Quantitative Aptitude PYQs

Shortcut Trick

Initial mixture ratio of Acid : Water = 3 : 2 (Total = 5 parts).

Since 5 parts = 50 L, we get 1 part = 10 L.

To make the new ratio 1 : 1, the amount of water must equal the amount of acid.

Acid remains constant at 3 parts, so water must increase from 2 parts to 3 parts.

Increase in water = 3 − 2 = 1 part = 10 L.

∴ The correct answer is 10 L.

Alternate Method

Initial Mixture (50 L)

Acid: 30 L

Water: 20 L

Final Mixture (Ratio 1:1)

Acid: 30 L

Water: 20 L

• 10 L Added

Given:

Total volume of initial mixture = 50 L

Initial ratio of Acid to Water = 3 : 2

Calculation:

⇒ Total ratio parts = 3 + 2 = 5 parts

⇒ 5 parts = 50 L

⇒ 1 part = 10 L

⇒ Initial volume of Acid = 3 × 10 = 30 L

⇒ Initial volume of Water = 2 × 10 = 20 L

Let the volume of water added be x liters.

⇒ New volume of Water = (20 + x) L

Since the new ratio of Acid to Water is 1 : 1, their volumes must be equal:

⇒ Acid volume = New Water volume

⇒ 30 = 20 + x

⇒ x = 30 − 20 = 10 L

∴ The correct answer is 10 L.

Additional Information

Constant Quantity Principle

When only one component (such as water) is added to a mixture, the total absolute amount of the other component (such as acid) remains strictly constant. Equating the constant component is the fastest way to solve.

Rule of Alligation

Used to find the specific ratio in which two ingredients or mixtures are combined to get a desired concentration. The formula is: (Quantity of Cheaper) / (Quantity of Dearer) = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper).

Successive Replacement

If from a vessel containing V liters of pure liquid, x liters are replaced by water 'n' times, the remaining pure liquid is calculated as: V × (1 − x / V)ⁿ.

Shortcut Trick

Let the initial cost price be CP₁ and the new cost price be CP₂.

The new selling price is increased by 25%, so SP₂ = 125% of (120% of CP₁) = 1.5 × CP₁.

Since the new profit is 30%, the new selling price also equals 130% of CP₂ = 1.3 × CP₂.

∴ 1.3 × CP₂ = 1.5 × CP₁ ⇒ CP₂ / CP₁ = 15 / 13.

Percentage increase in CP = (2 / 13) × 100% = 15.38%.

∴ The correct answer is 15.38%.

Alternate Method

Given:

Initial profit percentage = 20%

Increase in selling price = 25%

New profit percentage = 30%

CP₁ = 100SP₁ = 120SP₂ = 150CP₂ = 115.38SP₂ = 150+20%+25%+30% Profit+15.38%

Calculations:

⇒ Let the initial cost price (CP₁) be 100 units.

⇒ Initial selling price (SP₁) = 100 + 20% of 100 = 120 units.

⇒ New selling price (SP₂) increases by 25%, so SP₂ = 120 + (25 / 100) × 120 = 150 units.

⇒ The new profit is given as 30%, which means SP₂ is 130% of the new cost price (CP₂).

⇒ 1.30 × CP₂ = 150 units

⇒ CP₂ = 150 ÷ 1.30 = 1500 / 13 ≈ 115.38 units.

⇒ Increase in CP = (1500 / 13) − 100 = 200 / 13 units.

⇒ Percentage increase in CP = [(200 / 13) ÷ 100] × 100 = 200 / 13 % = 15.38%.

∴ The correct answer is 15.38%.

Additional Information

Profit and Loss Essentials

Profit is always calculated on the base Cost Price. The fundamental relation is: SP = CP × (1 + Profit% ÷ 100).

Successive Changes

When a value is changed multiple times sequentially, the net percentage change for two successive changes a% and b% is found using: (a + b + ab ÷ 100)%.

Shortcut Trick

Total expenditure of the group = Rs. 1200

Amount spent by the remaining 5 students = 1200 − 300 = Rs. 900

Average spending of the 5 students = 900 ÷ 5 = Rs. 180

∴ The correct answer is Rs. 180.

