Quantitative Aptitude PYQs

Shortcut Trick

For a surd of the form √(a − 2√b), find two numbers whose sum is a and product is b.

Since 6 + 5 = 11 and 6 × 5 = 30, we get √(11 − 2√30) = √6 − √5.

Its reciprocal is 1/(√6 − √5), which simplifies to its conjugate √6 + √5.

Substituting back: (√6 − √5) − (√6 + √5) = −2√5.

∴ The correct answer is −2√5.

Alternate Method

Given:

Expression = √(11 − 2√30) − 1/√(11 − 2√30)

Formula Used:

a² + b² − 2ab = (a − b)²

(a − b)(a + b) = a² − b²

Calculations:

⇒ Let us simplify the root term √(11 − 2√30).

⇒ We know that 11 = 6 + 5 and 30 = 6 × 5.

⇒ 11 − 2√30 = 6 + 5 − 2√(6 × 5)

⇒ 11 − 2√30 = (√6)² + (√5)² − 2(√6)(√5)

⇒ 11 − 2√30 = (√6 − √5)²

⇒ Taking the square root, √(11 − 2√30) = √6 − √5

⇒ Now, find the value of the reciprocal 1/√(11 − 2√30):

⇒ 1/(√6 − √5)

⇒ Rationalizing by multiplying the numerator and denominator by (√6 + √5):

⇒ (√6 + √5) / [(√6 − √5)(√6 + √5)]

⇒ (√6 + √5) / [(√6)² − (√5)²]

⇒ (√6 + √5) / (6 − 5) = √6 + √5

⇒ Substitute these values back into the original expression:

⇒ Expression = (√6 − √5) − (√6 + √5)

⇒ Expression = √6 − √5 − √6 − √5

⇒ Expression = −2√5

∴ The correct answer is −2√5.

Additional Information

Square Root of a Surd

To find the square root of a surd in the form (x ± 2√y), find two numbers 'a' and 'b' such that a + b = x and a × b = y. The simplified square root is √a ± √b.

Rationalization

The process of eliminating surds from the denominator by multiplying the numerator and denominator by its conjugate. The conjugate of √a − √b is √a + √b.

Algebraic Identity

Difference of squares: (a − b)(a + b) = a² − b², which is frequently used to simplify surd denominators during rationalization.

Shortcut Trick

The radius, the perpendicular distance, and half the chord form a right-angled triangle.

The given dimensions are multiples of the standard Pythagorean triplet (5, 12, 13).

Hypotenuse (radius) = 26 cm = 13 × 2

Perpendicular distance = 10 cm = 5 × 2

Thus, the half-chord must be = 12 × 2 = 24 cm.

The total length of the chord is 24 × 2 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Radius of the circle (OA) = 26 cm

Distance from center to the midpoint of the chord (OM) = 10 cm

O A B M

10 cm 26 cm 24 cm 24 cm

Formula Used:

Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ The line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord.

⇒ In right-angled ΔOMA, OA² = OM² + AM²

⇒ 26² = 10² + AM²

⇒ 676 = 100 + AM²

⇒ AM² = 676 − 100 = 576

⇒ AM = √576 = 24 cm

⇒ Since M is the midpoint of chord AB, the total length of the chord AB = 2 × AM

⇒ AB = 2 × 24 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Perpendicular from Center

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

Equal Chords Property

Chords that are equal in length are always equidistant from the center of the circle.

Pythagorean Triplets

A set of three positive integers a, b, and c that perfectly satisfy a² + b² = c². Commonly used triplets in geometry include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Shortcut Trick

According to the Inscribed Angle Theorem, the angle subtended by a chord at the center is double the angle subtended by it at any point on the remaining part of the circle.

⇒ Angle at circle = Angle at center ÷ 2

⇒ Angle at circle = 60° ÷ 2 = 30°

∴ The correct answer is 30°.

Alternate Method

Given:

Angle subtended by the chord at the center = 60°

Formula Used:

Angle subtended by an arc at the center = 2 × Angle subtended by it at the circumference in the same segment.

    60°  30°  O  A  B  P ∠APB = 30°

Calculations:

Let AB be the chord and O be the center of the circle.

⇒ ∠AOB = 60°

Let P be a point on the circumference in the same segment.

⇒ ∠APB = ∠AOB ÷ 2

⇒ ∠APB = 60° ÷ 2

⇒ ∠APB = 30°

∴ The correct answer is 30°.

