Quantitative Aptitude PYQs

Shortcut Trick

For similar figures, ratio of areas = (Ratio of perimeters)²

⇒ Area Ratio = (5 ÷ 7)² = 25 : 49

⇒ 25 ratio units = 75 cm²

⇒ 1 ratio unit = 3 cm²

⇒ Area of larger hexagon = 49 × 3 = 147 cm²

∴ The correct answer is 147 cm².

Alternate Method

Given:

Ratio of perimeters of two similar hexagons = 5 : 7

Area of the smaller hexagon = 75 cm²

Formula Used:

If two polygons are similar, the ratio of their areas is equal to the square of the ratio of their perimeters.

⇒ (Area₁ ÷ Area₂) = (Perimeter₁ ÷ Perimeter₂)²

Perimeter ratio = 5 Area = 75 cm²

Perimeter ratio = 7 Area = ?

Similar

Calculations:

⇒ Let Area₁ be the area of the smaller hexagon and Area₂ be the area of the larger hexagon.

⇒ Area₁ ÷ Area₂ = (5 ÷ 7)²

⇒ 75 ÷ Area₂ = 25 ÷ 49

⇒ 25 × Area₂ = 75 × 49

⇒ Area₂ = (75 × 49) ÷ 25

⇒ Area₂ = 3 × 49

⇒ Area₂ = 147 cm²

∴ The correct answer is 147 cm².

Additional Information

Properties of Similar Polygons

For any two similar polygons, the ratio of their corresponding sides, diagonals, and perimeters are all equal. The ratio of their areas strictly equals the square of the ratio of their corresponding linear dimensions.

Area of a Regular Hexagon

A regular hexagon is composed of 6 distinct equilateral triangles sharing a common center. Its area is calculated using Area = (3√3 ÷ 2)a², where a is the side length.

Ratio of Volumes for Similar 3D Shapes

Extending the property to 3D geometry, for two similar 3-dimensional figures (like similar cones or spheres), the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions: V₁ ÷ V₂ = (S₁ ÷ S₂)³.

Shortcut Trick

Ratio of pens sold at profit to pens sold at loss = 60 : 40 = 3 : 2.

Using the weighted average method:

⇒ Net profit/loss % = (n₁ × p₁ + n₂ × p₂) ÷ (n₁ + n₂)

⇒ Net profit % = (3 × 20% − 2 × 10%) ÷ (3 + 2)

⇒ Net profit % = (60% − 20%) ÷ 5 = 40% ÷ 5 = 8%

∴ The correct answer is 8% profit.

Alternate Method

Given:

Total number of pens = 100

Cost price of each pen = ₹10

60 pens sold at 20% profit, and the rest (100 − 60 = 40) sold at 10% loss.

60 Pens 40 Pens

+20% Profit

−10% Loss

Overall Net Profit % = ?

Formula Used:

Total Cost Price = Quantity × Rate

Profit = CP × Profit% ÷ 100

Loss = CP × Loss% ÷ 100

Net Profit % = (Net Profit ÷ Total CP) × 100

Calculations:

⇒ Total CP of 100 pens = 100 × 10 = ₹1000

⇒ CP of 60 pens = 60 × 10 = ₹600

⇒ Profit on 60 pens = 20% of 600 = (20 ÷ 100) × 600 = ₹120

⇒ CP of remaining 40 pens = 40 × 10 = ₹400

⇒ Loss on 40 pens = 10% of 400 = (10 ÷ 100) × 400 = ₹40

⇒ Overall Net Profit = Total Profit − Total Loss = 120 − 40 = ₹80

⇒ Overall Profit Percentage = (80 ÷ 1000) × 100 = 8%

∴ The correct answer is 8% profit.

Additional Information

Weighted Average Method

When parts of a quantity are sold at different profit or loss percentages, the net percentage can be found rapidly using the ratio of the parts: Net % = (n₁p₁ + n₂p₂) ÷ (n₁ + n₂).