Alternate Method

Given:

Total spending of 6 students = Rs. 1200

Amount spent by 1 student = Rs. 300

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Total Spending of 6 Students = Rs. 1200

1 Student: Rs. 300

5 Students: Rs. 900 Average = 900 ÷ 5 = Rs. 180

Calculation:

⇒ Amount spent by the other 5 students = Total spending − Amount spent by 1 student

⇒ Amount spent by the other 5 students = 1200 − 300 = Rs. 900

⇒ Since the remaining 5 students spent equally, their average spending is the total spent divided by 5.

⇒ Average spending = 900 ÷ 5 = Rs. 180

∴ The correct answer is Rs. 180.

Additional Information

Basic Average Formula

The average (or arithmetic mean) is calculated by adding all values and dividing by the count: Average = Sum of observations ÷ Total number of observations.

Combined Average

If two sets of sizes n₁ and n₂ have averages A₁ and A₂ respectively, Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Change in Average

If a value is added or removed and the new average changes by a specific amount, the impact can be seen directly on the total sum: New value = Old average ± (Total numbers × Change in average).

Shortcut Trick

Let the Market Price (MP) be 100%.

Selling Price after 15% discount = 100% − 15% = 85%.

According to the question, 85% = Rs. 7650

100% (Market Price) = (7650 ÷ 85) × 100 = 90 × 100 = 9000

∴ The correct answer is Rs. 9000.

Alternate Method

MP = ?

SP = 7650

-15% Discount

MP × 85% = SP

Given:

Selling Price (SP) = Rs. 7650

Discount = 15%

Formula Used:

SP = MP × (100 − Discount%) ÷ 100

Calculations:

⇒ 7650 = MP × (100 − 15) ÷ 100

⇒ 7650 = MP × 85 ÷ 100

⇒ MP = (7650 × 100) ÷ 85

⇒ MP = 90 × 100

⇒ MP = 9000

∴ The correct answer is Rs. 9000.

Additional Information

Marked Price and Discount

Discount is always offered on the Marked Price (or List Price) of an article. The relationship is simple: Selling Price = Marked Price − Discount.

Successive Discounts

The single equivalent discount for two successive discounts of x% and y% is given by the formula (x + y − xy ÷ 100)%.

CP to MP Ratio

The direct relationship between Cost Price and Marked Price is: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Shortcut Trick

For a Men OR b Women completing a work in D days, the time taken by c Men AND d Women is given by:

⇒ Required Days = D ÷ (c/a + d/b)

⇒ Required Days = 170 ÷ (20/10 + 20/15)

⇒ Required Days = 170 ÷ (2 + 4/3) = 170 ÷ (10/3)

⇒ Required Days = 170 × 3 ÷ 10 = 51 days

∴ The correct answer is 51 days.

Alternate Method

Given:

Time taken by 10 Men = 170 days

Time taken by 15 Women = 170 days

Let total work be the LCM of the given days (170 and 170) = 170 units.

⇒ Efficiency of 10 Men = 1 unit/day

⇒ Efficiency of 1 Man = 1/10 units/day

⇒ Efficiency of 15 Women = 1 unit/day

⇒ Efficiency of 1 Woman = 1/15 units/day

⇒ Combined efficiency of 20 Men and 20 Women = (20 × 1/10) + (20 × 1/15)

⇒ Combined efficiency = 2 + 4/3 = 10/3 units/day

⇒ Required Days = Total Work ÷ Combined Efficiency

⇒ Required Days = 170 ÷ (10/3)

⇒ Required Days = 170 × 3 ÷ 10 = 51 days

∴ The correct answer is 51 days.

Additional Information

MDH Rule (Chain Rule)

Formula: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where M is Men, D is Days, H is Hours, and W is Work done.

Total Work using Efficiency

Total Work = Efficiency × Number of Days. The ratio of efficiencies is inversely proportional to the ratio of days taken to complete the work.

Work and Wages

Wages are always distributed in direct proportion to the amount of work done by each person, or in the ratio of their efficiencies if they work for the same number of days.