Additional Information

Inscribed Angle Theorem

The angle subtended by an arc or chord at the center is double the angle subtended by it at any point on the remaining part of the circle.

Angles in the Same Segment

Angles subtended by the same arc at any points on the remaining part of the circle are always equal to each other.

Angle in a Semicircle

An angle subtended by a diameter at any point on the circumference is always a right angle (90°).

Cyclic Quadrilateral

If all four vertices of a quadrilateral lie on the circumference of a circle, the sum of its opposite angles is always equal to 180°.

Shortcut Trick

We know the standard trigonometric identity: sec²A − tan²A = 1

This expands to: (secA − tanA) × (secA + tanA) = 1

Thus, (secA − tanA) is always the reciprocal of (secA + tanA).

⇒ secA − tanA = 1 ÷ (5/2) = 2/5

∴ The correct answer is 2/5.

Alternate Method

Given:

secA + tanA = 5/2

Formula Used:

a² − b² = (a − b)(a + b)

sec²A − tan²A = 1

Calculations:

⇒ sec²A − tan²A = 1

⇒ (secA − tanA)(secA + tanA) = 1

Substituting the given value of (secA + tanA):

⇒ (secA − tanA) × (5/2) = 1

⇒ secA − tanA = 1 × (2/5)

⇒ secA − tanA = 2/5

∴ The correct answer is 2/5.

Additional Information

Fundamental Trigonometric Identities

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Reciprocal Pairs in Trigonometry

Just like (secθ − tanθ) and (secθ + tanθ) are reciprocals of each other, the pair (cosecθ − cotθ) and (cosecθ + cotθ) also share a direct reciprocal relationship.

Derivation

All squared trigonometric identities are derived directly from the Pythagorean theorem: P² + B² = H² by dividing the equation by P², B², or H² respectively.

Shortcut Trick

Hourly Wage = Total Earnings ÷ Total Hours

⇒ Hourly Wage = 52.50 ÷ 3.5

⇒ Multiply by 10 to clear decimals: 525 ÷ 35

⇒ 525 ÷ 35 = 15

⇒ As a fraction: 15/1

∴ The correct answer is 15/1.

Alternate Method

Given:

Total Earnings = 52.50

Total Time = 3.5 hours

Formula Used:

Hourly Wage = Total Earnings ÷ Total Time

Calculations:

⇒ Hourly Wage = 52.50 ÷ 3.5

⇒ To remove the decimal point, multiply the numerator and the denominator by 10.

⇒ Hourly Wage = (52.50 × 10) ÷ (3.5 × 10)

⇒ Hourly Wage = 525 ÷ 35

⇒ Simplify by dividing both terms by 5:

⇒ 525 ÷ 5 = 105

⇒ 35 ÷ 5 = 7

⇒ Hourly Wage = 105 ÷ 7 = 15

⇒ Expressing the integer 15 as a fraction gives 15/1.

∴ The correct answer is 15/1.

Additional Information

Unitary Method

It is a mathematical technique used to find the value of a single unit by dividing the total value by the number of units. Here, Value of 1 hour = Total Earnings ÷ Total Hours.

Decimal Division

When dividing by a decimal value, you can multiply both the dividend and the divisor by the same power of 10 to transform the divisor into a whole number before computing the final division.

Integers as Fractions

Any whole number or integer can easily be written in a fraction format by placing it over a denominator of 1. For example, x = x/1.

Shortcut Trick

Square the term (√6 − 1) directly using the identity (a − b)² = a² − 2ab + b².

LHS = (√6)² − 2(√6)(1) + 1² = 6 − 2√6 + 1 = 7 − 2√6

Equate this to the RHS:

7 − 2√6 = ? − 2√6

? = 7

∴ The correct answer is 7.

Alternate Method

Given:

(√6 − 1)² = ? − 2√6

Formula Used:

(a − b)² = a² − 2ab + b²

Calculations:

⇒ (√6 − 1)² = ? − 2√6

⇒ (√6)² − 2 × √6 × 1 + (1)² = ? − 2√6

⇒ 6 − 2√6 + 1 = ? − 2√6

⇒ 7 − 2√6 = ? − 2√6

⇒ 7 − 2√6 + 2√6 = ?

⇒ ? = 7

∴ The correct answer is 7.

