final value = Initial × Multiplier₁ × Multiplier₂. A 30% decrease has a multiplier of 0.70 and a 50% increase has a multiplier of 1.50.

Order Independence

The net percentage change is independent of the order. A 30% decrease followed by a 50% increase results in the same final value as a 50% increase followed by a 30% decrease.

Shortcut Trick

The relationship between the Volume (V) and Curved Surface Area (CSA) of a cylinder is: V = (CSA × r) ÷ 2

Given CSA = 616 cm² and radius (r) = 3.5 cm

V = (616 × 3.5) ÷ 2

V = 308 × 3.5 = 1078 cm³

∴ The correct answer is 1078 cm³.

Alternate Method

Given:

Radius (r) = 3.5 cm = 7/2 cm

Curved Surface Area (CSA) = 616 cm²

Formula Used:

Curved Surface Area of Cylinder = 2 × π × r × h

Volume of Cylinder = π × r² × h

Calculations:

⇒ CSA = 2 × (22/7) × 3.5 × h

⇒ 616 = 2 × (22/7) × (7/2) × h

⇒ 616 = 22 × h

⇒ h = 616 ÷ 22 = 28 cm

⇒ Volume = (22/7) × (3.5)² × 28

⇒ Volume = (22/7) × 3.5 × 3.5 × 28

⇒ Volume = 22 × 0.5 × 3.5 × 28

⇒ Volume = 11 × 3.5 × 28

⇒ Volume = 38.5 × 28 = 1078 cm³

∴ The correct answer is 1078 cm³.

Additional Information

Total Surface Area of Cylinder

The total surface area includes the curved surface and the areas of the two circular bases: TSA = 2 × π × r(r + h).

Volume Relation with Cone

If a cone and a cylinder have the same radius and height, the volume of the cylinder is three times the volume of the cone (V_cyl = 3 × V_cone).

Base Area of Cylinder

The area of the circular base is given by A = π × r². Thus, Volume = Base Area × Height.

Shortcut Trick

To find the Third Proportional (x) of two numbers a and b, use the formula: x = b² ÷ a.

Here, a = 6 and b = 12.

⇒ x = (12 × 12) ÷ 6

⇒ x = 144 ÷ 6 = 24.

∴ The correct answer is 24 chocolates.

Alternate Method

Given that 6, 12, and x are in continued proportion, the ratio between the first and second term is equal to the ratio between the second and third term.

Relationship used: Product of Extremes = Product of Means

⇒ 6 : 12 :: 12 : x

⇒ 6 ÷ 12 = 12 ÷ x

⇒ 6 × x = 12 × 12

⇒ 6x = 144

⇒ x = 144 ÷ 6

⇒ x = 24

∴ The correct answer is 24 chocolates.

Additional Information

Mean Proportional

The mean proportional (b) between two numbers a and c is given by the square root of their product: b = √(a × c).

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b = c : d), then d is the fourth proportional: d = (b × c) ÷ a.

Continued Proportion

Three quantities a, b, c are in continued proportion if a : b = b : c. This implies that the square of the middle term equals the product of the first and last terms (b² = ac).

Shortcut Trick

A reduction of 20% means the New Price is 80% of the Original Price.

New Price = 80% of 655

⇒ 0.8 × 655 = 524

∴ The correct answer is ₹524.

Alternate Method

Given:

Cost of the shirt (Original Price) = ₹655

Discount percentage during sale = 20%

Formula Used:

Discount Amount = (Discount % ÷ 100) × Original Price

New Price = Original Price − Discount Amount

Calculations:

⇒ Discount Amount = (20 ÷ 100) × 655

⇒ Discount Amount = 0.2 × 655 = ₹131

⇒ New Price = 655 − 131

⇒ New Price = 524

∴ The correct answer is ₹524.

Additional Information

Percentage Decrease

To find the final value after a percentage decrease, multiply the original value by (1 − r/100), where r is the rate of decrease.

Successive Discounts

If two discounts a% and b% are given, the effective discount is (a + b − ab/100)%.

Multiplying Factor

A 20% reduction corresponds to a multiplying factor of 0.80, whereas a 20% increase corresponds to 1.20.