Dishonest Dealer Concept

When a dealer sells goods at CP but uses false weight to cheat customers, the actual Profit % = (True Weight − False Weight) ÷ False Weight × 100.

Successive Discounts

If an article is offered at successive discounts of a% and b%, the single equivalent discount is given by: D = a + b − (a × b) ÷ 100.

Shortcut Trick

Net change in total due to the removal = Remaining results × Drop in Average

Net change = 10 × (75 − 72) = 10 × 3 = 30

Sum of the 2 removed results = (2 × Old Average) + Net change = (2 × 75) + 30 = 180

Average of these top 2 results = 180 ÷ 2 = 90

∴ The correct answer is 90.

Alternate Method

Given:

Average of all 12 results = 75

Average of remaining 10 results (excluding top 2) = 72

Total of 12 Results: 12 × 75 = 900

Total of 10 Results: 10 × 72 = 720

Top 2 Results

Sum = 180

Formula Used:

Total Sum = Average × Number of results

Calculations:

⇒ Total sum of all 12 results = 12 × 75 = 900

⇒ Total sum of the 10 remaining results = 10 × 72 = 720

⇒ Sum of the top 2 results = Total sum of 12 results − Total sum of 10 results

⇒ Sum of the top 2 results = 900 − 720 = 180

⇒ Average of the top 2 results = 180 ÷ 2 = 90

∴ The correct answer is 90.

Additional Information

Net Deviation Method for Averages

When items are removed and the group's average decreases, it means the removed items were larger than the original average. The total sum of removed items is calculated as: (Number of removed items × Old Average) + (Number of remaining items × Drop in Average).

Combined Average Formula

If two distinct groups have sizes n₁ and n₂ with averages A₁ and A₂ respectively, their combined weighted average is given by: A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Properties of Averages

If every observation in a dataset is systematically increased, decreased, multiplied, or divided by a constant number 'k', the average of the dataset will undergo the exact same mathematical operation with 'k'.

Shortcut Trick

You can find the final payable amount directly using successive multipliers.

Final Price = Marked Price × (100 − Discount%) / 100 × (100 + GST%) / 100

Final Price = 2000 × (80 / 100) × (118 / 100)

Final Price = 2000 × 0.80 × 1.18 = 1888

∴ The correct answer is Rs. 1888.

Alternate Method

Given:

Marked Price of the item = Rs. 2000

Discount = 20%

GST = 18% (charged on the discounted price)

Marked Price ₹2000

−20%

Discounted ₹1600

+18%

Final Price ₹1888

Calculations:

⇒ Discount Amount = 20% of 2000

⇒ Discount Amount = (20 ÷ 100) × 2000 = Rs. 400

⇒ Discounted Price = Marked Price − Discount Amount

⇒ Discounted Price = 2000 − 400 = Rs. 1600

⇒ GST Amount = 18% of Discounted Price

⇒ GST Amount = (18 ÷ 100) × 1600

⇒ GST Amount = 18 × 16 = Rs. 288

⇒ Final Payable Amount = Discounted Price + GST Amount

⇒ Final Payable Amount = 1600 + 288 = Rs. 1888

∴ The correct answer is Rs. 1888.

Additional Information

Effective Percentage Change

When a value undergoes two successive percentage changes x% and y%, the net effective change is given by: x + y + (x × y) ÷ 100. Decreases are taken as negative, while increases are taken as positive.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the single equivalent discount is: d₁ + d₂ − (d₁ × d₂) ÷ 100.

Taxes and Discounts Application Rule

Discounts are always calculated on the Marked Price (MP), and taxes like GST are always calculated on the final Discounted Price or Selling Price (SP) before tax, not on the original Marked Price.

Shortcut Trick

The length and width of a rectangle form a right-angled triangle with the diagonal.

The given sides 10 cm and 24 cm can be simplified by dividing by 2, yielding 5 and 12.

These values match the standard Pythagorean triple (5, 12, 13).

By multiplying the hypotenuse value back by 2, we directly get the diagonal: 13 × 2 = 26 cm.