Additional Information

Standard Algebraic Identities

(a + b)² = a² + 2ab + b² | (a − b)² = a² − 2ab + b² | a² − b² = (a − b)(a + b)

Properties of Surds

The square of a square root returns the base number: (√a)² = a. Similar surds can be added or subtracted directly.

BODMAS Rule

Simplification strictly follows the sequence: Bracket, Order (Powers/Roots), Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

The interest for the n^th year in compound interest follows a geometric progression.

Interest for the 1^st year = 12% of 74000 = ₹8880

Interest for the 3^rd year = (1^st year Interest) × (1 + R/100)ⁿ⁻¹

⇒ Interest for 3^rd year = 8880 × (1 + 12/100)²

⇒ 8880 × (1.12)² = 8880 × 1.2544 = 11139.072 ≈ ₹11139.07

∴ The correct answer is ₹11139.07.

Alternate Method

Given:

Principal (P) = ₹74,000

Rate of Interest (R) = 12% per annum

Formula Used:

Amount after n years = P × (1 + R/100)ⁿ

Interest for the n^th year = Amount after (n−1) years × (R/100)

Start End Year 1 End Year 2 3^rd Yr Int. ₹74,000 ₹82,880 ₹92,825.6 ₹11,139.1 +12% +8880 +12% +9945.6 12% Int. Progression of principal amount and accumulated interest

Calculations:

⇒ Amount after 1 year (A₁) = 74000 × (1 + 12/100)

⇒ A₁ = 74000 × 1.12 = ₹82,880

⇒ Amount after 2 years (A₂) = 82880 × (1 + 12/100)

⇒ A₂ = 82880 × 1.12 = ₹92,825.6

⇒ Interest for the 3^rd year = 12% of Amount after 2 years

⇒ I₃ = 92825.6 × (12/100)

⇒ I₃ = 11139.072

∴ The correct answer is ₹11139.07.

Additional Information

Compound Interest Formula

Total Amount A = P(1 + R/100)ⁿ, where P is the principal, R is the rate of interest, and n is the time period.

Successive Interest Growth

In compound interest, the interest amount for each subsequent year behaves like a geometric progression with a common ratio of (1 + R/100).

Simple vs Compound Interest

For the 1^st year, simple and compound interest are identical. From the 2^nd year onwards, compound interest is always greater because interest is earned on previously accumulated interest.

Shortcut Trick

When Simple Interests are equal for two different sums:

P₁ : P₂ = (R₂ × T₂) : (R₁ × T₁)

Ratio of A to B = (4 × 3) : (6 × 2)

Ratio of A to B = 12 : 12 = 1:1

∴ The correct answer is 1:1.

Alternate Method

Given:

For Principal A: Rate (R₁) = 6% p.a., Time (T₁) = 2 years

For Principal B: Rate (R₂) = 4% p.a., Time (T₂) = 3 years

Simple interest generated from A and B are equal.

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Case 1 Principal = A R = 6%, T = 2 yr SI = 12% of A Case 2 Principal = B R = 4%, T = 3 yr SI = 12% of B = 12% of A = 12% of B → A : B = 1 : 1

Calculations:

⇒ SI on A = (A × 6 × 2) ÷ 100

⇒ SI on B = (B × 4 × 3) ÷ 100

Since the simple interest is equal in both cases:

⇒ (A × 12) ÷ 100 = (B × 12) ÷ 100

⇒ 12A = 12B

⇒ A ÷ B = 1 ÷ 1

⇒ A : B = 1:1

∴ The correct answer is 1:1.

Additional Information

Equal Simple Interest Distribution

When a sum is divided into parts such that their simple interests are equal, the ratio of their principals is P₁ : P₂ = (1 ÷ R₁T₁) : (1 ÷ R₂T₂).

Equal Amount Distribution

When a sum is divided into parts such that their final amounts are equal, the ratio of principals is P₁ : P₂ = 1 ÷ (100 + R₁T₁) : 1 ÷ (100 + R₂T₂).

Compound Ratio

When evaluating multiplicative factors in equations like 12A = 12B, solving for proportional components is simplified using the inverse ratio property A : B = B's coefficient : A's coefficient.

Shortcut Trick

A real number that is not rational is called an irrational number.

Square roots of non-perfect squares are always irrational.

Since 7 is not a perfect square, its square root cannot be written as a simple fraction.

The numbers 3/4, 5, and −2 are clearly rational as they can easily be expressed in the form p/q.

∴ The correct answer is √7.

