Quantitative Aptitude PYQs

Shortcut Trick

Total percentage interest = Rate × Time = 8% × 4 = 32%.

According to the question, 32% of the Principal = ₹1,320.

1% of the Principal = 1320 ÷ 32 = 41.25.

100% (Original Sum) = 41.25 × 100 = ₹4,125.

∴ The correct answer is ₹4,125.

Alternate Method

Given:

Simple Interest (SI) = ₹1,320

Rate of Interest (R) = 8% per annum

Time (T) = 4 years

Formula Used:

Principal (P) = (Simple Interest × 100) ÷ (Rate × Time)

Calculations:

⇒ P = (1320 × 100) ÷ (8 × 4)

⇒ P = 132000 ÷ 32

⇒ P = 4125

∴ The correct answer is ₹4,125.

Additional Information

Amount in Simple Interest

The total Amount (A) is the sum of the Principal and the Simple Interest: A = P + SI.

Compound Interest (CI)

Unlike Simple Interest, Compound Interest is calculated on the Principal plus the interest accumulated over previous periods: A = P(1 + r/100)ⁿ.

Difference between SI and CI

For a period of 1 year at the same rate, Simple Interest and Compound Interest are always equal, provided the interest is compounded annually.

Rule of 72

A quick way to estimate how long it takes to double your money with Compound Interest: Time ≈ 72 ÷ Interest Rate.

Shortcut Trick

Ratio of Profit = Ratio of Investments (since time is the same).

Profit Ratio of P : Q = 3000 : 600 = 5 : 1.

Total units = 5 + 1 = 6 units → ₹54,846.

P's share (5 units) = (54,846 ÷ 6) × 5 = 9,141 × 5 = ₹45,705.

∴ The correct answer is ₹45,705.

Alternate Method

Given:

Investment of P = ₹3,000

Investment of Q = ₹600

Total Profit at the end of 1 year = ₹54,846

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Profit Ratio (P : Q) = 3000 : 600

⇒ Profit Ratio (P : Q) = 5 : 1

⇒ Sum of ratios = 5 + 1 = 6

⇒ P's share in profit = (5 / 6) × 54,846

⇒ P's share = 5 × 9,141

⇒ P's share = ₹45,705

∴ The correct answer is ₹45,705.

Additional Information

Basic Rule of Partnership

Profit is distributed in the ratio of the product of Capital and Time Period (P = C × T).

Variable Time Investment

If P invests for T₁ months and Q for T₂ months, their profit ratio is (C₁ × T₁) : (C₂ × T₂).

Active vs. Sleeping Partner

An active partner may receive an additional salary or commission from the total profit before the remaining profit is shared according to the investment ratio.

Shortcut Trick

Total Area = Area₁ + Area₂ + Area₃

Area₃ = 1/2 × 12 × 38 = 6 × 38 = 228 cm²

Summing all areas ⇒ 24 + 30 + 228 = 282 cm²

∴ The correct answer is 282 cm².

Alternate Method

Given:

Area of Triangle 1 = 24 cm²

Area of Triangle 2 = 30 cm²

Base of Triangle 3 = 12 cm, Height of Triangle 3 = 38 cm

Formula Used:

Area of a Triangle = (1/2) × base × height

Total Area of Pentagon = Area of Triangle 1 + Area of Triangle 2 + Area of Triangle 3

Calculations:

⇒ Area of first triangle = 24 cm²

⇒ Area of second triangle = 30 cm²

⇒ Area of third triangle = (1/2) × 12 × 38

⇒ Area of third triangle = 6 × 38 = 228 cm²

⇒ Total Area of Pentagon = 24 + 30 + 228

⇒ Total Area of Pentagon = 54 + 228 = 282 cm²

∴ The correct answer is 282 cm².

Additional Information

Polygon Triangulation

Any convex polygon with n sides can be divided into (n − 2) triangles by drawing all possible diagonals from a single vertex.

Additive Property of Area

The total area of a geometric figure is the sum of the areas of its non-overlapping constituent parts.

Sum of Interior Angles

The sum of the interior angles of a pentagon (5 sides) is (5 − 2) × 180° = 540°.

Shortcut Trick

To find the net percentage change for successive increases of a% and b%, use the formula: a + b + (a × b)/100

Here, a = 9% and b = 13%

Total percentage increase = 9 + 13 + (9 × 13)/100

Total percentage increase = 22 + 117/100 = 22 + 1.17 = 23.17%

∴ The correct answer is 23.17%.

Alternate Method

Given:

First increase = 9%

Second increase = 13%

Calculations:

⇒ Let the initial price of the article be 100.