Shortcut Trick

Side of the square = Perimeter ÷ 4

⇒ Side = 100 ÷ 4 = 25 cm

Area of square = (Side)²

⇒ Area = 25 × 25 = 625 cm²

∴ The correct answer is 625 cm².

Alternate Method

Given:

Perimeter of the square = 100 cm

Formula Used:

Perimeter of a square = 4 × a (where a is the side length)

Area of a square = a²

Side (a) = 25 cm Area = 625 sq cm Perimeter = 100 cm

a = 25 cm

Calculations:

⇒ 4 × a = 100

⇒ a = 100 ÷ 4

⇒ a = 25 cm

⇒ Area = 25 × 25

⇒ Area = 625 cm²

∴ The correct answer is 625 cm².

Additional Information

Diagonal of a Square

The length of the diagonal (d) of a square is given by d = a√2, where a is the side length of the square.

Area using Diagonal

The area can also be calculated directly from the diagonal length using the formula: Area = d² ÷ 2.

Incircle and Circumcircle

The radius of a circle inscribed within a square is a ÷ 2, and the radius of the circumscribed circle passing through its vertices is a ÷ √2.

Shortcut Trick

To find the value of an included observation, use the direct deviation shortcut:

New Score = Old Average + (Total New Observations × Change in Average)

⇒ Score on 8^th test = 72 + (8 × 1)

⇒ Score = 72 + 8 = 80

∴ The correct answer is 80 marks.

Alternate Method

Given:

⇒ Average score of 7 tests = 72

⇒ Average score after 8 tests = 73

Formula Used:

⇒ Total Score = Average × Number of tests

Deviation Method (See-Saw Balance) Net Deviation = 0 ⇒ 7 × (−1) + 1 × (x − 73) = 0 ⇒ x = 80

Avg = 73

7 Tests Avg: 72

8^th Test Score: 80

−1

Deviation: +7

Calculations:

⇒ Total score of 7 tests = 7 × 72 = 504

⇒ Total score of 8 tests = 8 × 73 = 584

⇒ Score on the 8^th test = Total score of 8 tests − Total score of 7 tests

⇒ Score on the 8^th test = 584 − 504 = 80

∴ The correct answer is 80 marks.

Additional Information

1. Inclusion of a New Value

When a new value is added to a group: New Value = Old Average + [New Total Count × Increase in Average]. If the average decreases, use a negative sign for the change.

2. Exclusion of a Value

When a value is removed from a group: Excluded Value = Old Average − [New Total Count × Increase in Average]. This logic reverses the inclusion formula.

3. Combined Average

To find the average of two groups combined: A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n represents the item count and A represents the average of each respective group.

Shortcut Trick

The purchase amount of Rs. 25,000 exceeds the Rs. 10,000 threshold, so a flat 30% discount is applied.

Amount after 30% discount = 70% of 25,000 = Rs. 17,500.

Since Rs. 17,500 is greater than Rs. 15,000, an extra Rs. 1,000 discount is triggered.

Final amount to be paid = 17,500 − 1,000 = Rs. 16,500.

∴ The correct answer is Rs. 16,500.

Alternate Method

Given:

Total Purchase Amount = Rs. 25,000

Condition 1: 30% discount if the total purchase is above Rs. 10,000

Condition 2: Additional Rs. 1,000 off if the total after the first discount is above Rs. 15,000

Purchase Rs. 25,000

• 30% Off

After 30% Rs. 17,500

• Rs. 1000

Final Pay Rs. 16,500

Calculations:

⇒ The initial purchase value is Rs. 25,000.

⇒ Since 25,000 > 10,000, Condition 1 is satisfied and the 30% discount is applied.

⇒ First Discount = 30% of 25000

⇒ Discounted Amount = 25000 × (70 ÷ 100)

⇒ Discounted Amount = Rs. 17,500

⇒ Next, we check Condition 2. The discounted amount is Rs. 17,500, which is strictly greater than Rs. 15,000.

⇒ Since Condition 2 is satisfied, the customer gets the additional discount of Rs. 1,000.