Alternate Method

Given:

Options: 3/4, 5, √7, −2

Definition Used:

⇒ Rational Number: A number that can be expressed as p/q, where p and q are integers and q ≠ 0.

⇒ Irrational Number: A real number that is not rational.

Calculations:

⇒ Option 1: 3/4 is already in the form p/q, so it is a rational number.

⇒ Option 2: 5 can be written as 5/1, which is in the form p/q, so it is a rational number.

⇒ Option 4: −2 can be written as −2/1, which is in the form p/q, so it is a rational number.

⇒ Option 3: √7. Since 7 is not a perfect square, its square root evaluates to a non-terminating and non-repeating decimal.

⇒ Thus, √7 cannot be expressed as a simple fraction p/q.

⇒ Therefore, √7 is a real number but NOT rational.

∴ The correct answer is √7.

Additional Information

Real Numbers

Real numbers consist of all rational and irrational numbers. They can be geometrically represented on a continuous number line.

Rational Numbers

Numbers that can be written as a ratio of two integers p/q (where q ≠ 0). Their decimal forms are either terminating or repeating.

Irrational Numbers

Numbers that cannot be written as a simple fraction of integers. Their decimal expansions are non-terminating and non-repeating. Common examples include π, √2, and √3.

Perfect Squares

The square root of a perfect square integer (like √16 = 4) is always rational. Conversely, the square root of any non-perfect square integer is always irrational.

Shortcut Trick

Speed = Distance ÷ Time

Speed in m/s = 500 ÷ (5 × 60) = 500 ÷ 300 = 5/3 m/s

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

Speed in km/h = (5/3) × (18/5) = 18 ÷ 3 = 6 km/h

∴ The correct answer is 6 km/h.

Alternate Method

Given:

Distance covered by Manoj = 500 m

Time taken = 5 minutes = 5 × 60 = 300 seconds

Distance = 500 m Time = 5 minutes (300 s) Speed = ? km/h

Formula Used:

Speed = Distance ÷ Time

Conversion rule: x m/s = x × (18/5) km/h

Calculations:

⇒ Speed in m/s = 500 ÷ 300

⇒ Speed = 5/3 m/s

⇒ Converting to km/h: Speed = (5/3) × (18/5) km/h

⇒ Speed = (5 × 18) ÷ (3 × 5)

⇒ Speed = 90 ÷ 15 = 6 km/h

∴ The correct answer is 6 km/h.

Additional Information

Unit Conversion Rule

1 km/h = (1000 m) ÷ (3600 s) = 5/18 m/s.

To convert m/s to km/h, multiply by the inverse fraction, which is 18/5.

Average Speed Concept

Average Speed = Total Distance ÷ Total Time.

When an object covers two equal distances at speeds a and b, the Average Speed = 2ab ÷ (a + b).

Relative Speed Concept

When two bodies travel in the same direction, Relative Speed = |a − b|.

When two bodies travel in opposite directions, Relative Speed = a + b.

Shortcut Trick

Total Volume = Volume of Cube + Volume of Pyramid

⇒ Total Volume = (side)³ + (1/3) × (√3 / 4 × base²) × height

⇒ Total Volume = 10³ + (1/3) × (√3 / 4 × 100) × 14

⇒ Total Volume = 1000 + (350√3 / 3) ≈ 1000 + 202.07

⇒ Total Volume = 1202.07 cm³

∴ The correct answer is 1202.07 cm³.

Alternate Method

Given:

Side of the cube (a) = 10 cm

Side of the equilateral triangular base of the pyramid (b) = 10 cm

Height of the pyramid (h) = 14 cm

Formula Used:

Volume of a cube = a³

Area of an equilateral triangle = (√3 / 4) × b²

Volume of a pyramid = (1/3) × Base Area × height

10 cm 10 cm h = 14 cm Equilateral Base = 10 cm Composite Solid: Cube + Pyramid

Calculations:

⇒ Volume of the cube = 10 × 10 × 10 = 1000 cm³

⇒ Area of the pyramid's base = (√3 / 4) × 10² = 25√3 cm²

⇒ Volume of the pyramid = (1/3) × 25√3 × 14 = (350√3) / 3 cm³

⇒ Substituting the approximate value of √3 ≈ 1.732:

⇒ Volume of the pyramid ≈ (350 × 1.732) / 3

⇒ Volume of the pyramid ≈ 606.2 / 3 ≈ 202.07 cm³

⇒ Total Volume = Volume of the cube + Volume of the pyramid

⇒ Total Volume = 1000 + 202.07 = 1202.07 cm³

∴ The correct answer is 1202.07 cm³.