⇒ Price after first increase (9%) = 100 + (9% of 100) = 109.

⇒ Price after second increase (13%) = 109 + (13% of 109).

⇒ Price after second increase = 109 + (13 × 109)/100 = 109 + 14.17 = 123.17.

⇒ Net increase in price = 123.17 − 100 = 23.17.

⇒ Percentage increase = (23.17 / 100) × 100 = 23.17%.

∴ The correct answer is 23.17%.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the effective change is [x + y + (xy/100)]%.

Successive Discounts

For two successive discounts of d₁% and d₂%, the single equivalent discount is [d₁ + d₂ − (d₁ × d₂)/100]%.

Multiplying Factor Method

Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100), where r₁ and r₂ are the percentage increase rates.

Shortcut Trick

The number N must divide (228 − 18) = 210.

According to the remainder rule, the divisor N must be greater than the remainder 18 (N > 18).

List factors of 210 in descending order: 210, 105, 70, 42, 35, 30, 21, 15...

The largest two-digit factor that satisfies N > 18 is 70.

∴ The correct answer is 70.

Alternate Method

Given: Dividend = 228, Remainder = 18.

Formula Used: Dividend = (Divisor × Quotient) + Remainder

⇒ Let the divisor be N and the quotient be q.

⇒ 228 = (N × q) + 18

⇒ N × q = 228 − 18 = 210

⇒ This implies N is a factor of 210.

⇒ Factors of 210 = 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.

⇒ Two-digit factors of 210 are 10, 14, 15, 21, 30, 35, 42, 70.

⇒ Since the remainder is 18, N must be greater than 18.

⇒ Possible values for N = 21, 30, 35, 42, 70.

⇒ The greatest two-digit value of N is 70.

∴ The correct answer is 70.

Additional Information

Euclid's Division Lemma

For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

Divisor and Remainder Relationship

The divisor is always strictly greater than the remainder. If a number divides another with a remainder r, then the divisor must be a factor of (Dividend − r).

Finding Factors

To find all factors of a number, check divisibility by integers from 1 up to the square root of that number.

Shortcut Trick

Combine all terms of x and constants in the Angle Sum Property equation:

⇒ (3x + 6x + 4x) + (−7 − 14 + 6) = 180

⇒ 13x − 15 = 180 ⇒ 13x = 195 ⇒ x = 15

Required sum ∠A + ∠C = (3x − 7) + (4x + 6) = 7x − 1

⇒ 7(15) − 1 = 105 − 1 = 104°

∴ The correct answer is 104°.

Alternate Method

Given:

∠A = (3x − 7)°

∠B = 2∠A = 2(3x − 7) = (6x − 14)°

∠C = (4x + 6)°

According to the Angle Sum Property of a triangle:

⇒ ∠A + ∠B + ∠C = 180°

⇒ (3x − 7) + (6x − 14) + (4x + 6) = 180

⇒ 13x − 15 = 180

⇒ 13x = 180 + 15

⇒ 13x = 195

⇒ x = 195 ÷ 13

⇒ x = 15

Now, substitute the value of x to find the angles:

⇒ ∠A = 3(15) − 7 = 45 − 7 = 38°

⇒ ∠C = 4(15) + 6 = 60 + 6 = 66°

Sum of ∠A and ∠C:

⇒ ∠A + ∠C = 38° + 66° = 104°

∴ The correct answer is 104°.

Additional Information

Angle Sum Property

The sum of the interior angles of any triangle is always 180°. This is a fundamental property in Euclidean geometry.

Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. For instance, Exterior ∠ACD = ∠A + ∠B.

Types of Triangles by Angles

An Acute triangle has all angles < 90°, a Right triangle has one angle = 90°, and an Obtuse triangle has one angle > 90°.

Shortcut Trick

In both cases, the relative speed (S_F − S_S) remains the same.

Time to pass total distance (L_F + L_S) = 36 sec.

Time for passenger in faster train to pass length L_S = 12 sec.

⇒ (L_F + L_S) : L_S = 36 : 12 = 3 : 1

⇒ L_F : L_S = (3 − 1) : 1 = 2 : 1

∴ The correct answer is 2 : 1.

Alternate Method

Given:

Time taken to pass slower train completely = 36 sec.

Time taken by passenger to pass slower train = 12 sec.

Let length of faster train be L_F and slower train be L_S.

Let speeds be S_F and S_S. Relative speed V_R = S_F − S_S.