⇒ Final Amount Paid = 17500 − 1000

⇒ Final Amount Paid = Rs. 16,500

∴ The correct answer is Rs. 16,500.

Additional Information

Successive Discounts

When two discounts d₁% and d₂% are given one after another, the effective or equivalent discount is calculated using the formula: Effective Discount% = (d₁ + d₂ − d₁×d₂ ÷ 100)%.

Selling Price vs Marked Price

Discount is always calculated on the marked price of the items. The mathematical relationship is: Selling Price = Marked Price × (100 − Discount%) ÷ 100.

Conditional Flat Discounts

These are dynamic discounts mapped to specific spending thresholds. They encourage larger purchases by offering higher percentage deductions or flat absolute cashbacks when the invoice total crosses specific levels.

Shortcut Trick

Initial salt = 60% of 100 L = 60 L. When water is added, the salt quantity remains constant.

To reduce concentration to 40%, the new total volume must be = 60 × (100 ÷ 40) = 150 L.

Water added = 150 − 100 = 50 L. The mixture now has 90 L water and 60 L salt.

To increase concentration back to 50%, the amount of salt must equal the amount of water.

Since water is 90 L, salt must also become 90 L. Pure salt to add = 90 − 60 = 30 L.

∴ The correct answer is 50 L of water, then 30 L of salt.

Alternate Method

Given:

Initial solution = 100 L

Initial salt concentration = 60%

Concept Used:

When water is added, the amount of salt remains constant. When pure salt is added, the amount of water remains constant.

100 L Salt: 60 L Water: 40 L 60% Salt + 50 L W 150 L Salt: 60 L Water: 90 L 40% Salt + 30 L S 180 L Salt: 90 L Water: 90 L 50% Salt Step-by-step Mixture Process

Calculations:

⇒ Initial amount of salt = 60% of 100 = 60 L

⇒ Initial amount of water = 100 − 60 = 40 L

⇒ Let water added be w. The new salt concentration is 40%.

⇒ [60 ÷ (100 + w)] × 100 = 40

⇒ 6000 = 4000 + 40w

⇒ 40w = 2000 ⇒ w = 50 L

⇒ Now, total volume = 150 L (Salt = 60 L, Water = 90 L).

⇒ Let salt added be s. The final salt concentration is 50%.

⇒ [(60 + s) ÷ (150 + s)] × 100 = 50

⇒ 6000 + 100s = 7500 + 50s

⇒ 50s = 1500 ⇒ s = 30 L

∴ The correct answer is 50 L of water, then 30 L of salt.

Additional Information

Constant Component Rule in Mixtures

When a pure liquid (e.g., water) is added to a mixture, the quantity of the other component (e.g., salt) remains constant. Equating the constant component's quantity before and after helps find the new overall volume directly.

50% Concentration Rule

For a mixture of two components to have a 50% concentration of one component, both components must be present in exactly equal quantities (a 1:1 ratio).

Successive Replacement Formula

If from V litres of pure liquid, x litres are drawn and replaced by water n times, the remaining pure liquid = V × (1 − x ÷ V)ⁿ.

Shortcut Trick

Given sinA = 4/5, we know Perpendicular = 4 and Hypotenuse = 5.

Base = √(5² − 4²) = 3.

Since A is in the 2^nd quadrant, tanA is negative, so tanA = −4/3.

Expression: (1 + tanA)(1 − tanA) = 1 − tan²A

⇒ 1 − (−4/3)² = 1 − 16/9 = −7/9.

∴ The correct answer is −7/9.

Alternate Method

Given:

sinA = 4/5

Angle A lies in the 2^nd quadrant.

Formula Used:

sinA = Perpendicular / Hypotenuse

tanA = Perpendicular / Base

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

(a + b)(a − b) = a² − b²

A Base = 3 Perpendicular = 4 Hypotenuse = 5

Calculations:

⇒ Let Perpendicular (P) = 4 and Hypotenuse (H) = 5.

⇒ (Base)² = H² − P²

⇒ Base = √(5² − 4²) = √(25 − 16) = √9 = 3

⇒ tanA = P / Base = 4 / 3 (magnitude)

⇒ Since A lies in the 2^nd quadrant, only sine and cosecant are positive. Thus, tangent is negative.