∴ The correct answer is 26 cm.

Alternate Method

Given:

Length of the rectangle (L) = 24 cm

Width of the rectangle (W) = 10 cm

Formula Used:

Diagonal of a rectangle (d) = √(L² + W²)

Length = 24 cm Width = 10 cm Diagonal (d)

Calculations:

⇒ d = √(24² + 10²)

⇒ d = √(576 + 100)

⇒ d = √(676)

⇒ d = 26 cm

∴ The correct answer is 26 cm.

Additional Information

Area of a Rectangle

The area is calculated as the product of its length and width. Area = L × W.

Perimeter of a Rectangle

The perimeter represents the total outer boundary of the shape. Perimeter = 2 × (L + W).

Properties of Diagonals

In any rectangle, the two diagonals are exactly equal in length and bisect each other, but they only intersect at exactly 90 degrees if the rectangle is a perfect square.

Shortcut Trick Purchase value = ₹1200, which is > ₹1000, so the flat ₹150 discount is applicable. Discount % = (Discount Amount ÷ Marked Price) × 100 Discount % = (150 ÷ 1200) × 100 Discount % = (1 ÷ 8) × 100 = 12.5% ∴ The correct answer is 12.5%.

Alternate Method

Marked Price (MP) = ₹1200

Selling Price = ₹1050

Discount = ₹150

Paid Amount Off

Given: Marked Price (MP) of the item = ₹1200 Condition for discount: Purchase value > ₹1000 Flat discount amount = ₹150 Formula Used: Discount Percentage = (Discount Amount ÷ Marked Price) × 100 Calculations: ⇒ Here, the MP (₹1200) is greater than ₹1000, so the flat discount of ₹150 is valid. ⇒ Discount % = (150 ÷ 1200) × 100 ⇒ Discount % = (15 ÷ 120) × 100 ⇒ Discount % = (1 ÷ 8) × 100 ⇒ Discount % = 12.5% ∴ The correct answer is 12.5%.

Additional Information Marked Price & Discount Discount is always calculated on the Marked Price (MP) of an article, not on the Cost Price (CP). Successive Discount If two successive discounts of a% and b% are given, the equivalent single discount % = [a + b − (a × b) ÷ 100]%. Selling Price Formula Selling Price (SP) = Marked Price × (100 − Discount %) ÷ 100.

Shortcut Trick

Efficiency of Sudarshan = 50 ÷ 10 = 5 pages/hour.

Combined efficiency of Sudarshan & Prakash = 300 ÷ 40 = 7.5 pages/hour.

Efficiency of Prakash = 7.5 − 5 = 2.5 pages/hour.

Time required for Prakash to copy 25 pages = 25 ÷ 2.5 = 10 hours.

∴ The correct answer is 10 hours.

Alternate Method

Given:

Sudarshan copies 50 pages in 10 hours.

Sudarshan and Prakash together copy 300 pages in 40 hours.

Formula Used:

Efficiency = Total Work ÷ Time

Time = Total Work ÷ Efficiency

Calculations:

⇒ Time taken by Sudarshan to complete 300 pages = (300 ÷ 50) × 10 = 60 hours.

⇒ From the table, Efficiency of Sudarshan = 5 pages/hour.

⇒ Combined Efficiency = 7.5 pages/hour.

⇒ Efficiency of Prakash = Combined Efficiency − Sudarshan's Efficiency

⇒ Efficiency of Prakash = 7.5 − 5 = 2.5 pages/hour.

⇒ Time required by Prakash for 25 pages = 25 ÷ 2.5 = 10 hours.

∴ The correct answer is 10 hours.

Additional Information

Efficiency and Time

Time taken to complete a job is inversely proportional to the efficiency. Time = Total Work ÷ Efficiency.

Combined Work Rule

When two or more individuals work together, their combined rate of work (efficiency) is the sum of their individual rates.

Unitary Method

It involves finding the value of one unit (like pages per hour) and then using it to calculate the required value for a specific quantity.