Formula: Distance = Relative Speed × Time

⇒ Case 1: L_F + L_S = (S_F − S_S) × 36 --- (1)

⇒ Case 2: L_S = (S_F − S_S) × 12 --- (2)

⇒ From (2), (S_F − S_S) = L_S / 12

⇒ Substitute in (1): L_F + L_S = (L_S / 12) × 36

⇒ L_F + L_S = 3 × L_S

⇒ L_F = 3L_S − L_S = 2L_S

⇒ L_F / L_S = 2 / 1

∴ The correct answer is 2 : 1.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is (S₁ − S₂). If they move in opposite directions, relative speed is (S₁ + S₂).

Train Crossing a Point Object

When a train passes a passenger, a pole, or a stationary man, the distance covered is exactly equal to the length of the train (or the length of the train being observed).

Unit Conversion

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

Shortcut Trick

Assume total work as LCM(12, 6) = 12 units.

Efficiency: A = 1 unit/day, B = 2 units/day.

Combined efficiency of A + B = 3 units/day.

Work done in 2 days = 3 × 2 = 6 units.

Remaining work = 12 − 6 = 6 units, completed by B in 6 ÷ 2 = 3 days.

Total time taken = 2 + 3 = 5 days.

∴ The correct answer is 5 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 6 days

Formula Used:

Efficiency = Total Work ÷ Time

⇒ Total Work = 12 units.

⇒ Work done by A and B in 2 days = (1 + 2) × 2 = 6 units.

⇒ Remaining work to be completed by B = 12 − 6 = 6 units.

⇒ Time taken by B to finish remaining work = 6 ÷ 2 = 3 days.

⇒ Total time for the whole work = Work together + Work by B alone = 2 + 3 = 5 days.

∴ The correct answer is 5 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work. If the ratio of time is x:y, the ratio of efficiency is y:x.

Work and Wages

Wages are generally distributed in the ratio of the work done by each individual, which is the product of their efficiency and the time they worked.

Combined Work Formula

If A can do a work in x days and B in y days, then together they can do it in (x × y) ÷ (x + y) days.

Shortcut Trick

Total amount to be divided after deductions = 5278 − (67 + 61 + 68) = ₹5,082.

Sum of ratio parts = 8 + 20 + 5 = 33 units.

Value of 1 unit = 5082 ÷ 33 = 154.

X's share after deduction = 8 units = 8 × 154 = ₹1,232.

∴ The correct answer is ₹1,232.

Alternate Method

Given:

Total amount = ₹5,278

Deductions from X, Y, and Z = ₹67, ₹61, and ₹68

Ratio after deduction = 8 : 20 : 5

Calculations:

⇒ Sum of deductions = 67 + 61 + 68 = ₹196

⇒ Amount remaining after deduction = 5278 − 196 = ₹5,082

⇒ Let the parts be 8k, 20k, and 5k

⇒ 8k + 20k + 5k = 5082

⇒ 33k = 5082

⇒ k = 5082 ÷ 33 = 154

⇒ X's share after deduction = 8k = 8 × 154 = 1232

∴ The correct answer is ₹1,232.

Additional Information

Ratio and Proportion

To divide a quantity S in the ratio a : b : c, the parts are [a/(a+b+c)] × S, [b/(a+b+c)] × S, and [c/(a+b+c)] × S respectively.

Deduction Rule

When fixed amounts are subtracted from the original shares to get a new ratio, always subtract the sum of those amounts from the total capital first.

Unitary Method in Ratios

Finding the value of '1 unit' by dividing the total quantity by the sum of ratio terms is the fastest way to calculate individual components.

Shortcut Trick

The third proportional (x) to two numbers a and b is calculated using the direct formula: x = b² ÷ a.

Given: a = 8 and b = 48.2.

Substituting the values: x = (48.2 × 48.2) ÷ 8

x = 2323.24 ÷ 8 = 290.405.

∴ The correct answer is 290.405.

Alternate Method

Given:

First number (a) = 8

Second number (b) = 48.2

Formula Used:

Three numbers a, b, and x are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third: a : b = b : x.

Calculations:

⇒ 8 ÷ 48.2 = 48.2 ÷ x ⇒ 8 × x = 48.2 × 48.2 ⇒ 8x = 2323.24 ⇒ x = 2323.24 ÷ 8 ⇒ x = 290.405

∴ The correct answer is 290.405.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b). It is the middle term when three numbers are in continued proportion.

Fourth Proportional

For four numbers a, b, c, d to be in proportion (a : b :: c : d), the fourth proportional is given by d = (b × c) ÷ a.

Continued Proportion

Numbers a, b, c, d... are said to be in continued proportion if a/b = b/c = c/d and so on.

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.