⇒ tanA = −4/3

⇒ We need to find the value of (1 + tanA)(1 − tanA).

⇒ Using algebraic identity: (1 + tanA)(1 − tanA) = 1 − tan²A

⇒ 1 − (−4/3)²

⇒ 1 − (16/9)

⇒ (9 − 16) / 9 = −7/9

∴ The correct answer is −7/9.

Additional Information

ASTC Rule (Quadrant Signs)

1^st Quad: All positive | 2^nd Quad: sin, cosec positive | 3^rd Quad: tan, cot positive | 4^th Quad: cos, sec positive.

Pythagorean Triplets

Sets of integers satisfying a² + b² = c², such as (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Algebraic Identities in Trigonometry

Using standard formulas like (a + b)(a − b) = a² − b² simplifies complex trigonometric expressions significantly.

Shortcut Trick

The standard conversion value of 1 radian is approximately 57.3°.

To convert an angle from radians to degrees, multiply the given value by 57.3°.

Angle in degrees = 0.5 × 57.3°

⇒ Angle = 28.65°

∴ The correct answer is 28.65°.

Alternate Method

Given:

Angle in radians = 0.5 rad

Formula Used:

Angle in degrees = Angle in radians × (180° / π)

Calculations:

⇒ Angle in degrees = 0.5 × (180° / π)

⇒ Angle in degrees = 90° / π

Substituting the approximate value of π ≈ 3.1416:

⇒ Angle in degrees = 90° / 3.1416

⇒ Angle in degrees ≈ 28.6478°

⇒ Rounding off to two decimal places, we get 28.65°.

∴ The correct answer is 28.65°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply it by (π / 180°). For example, 90° = 90 × (π / 180) = π/2 radians.

Definition of Radian

1 radian is the angle subtended at the center of a circle by an arc whose length is exactly equal to the radius of the circle.

Common Angles

Key angular relationships frequently used: 180° = π radians, 360° = 2π radians, 60° = π/3 radians, and 45° = π/4 radians.

Shortcut Trick

When sin A = cos B, the sum of angles A and B is always 90°.

Given: A = x and B = (x + 20°)

⇒ x + (x + 20°) = 90°

⇒ 2x = 70°

⇒ x = 35°

∴ The correct answer is 35°.

Alternate Method

Given:

sin x = cos(x + 20°)

x

x + 20° A B C In right triangle: sin(C) = cos(A) = AB / AC

Formula Used:

cos θ = sin(90° − θ)

Calculations:

⇒ sin x = sin(90° − (x + 20°))

Comparing the angles on both sides:

⇒ x = 90° − x − 20°

⇒ x + x = 70°

⇒ 2x = 70°

⇒ x = 35°

∴ The correct answer is 35°.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Trigonometric ratios of complementary angles are related as: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Basic Trigonometric Identities

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Specific Angle Values

The values at common angles are frequently tested: sin 30° = cos 60° = 1/2, sin 45° = cos 45° = 1/√2, sin 60° = cos 30° = √3/2.

Shortcut Trick The orthocenter is the point where all three altitudes of a triangle intersect. For an acute-angled triangle, the orthocenter lies inside the triangle. For a right-angled triangle, it lies exactly on the right-angled vertex. For an obtuse-angled triangle (one interior angle > 90°), the orthocenter always lies outside the triangle. ∴ The correct answer is Outside the triangle.

Alternate Method Given: A triangle having one interior angle greater than 90°.

Formula / Concept Used: ⇒ Orthocenter: The exact point of intersection of the three altitudes of a triangle. ⇒ Altitude: A perpendicular line segment dropped from a vertex to the opposite side (or its extension).