Shortcut Trick

Let the second number be 100 units.

The first number is 20% more = 120 units, and the third is 20% less = 80 units.

Average = (120 + 100 + 80) ÷ 3 = 300 ÷ 3 = 100 units.

Since the given average is 35, we get 100 units = 35.

The second number is exactly 100 units, which equals 35.

∴ The correct answer is 35.

Alternate Method

Given:

The average of three numbers = 35

The first number is 20% more than the second number.

The third number is 20% less than the second number.

Formula Used:

Average = Sum of all observations ÷ Total number of observations

First No. (1.2x)

+20%

Second No. (x)

Base

Third No. (0.8x)

−20%

Average (x)

Average = (1.2x + x + 0.8x) ÷ 3 = x

Calculations:

⇒ Let the second number be x.

⇒ First number = x + (20% of x) = 1.2x

⇒ Third number = x − (20% of x) = 0.8x

⇒ Sum of all three numbers = 1.2x + x + 0.8x = 3x

⇒ Average = Sum of numbers ÷ 3

⇒ Average = 3x ÷ 3 = x

⇒ Since the given average is 35, we have x = 35.

∴ The correct answer is 35.

Additional Information

Average of Symmetric Values

When values are symmetrically distributed around a central base value (such as x − k, x, and x + k), their arithmetic mean directly equals the central value x without requiring complex calculations.

Percentage Increase and Decrease

If a given quantity increases by R%, its new value becomes (100 + R)% of itself. Conversely, if it decreases by R%, it becomes (100 − R)% of the original value.

Combined Average Formula

The overall or combined average of two groups is calculated using the weighted sum of their elements: A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n represents the number of elements and A represents the group average.

Shortcut Trick

The standard trigonometric identity is sec²A − tan²A = 1.

Expanding the difference of squares gives: (secA − tanA) × (secA + tanA) = 1.

This means secA + tanA and secA − tanA are always reciprocals of each other.

Given that secA − tanA = 1/3, we can directly invert this to find secA + tanA = 3.

∴ The correct answer is 3.

Alternate Method

Given:

secA − tanA = 1/3

Formula Used:

sec²A − tan²A = 1

a² − b² = (a − b)(a + b)

Calculations:

⇒ sec²A − tan²A = 1

⇒ (secA − tanA) × (secA + tanA) = 1

Substitute the given value of secA − tanA into the equation:

⇒ (1/3) × (secA + tanA) = 1

⇒ secA + tanA = 1 × 3

⇒ secA + tanA = 3

∴ The correct answer is 3.

Additional Information

Pythagorean Trigonometric Identities

There are three foundational identities in trigonometry: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Reciprocal Property

Derived directly from the identity sec²θ − tan²θ = 1, the terms (secθ − tanθ) and (secθ + tanθ) will always behave as multiplicative inverses or reciprocals. The exact same logic applies to cosecθ and cotθ.

Algebraic Difference of Squares

The expression a² − b² = (a − b)(a + b) is universally applicable and frequently used to simplify higher power trigonometric equations without solving for individual angles.

Shortcut Trick

Initial ratio of Oil : Kerosene = 2 : 1 (Total = 3 units → 30 L)

Final ratio of Oil : Kerosene = 1 : 2

Since the quantity of oil remains unchanged, multiply the final ratio by 2 to equalise the oil parts → 2 : 4

Increase in kerosene parts = 4 − 1 = 3 units.

Since 3 units initially correspond to the total mixture of 30 L, the added kerosene is also 30 L.

∴ The correct answer is 30 L.

Alternate Method

Given:

Total initial mixture = 30 L

Initial ratio of Oil : Kerosene = 2 : 1

Final ratio of Oil : Kerosene = 1 : 2

Method Used:

When a pure component (kerosene) is added, the quantity of the other component (oil) remains completely unaffected. We equate the oil amount in both states to solve for the added quantity.