A B C (>90°)

Orthocenter Outside Triangle

Calculations: ⇒ Let ΔABC be an obtuse-angled triangle where the interior angle at vertex C is greater than 90°. ⇒ Due to the obtuse angle, the altitude drawn from vertex A must fall on the extended side BC, meeting it at a point entirely outside the bounded region. ⇒ Similarly, the altitude drawn from vertex B falls on the extended side AC, also meeting it at an external point. ⇒ When these two external altitudes are extended further downward, they intersect with the extended altitude from vertex C at a common point. ⇒ Because two of the altitudes lie completely outside the area of the triangle, their intersection point (the Orthocenter) must also lie completely outside the triangle. ∴ The correct answer is Outside the triangle.

Additional Information Orthocenter of an Acute Triangle For any acute-angled triangle (where all interior angles are < 90°), all three altitudes fall inside the boundaries, placing the orthocenter permanently inside the triangle. Orthocenter of a Right-Angled Triangle For a right-angled triangle, the two perpendicular legs forming the 90° angle naturally act as altitudes. Hence, the orthocenter rests exactly on the right-angled vertex. Circumcenter Shift Rule The circumcenter (intersection of perpendicular bisectors) follows a similar rule: Inside for acute triangles, at the midpoint of the hypotenuse for right triangles, and entirely outside for obtuse triangles. Centroid and Incenter Regardless of the triangle classification (acute, right, or obtuse), both the centroid (intersection of medians) and the incenter (intersection of angle bisectors) consistently remain safely inside the triangle.

Shortcut Trick

The radius, distance from the center, and half of the chord form a right-angled triangle.

We can use the standard Pythagorean triplet (3, 4, 5) scaled by 3 → (9, 12, 15).

Since the hypotenuse (radius) is 15 cm and the perpendicular (distance) is 9 cm, the base (half-chord) must be 12 cm.

Total length of the chord = 2 × 12 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Radius (r) = 15 cm

Distance of chord from center (d) = 9 cm

Formula Used:

The perpendicular drawn from the center of a circle to a chord bisects the chord.

Using Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

O

A B

M 9 cm

15 cm x

Calculations:

⇒ Let the half-length of the chord be x.

⇒ r² = d² + x²

⇒ 15² = 9² + x²

⇒ 225 = 81 + x²

⇒ x² = 225 − 81

⇒ x² = 144

⇒ x = √144 = 12 cm

⇒ Full length of the chord = 2 × x

⇒ Length = 2 × 12 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Perpendicular Bisector Theorem

The perpendicular drawn from the center of a circle to a chord bisects the chord into two equal halves.

Properties of Chords

Chords of equal length in a circle are equidistant from the center. Conversely, chords equidistant from the center are equal in length.

Diameter

The longest chord of a circle passes through the center and is known as the diameter (Length = 2 × Radius).

Shortcut Trick

The radius, tangent length, and distance from the center form a right-angled triangle.

We can apply the direct formula: r = √(d² − t²)

⇒ r = √((√125)² − 10²)

⇒ r = √(125 − 100)

⇒ r = √25 = 5 cm

∴ The correct answer is 5 cm.

Alternate Method

Given:

Distance from point P to the center (OP) = √125 cm

Length of the tangent from point P (PT) = 10 cm

O

P

T √125 cm 10 cm r

Formula Used:

By applying Pythagoras Theorem in right-angled ΔOTP:

OP² = OT² + PT²

Calculation:

A tangent to a circle is always perpendicular to the radius at the point of tangency, so ∠OTP = 90°.

⇒ (√125)² = r² + 10²

⇒ 125 = r² + 100

⇒ r² = 125 − 100

⇒ r² = 25

⇒ r = 5 cm

∴ The correct answer is 5 cm.

Additional Information

Secant-Tangent Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the tangent length equals the product of the lengths of the external secant segment and the entire secant segment.

Properties of Tangents

1. Only two tangents can be drawn from an external point to a circle, and their lengths are always equal.
2. A tangent to a circle is always perpendicular to the radius drawn to the point of tangency.

Direct Common Tangent (DCT)

For two circles with distance between centers d and radii r₁, r₂, Length = √(d² − (r₁ − r₂)²).

Shortcut Trick

In compound interest, if a sum becomes n times in T years, it becomes n^m times in m × T years.

Given, the sum doubles (n = 2) in 15 years (T = 15).

To become 8 times, we write 8 as 2³, so the multiplier power m = 3.

Required time = 15 × 3 = 45 years.

∴ The correct answer is 45 years.

Alternate Method

Given:

Amount becomes 2P in T = 15 years.

We need the time for the amount to become 8P.

Formula Used:

A = P(1 + r/100)^T

      ×2 ×2 ×2     P 2P 4P 8P 0 yrs 15 yrs 30 yrs 45 yrs Total time for 8P = 15 + 15 + 15 = 45 years

Calculations:

⇒ For the first case:

⇒ 2P = P(1 + r/100)¹⁵

⇒ 2 = (1 + r/100)¹⁵

⇒ For the second case:

⇒ 8P = P(1 + r/100)^T

⇒ 8 = (1 + r/100)^T

⇒ We can express 8 as 2³:

⇒ (2)³ = (1 + r/100)^T

⇒ Substituting the value of 2 from the first equation:

⇒ [(1 + r/100)¹⁵]³ = (1 + r/100)^T

⇒ (1 + r/100)⁴⁵ = (1 + r/100)^T

⇒ Comparing the powers on both sides:

⇒ T = 45 years

∴ The correct answer is 45 years.

Additional Information

Compound Interest Growth

In compound interest, the principal grows in a geometric progression (GP). This means the sum is multiplied by a constant factor over equal time intervals.

Simple vs Compound Interest

Unlike CI which grows multiplicatively, Simple Interest (SI) grows in an arithmetic progression (AP), meaning a constant amount is added in equal time intervals.

Rule of 72

To estimate the number of years required to double your money at a given compound interest rate, divide 72 by the annual interest rate (T ≈ 72 ÷ r).

Shortcut Trick

Integers are whole numbers without any fractional or decimal parts.

Rational numbers are any numbers that can be written exactly in the form p/q.

Non-perfect square roots (like √2) cannot be expressed as exact fractions, making them irrational.

Therefore, √2 is neither an integer nor a rational number.

∴ The correct answer is √2.

Alternate Method

Given:

Numbers to verify: -4, 1.5, √2, 7

Definitions Used:

⇒ Integer: A whole number that can be positive, negative, or zero (e.g., -2, 0, 5).

⇒ Rational Number: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

⇒ Irrational Number: A number that cannot be written as a simple fraction, meaning its decimal expands forever without repeating.

Calculations:

⇒ -4 is a negative integer and can be written as -4/1 (which makes it Rational).

⇒ 1.5 is a terminating decimal and can be written as 15/10 or 3/2 (which makes it Rational).

⇒ √2 is approximately 1.41421... with non-terminating and non-repeating decimals. It cannot be written as p/q, so it is an Irrational number.

⇒ 7 is a positive integer and can be written as 7/1 (which makes it Rational).

⇒ Therefore, √2 is the only number among the options that is neither an Integer nor a Rational number.

∴ The correct answer is √2.

Additional Information

Rational Numbers

Numbers that can be represented as a ratio of two integers p/q where q is not equal to 0. Example: 1/2, 4, 0.75.

Irrational Numbers

Real numbers that cannot be represented as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Example: π, √2, √3.

Integers

The set of whole numbers and their opposites, including zero. They do not have decimal or fractional parts. Example: -3, 0, 42.

Real Numbers

Any continuous quantity that can represent a distance along a line. The set of real numbers encompasses both rational and irrational numbers.

Shortcut Trick

Total time of journey = 6:00 a.m. to 2:30 p.m. = 8.5 hours.

Distance = Speed × Time

Distance = 60 × 8.5 = 510 km.

∴ The correct answer is 510 km.

Alternate Method

Mumbai 6:00 a.m. Kolhapur 2:30 p.m.  Train  Speed = 60 km/h  Time = 8.5 hours Distance = ?

Given:

Departure time from Mumbai = 6:00 a.m.

Arrival time at Kolhapur = 2:30 p.m.

Speed of the train = 60 km/h

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Total time of journey = Time from 6:00 a.m. to 12:00 noon + Time from 12:00 noon to 2:30 p.m.