Initial Mixture (Ratio 2:1)

Oil: 20 L Ker: 10 L → Total = 30 L

Final Mixture (Ratio 1:2)

Oil: 20 L 10 L

• x L Added

∴ Final Kerosene = 10 + x = 40 L ⇒ x = 30 L

Calculations:

⇒ Quantity of oil initially = (2 ÷ 3) × 30 = 20 L

⇒ Quantity of kerosene initially = (1 ÷ 3) × 30 = 10 L

⇒ Let the quantity of kerosene added be x L.

⇒ New quantity of kerosene = (10 + x) L

⇒ New mixture ratio = Oil ÷ Kerosene = 20 ÷ (10 + x) = 1 ÷ 2

⇒ 20 × 2 = 1 × (10 + x)

⇒ 40 = 10 + x

⇒ x = 30 L

∴ The correct answer is 30 L.

Additional Information

Constant Component Principle

When a pure ingredient is added to or evaporated from a mixture, the total quantity of the other ingredient(s) remains strictly unchanged. Establishing an equation based on this unchanged quantity is the fastest way to solve.

Rule of Alligation

An alternative technique to find the exact ratio in which two mixtures or specific ingredients of varied concentrations must be mixed together to produce a desired final concentration.

Successive Replacement

If an initial volume V of pure liquid has x units removed and replaced with water repeatedly for n times, the final volume of pure liquid left is evaluated using the formula: Final Quantity = V × (1 − x ÷ V)ⁿ.

Shortcut Trick

To convert an angle from radians to degrees directly, substitute π = 180°.

⇒ Given angle = 3π/4

⇒ Angle = (3 × 180°) / 4

⇒ Angle = 3 × 45°

⇒ Angle = 135°

∴ The correct answer is 135°.

Alternate Method

Given:

Angle in radians = 3π/4

Formula Used:

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

135° 0° 90° (π/2) 180° (π)

Calculations:

⇒ Angle in degrees = (3π/4) × (180° / π)

⇒ Angle in degrees = (3 / 4) × 180°

⇒ Angle in degrees = 3 × (180° / 4)

⇒ Angle in degrees = 3 × 45°

⇒ Angle in degrees = 135°

∴ The correct answer is 135°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply the given angle by (π / 180°).

Common Angles

Important conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π.

Relationship between Arc Length and Angle

Length of an arc (s) is given by the formula s = r × θ, where r is the radius and θ is the central angle measured strictly in radians.

Shortcut Trick

For trigonometric ratios, if sin(A) = cos(B), then their angles are complementary, meaning A + B = 90°.

⇒ x + (3x − 30°) = 90°

⇒ 4x = 120°

⇒ x = 30°

∴ The correct answer is 30°.

Alternate Method

x 3x − 30° 90° Complementary Angles: A + B = 90°

Given:

sin(x) = cos(3x − 30°)

Formula Used:

Complementary angle identity: sin(θ) = cos(90° − θ)

Calculation:

By substituting the identity into the given equation, we can rewrite the sine function in terms of cosine:

⇒ cos(90° − x) = cos(3x − 30°)

Since the trigonometric ratios are equal and we assume acute angles, we can directly equate their arguments:

⇒ 90° − x = 3x − 30°

Rearranging the terms to isolate x on one side:

⇒ 90° + 30° = 3x + x

⇒ 120° = 4x

⇒ x = 120° ÷ 4

⇒ x = 30°

∴ The correct answer is 30°.

Additional Information

Complementary Angles

Two angles are complementary if their sum is exactly 90°. In right-angled triangles, the two non-right angles are always complementary.

Trigonometric Co-function Identities

The prefix 'co-' in cosine stands for 'complementary'. The core relations mapping these are: sin(θ) = cos(90° − θ), tan(θ) = cot(90° − θ), and sec(θ) = csc(90° − θ).

Pythagorean Identities

The fundamental square identities form the basis of trigonometric proofs: sin²(θ) + cos²(θ) = 1, 1 + tan²(θ) = sec²(θ), and 1 + cot²(θ) = csc²(θ).

Shortcut Trick The circumcenter of a triangle is the intersection of the perpendicular bisectors of its sides. For an acute-angled triangle, it lies completely inside the triangle. For a right-angled triangle, it lies exactly on the midpoint of the hypotenuse. For an obtuse-angled triangle, it lies completely outside the triangle, opposite the longest side. ∴ The correct answer is Outside the triangle.

Alternate Method Given: A triangle containing an obtuse angle (angle > 90°). Concept Used: The circumcenter is equidistant from all three vertices and is found by the intersection of the perpendicular bisectors of the sides.

O (Circumcenter)

90°

A B C

Longest Side

⇒ In an obtuse-angled triangle, the angle subtended by the longest side is greater than 90°. ⇒ The perpendicular bisectors of the two smaller sides diverge away from the interior of the triangle. ⇒ Therefore, they intersect at a single point that is completely outside the boundaries of the triangle. ⇒ The circumcircle drawn with this intersection point as the center will completely enclose the triangle, but its center remains outside. ∴ The correct answer is Outside the triangle.

Additional Information Incenter Location The point of intersection of the angle bisectors always lies completely inside the triangle, regardless of the triangle's type. Orthocenter Location The point of intersection of the altitudes lies inside for acute triangles, at the 90° vertex for right triangles, and outside for obtuse triangles. Centroid Location The point of intersection of the medians (acting as the center of mass) is always located strictly inside the triangle.

Shortcut Trick

According to the fundamental properties of a circle, equal chords are always equidistant from the center.

Given that chord AB = chord CD.

The perpendicular distance of AB from the center O is 8 cm.

Therefore, the distance of CD from the center O will directly be equal to 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

O A B C D P Q

8 cm ?

AB = CD OQ = OP = 8 cm

Given:

⇒ Chords AB and CD are equal in length (AB = CD).

⇒ The perpendicular distance from center O to chord AB is OP = 8 cm.

Concept Used:

⇒ In a circle, if two chords are exactly equal in length, then they are situated at an equal perpendicular distance from the center.

Calculations:

⇒ Let the perpendicular distance from center O to chord CD be OQ.

⇒ Since AB = CD, their respective distances from the center O must be identical.

⇒ Distance OQ = Distance OP

⇒ OQ = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Perpendicular from Center

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

Chord Distance Property

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

Chord Size and Distance

The larger the chord, the closer it lies to the center of the circle. The diameter is the longest possible chord and its distance from the center is 0.

Intersecting Chords Theorem

If two chords AB and CD intersect internally at a point P, the product of their segments is equal: AP × PB = CP × PD.

Shortcut Trick

By the fundamental properties of circles, the radius drawn to the point of tangency is always perpendicular to the tangent line.

Since the diameter is a straight line that contains the radius, it is also perpendicular to the tangent at the point of contact.

The length of the diameter (10 cm) is given only to distract from this universal geometric truth.

∴ The correct answer is 90°.

Alternate Method

Given:

Diameter of the circle = 10 cm

A straight line is tangent to the circle.

Center O Diameter = 10 cm Tangent Point of Contact P 90°

Concept Used:

According to the Tangent-Radius Theorem, a tangent to a circle is orthogonal (perpendicular) to the radius at the exact point of contact.

Calculations:

⇒ Let the center of the given circle be O and the specific point of tangency be P.

⇒ The line segment OP represents the radius of the circle at the tangent contact point.

⇒ By the theorem, the angle formed between the tangent line and the radius OP is exactly 90°.

⇒ The diameter is a straight line segment that directly passes through the center O and the point P, meaning it structurally aligns with the radius line.

⇒ Therefore, the angle between the tangent line and the diameter at the point of contact remains perfectly 90°.

∴ The correct answer is 90°.

Additional Information

Tangent-Radius Theorem

A tangent to a circle is strictly perpendicular to the radius drawn precisely through the point of contact. This geometrical property applies universally regardless of the dimensional size of the circle.