⇒ Total time = 6 hours + 2.5 hours = 8.5 hours

⇒ Distance traveled = 60 km/h × 8.5 hours

⇒ Distance = 510 km

∴ The correct answer is 510 km.

Additional Information

Average Speed

When a journey involves multiple segments, Average Speed = Total Distance ÷ Total Time.

Relative Speed

If two trains are moving in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂). If in the same direction, it's the difference (S₁ − S₂).

Unit Conversion

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

Shortcut Trick

Number of containers = Volume of the Spherical Tank ÷ Volume of 1 Cylindrical Container

⇒ N = [(4/3) × π × R³] ÷ [π × r² × h]

⇒ N = [(4/3) × 6 × 6 × 6] ÷ [1 × 1 × 12]

⇒ N = 288 ÷ 12 = 24

∴ The correct answer is 24 containers.

Alternate Method

  R = 6 m Spherical Tank   Fill          r = 1 m  h = 12 m Cylindrical Container

Given:

Radius of the spherical tank (R) = 6 m

Radius of the cylindrical container (r) = 1 m

Height of the cylindrical container (h) = 12 m

Formula Used:

Volume of a sphere = (4/3) × π × R³

Volume of a cylinder = π × r² × h

Calculations:

⇒ Volume of the spherical tank = (4/3) × π × (6)³

⇒ Volume of the spherical tank = (4/3) × π × 216 = 288π m³

⇒ Volume of one cylindrical container = π × (1)² × 12 = 12π m³

⇒ Number of containers that can be filled = Volume of sphere ÷ Volume of one cylinder

⇒ Number of containers = 288π ÷ 12π

⇒ Number of containers = 24

∴ The correct answer is 24 containers.

Additional Information

Volume of a Cone

The volume of a right circular cone is given by V = (1/3) × π × r² × h, where r is the radius and h is the vertical height.

Surface Area of a Sphere

The total surface area of a complete sphere is calculated as A = 4 × π × r², where r is the radius.

Volume of a Hemisphere

A hemisphere is exactly half of a sphere. Its volume is given by V = (2/3) × π × r³.

Curved Surface Area of a Cylinder

The curved (or lateral) surface area of a cylinder is calculated using CSA = 2 × π × r × h.

Shortcut Trick

For the largest sphere inscribed in a cube of side a, the radius is r = a ÷ 2.

Volume of empty space = Volume of cube − Volume of sphere

⇒ Volume = a³ − (4/3) × π × (a/2)³

⇒ Volume = 8³ − (4/3) × π × 4³

⇒ Volume = 512 − 256π/3 cm³

∴ The correct answer is 512 − 256π/3 cm³.

Alternate Method

 r = 4 cm    side a = 8 cm Cross Section of Sphere inside a Cube

Given:

⇒ Side of the cube (a) = 8 cm

Formula Used:

⇒ Volume of a cube = a³

⇒ Volume of a sphere = (4/3) × π × r³

Calculations:

⇒ The largest sphere inscribed in a cube will have its diameter equal to the edge length of the cube.

⇒ Diameter of the sphere (d) = 8 cm

⇒ Radius of the sphere (r) = 8 ÷ 2 = 4 cm

⇒ Volume of the cube = 8³ = 512 cm³

⇒ Volume of the sphere = (4/3) × π × 4³

⇒ Volume of the sphere = (4/3) × π × 64 = 256π/3 cm³

⇒ Empty space volume = Volume of cube − Volume of sphere

⇒ Empty space volume = 512 − 256π/3 cm³

∴ The correct answer is 512 − 256π/3 cm³.

Additional Information

Largest Sphere in a Cube

When a sphere is inscribed within a perfect cube, the sphere touches all six inner faces, meaning its diameter equals the side length of the cube (d = a).

Volume Formulas

Volume of Cube = a³ | Volume of Sphere = (4/3) × π × r³

Total Surface Area Formulas

TSA of Cube = 6 × a² | TSA of Sphere = 4 × π × r²

Largest Cube in a Sphere

Conversely, if a cube is inscribed inside a sphere, the body diagonal of the cube acts as the diameter of the sphere (a√3 = 2r).