Tangents from an External Point

The defined lengths of the two tangents drawn from any common external point to a specific circle are always perfectly equal.

Secant of a Circle

A secant is a continuous straight line that effectively intersects a circle at two distinct geometrical points, entirely unlike a tangent line which uniquely touches the circle's boundary at exactly one coordinate point.

Shortcut Trick

For half-yearly compounding, new rate = 20% ÷ 2 = 10% and cycles = 2 × 2 = 4.

The effective compound interest rate for 4 cycles at 10% is widely known to be 46.41%.

Total CI = 46.41% of Principal

Total CI = 46.41% × 5000 = 46.41 × 50 = 2320.5

∴ The correct answer is Rs. 2320.5.

Alternate Method

Given:

Principal (P) = Rs. 5000

Annual Rate (R) = 20%

Time (T) = 2 years

Formula Used:

A = P × (1 + r/100)ⁿ

Where r = rate per period, n = number of periods, and CI = A − P

0m 5000

6m

12m

18m

24m 7320.5

+10%

+10%

+10%

+10%

Total CI = 7320.5 − 5000 = Rs. 2320.5

Calculations:

⇒ Compounded half-yearly, so rate per period (r) = 20% ÷ 2 = 10%

⇒ Number of half-years (n) = 2 × 2 = 4

⇒ Amount (A) = 5000 × (1 + 10/100)⁴

⇒ A = 5000 × (11/10) × (11/10) × (11/10) × (11/10)

⇒ A = 5000 × 1.4641

⇒ A = 7320.5

⇒ Compound Interest (CI) = A − P

⇒ CI = 7320.5 − 5000 = 2320.5

∴ The correct answer is Rs. 2320.5.

Additional Information

Quarterly Compounding

When interest is compounded quarterly, the rate is divided by 4 (R/4) and the time periods are multiplied by 4 (4T).

Effective Rate for 2 Cycles

The effective compound interest rate for 2 continuous cycles at R% can be found using the successive percentage formula: R + R + (R×R)/100.

Difference between CI and SI

For exactly 2 years, the difference is given by CI − SI = P(R/100)². For exactly 3 years, it is CI − SI = P(R/100)²(3 + R/100).

Shortcut Trick

Integers represent the complete set of positive counting numbers, their negative counterparts, and zero, without any fractional or decimal parts.

Since the given values -5, 0, and 6 exactly match these conditions, they belong to the set of integers.

∴ The correct answer is Integers.

Alternate Method

Given:

A set of numbers: -5, 0, and 6

Classification Analysis:

⇒ Natural Numbers: Includes only positive counting numbers (1, 2, 3...). It excludes 0 and -5.

⇒ Whole Numbers: Includes all natural numbers along with 0 (0, 1, 2, 3...). It excludes negative numbers like -5.

⇒ Irrational Numbers: Includes non-terminating, non-repeating decimals like π or √2.

⇒ Integers: Includes all positive whole numbers, their negative counterparts, and 0 (..., -3, -2, -1, 0, 1, 2, 3...).

Conclusion:

⇒ -5 is a negative integer.

⇒ 0 is a neutral integer (neither positive nor negative).

⇒ 6 is a positive integer.

⇒ Therefore, all three numbers completely satisfy the properties of Integers.

∴ The correct answer is Integers.

Additional Information

Rational Numbers

Any number that can be expressed as a fraction p/q where p and q are integers and q is not equal to 0. Every integer is a rational number because it can be written as p/1.

Real Numbers

The universal set combining all rational and irrational numbers. All integers, whole numbers, and natural numbers fall under the category of real numbers.

Even and Odd Integers

An integer is even if it is completely divisible by 2 (e.g., -4, 0, 6). It is odd if it leaves a remainder of 1 when divided by 2 (e.g., -5, 3, 7).

Shortcut Trick

When the distance is constant, Speed and Time are inversely proportional.

Ratio of Speeds (S₁ : S₂) = 50 : 40 = 5 : 4

Ratio of Times (T₁ : T₂) = 4 : 5

Given 4 units = 8 hours ⇒ 1 unit = 2 hours.

Required Time (5 units) = 5 × 2 = 10 h.

∴ The correct answer is 10 h.

Alternate Method

Given:

Speed of the car (S₁) = 50 km/h

Time taken (T₁) = 8 h

New speed (S₂) = 40 km/h

Total Distance = 400 km

Case 1: Speed = 50 km/h | Time = 8 h

Case 2: Speed = 40 km/h | Time = ?

Formula Used:

Distance = Speed × Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = 50 × 8 = 400 km

⇒ Time taken at 40 km/h = 400 ÷ 40

⇒ Time taken = 10 h

∴ The correct answer is 10 h.

Additional Information

Average Speed

When equal distances are covered at speeds x and y, the Average Speed for the entire journey is given by (2 × x × y) ÷ (x + y).

Relative Speed

If two objects are moving in opposite directions with speeds x and y, their relative speed is (x + y). If they move in the same direction, it is (x − y).

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

The volume of a sphere is directly proportional to the cube of its radius (V ∝ r³).

If the radius is doubled, the new volume becomes 2³ = 8 times the original volume.

New Volume = 8 × 288π

New Volume = 2304π cm³

∴ The correct answer is 2304π cm³.

Alternate Method

Given:

Volume of the original sphere = 288π cm³

Formula Used:

Volume of a sphere = (4/3)πr³

r V = 288π cm³

r × 2

2r New V = ?

Calculations:

⇒ (4/3)πr³ = 288π

⇒ (4/3)r³ = 288

⇒ r³ = 288 × 3 ÷ 4

⇒ r³ = 72 × 3 = 216

⇒ r = 6 cm

⇒ New radius (R) = 2 × 6 = 12 cm

⇒ New Volume = (4/3)πR³

⇒ New Volume = (4/3)π(12)³

⇒ New Volume = (4/3)π × 1728

⇒ New Volume = 4 × 576π

⇒ New Volume = 2304π cm³

∴ The correct answer is 2304π cm³.

Additional Information

Surface Area of a Sphere

The total surface area of a complete sphere is given by the formula 4πr².

Volume of a Hemisphere

A hemisphere is exactly half of a sphere. Its volume is derived directly as (2/3)πr³.

Scaling Effect on 3D Shapes

If all dimensions of a 3D figure are scaled by a factor of k, its surface area changes by a factor of k² and its volume changes by a factor of k³.

Shortcut Trick

Total exact tiles = (800 × 600) ÷ (40 × 40)

Total exact tiles = 480,000 ÷ 1,600 = 300

Total required with 10% extra = 300 × 1.1 = 330

∴ The correct answer is 330 tiles.

Alternate Method

Given:

Length of room (L) = 8 m = 800 cm

Width of room (W) = 6 m = 600 cm

Side of square tile = 40 cm

Extra tiles required = 10%

Length = 800 cm Width = 600 cm Area = 480,000 cm²

40 cm 40 cm Tile

Required = (Floor Area ÷ Tile Area) + 10% Extra

Calculations:

⇒ Area of the floor = 800 × 600 = 480,000 cm²

⇒ Area of one square tile = 40 × 40 = 1,600 cm²

⇒ Exact number of tiles needed = 480,000 ÷ 1,600

⇒ Exact number of tiles = 300

⇒ Extra tiles for cutting and breakage = 10% of 300

⇒ Extra tiles = (10 ÷ 100) × 300 = 30

⇒ Total tiles needed = 300 + 30 = 330

∴ The correct answer is 330 tiles.

Additional Information

Area and Perimeter of a Rectangle

Area = Length × Width | Perimeter = 2 × (Length + Width)

Area and Perimeter of a Square

Area = Side² | Perimeter = 4 × Side

Unit Conversions

1 meter (m) = 100 centimeters (cm) | 1 square meter (m²) = 10,000 square centimeters (cm²)

Percentage Increase

Final Value = Initial Value × (1 + Rate ÷ 100)