Additional Information

Volume of a Pyramid

Volume = (1/3) × Base Area × Height. This formula is universally applicable to any pyramid, regardless of the 2D shape of its base (square, triangle, hexagon, etc.).

Area of an Equilateral Triangle

Area = (√3 ÷ 4) × a², where a is the length of the side. In an equilateral triangle, all three sides are equal in length, and each internal angle measures 60°.

Volume of a Cube

Volume = a³, where a is the side length. A cube is a special type of right prism where all six faces are identical squares.

Shortcut Trick

Profit ratio is directly proportional to Capital × Time.

Since the time period is equal (12 months), Profit Ratio = 60,000 : 40,000 = 3 : 2

Remaining profit to be divided = Rs. 90,000

B's share = (2 ÷ 5) × 90,000 = Rs. 36,000

∴ The correct answer is Rs. 36,000.

Alternate Method

Given:

A's investment = Rs. 60,000

B's investment = Rs. 40,000

Time period for both = 12 months

Remaining profit to be distributed based on capital = Rs. 90,000

Partner A Inv: Rs. 60,000 Time: 12 months Ratio = 3 Partner B Inv: Rs. 40,000 Time: 12 months Ratio = 2 Remaining Profit for Distribution = Rs. 90,000

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Calculations:

⇒ Profit Ratio (A : B) = (60000 × 12) : (40000 × 12)

⇒ Profit Ratio (A : B) = 60000 : 40000

⇒ Profit Ratio (A : B) = 3 : 2

⇒ Total ratio parts = 3 + 2 = 5

⇒ B's share of the remaining profit = (2 ÷ 5) × Total Remaining Profit

⇒ B's share = (2 ÷ 5) × 90,000

⇒ B's share = 2 × 18,000 = Rs. 36,000

∴ The correct answer is Rs. 36,000.

Additional Information

Profit Sharing Ratio

Profit is divided among partners in the ratio of the product of their Capital and Time period. (Profit Ratio = C₁×T₁ : C₂×T₂).

Working Partner

A partner who manages the business is called an active or working partner. They generally receive a fixed salary or a certain percentage of the overall profit before the remaining profit is distributed.

Sleeping Partner

A partner who only invests capital and does not take part in managing the business is called a sleeping partner. They only receive a share of the remaining profit proportional to their investment.

Shortcut Trick

Area of triangle = 1/2 × Base × Height = 1/2 × 40 × 30 = 600 m².

Area of rectangle = Length × Breadth = 40 × 25 = 1000 m².

Combined Area = 600 + 1000 = 1600 m².

Percentage = (600 ÷ 1600) × 100 = (3 ÷ 8) × 100 = 37.5%.

∴ The correct answer is 1600 m² and 37.5%.

Alternate Method

Given:

Base of the triangle = 40 m

Height of the triangle = 30 m

Length of the rectangle = 40 m

Breadth of the rectangle = 25 m

Formulas Used:

Area of a Triangle = 1/2 × Base × Height

Area of a Rectangle = Length × Breadth

Percentage = (Part Value ÷ Total Value) × 100

 40 m 25 m 30 m Area = 600 sq m Area = 1000 sq m

Calculations:

⇒ Area of the triangle = 1/2 × 40 × 30

⇒ Area of the triangle = 20 × 30 = 600 m²

⇒ Area of the rectangle = 40 × 25 = 1000 m²

⇒ Total combined area = 600 + 1000 = 1600 m²

⇒ Percentage of the triangle's area in total area = (600 ÷ 1600) × 100

⇒ Percentage = (3 ÷ 8) × 100 = 37.5%

∴ The correct answer is 1600 m² and 37.5%.

Additional Information

Area of 2D Shapes

The area of a planar figure refers to the total space enclosed within its boundary. For standard polygons like triangles and rectangles, direct mathematical formulas are applied based on their distinct dimensions.

Types of Triangles Based on Area

For an equilateral triangle, area = (√3 ÷ 4) × side². For scalene triangles, Heron's formula is widely utilized: Area = √[s(s − a)(s − b)(s − c)], where s represents the semi-perimeter.

Percentages in Geometry

Comparing the region of two interlinked figures as a percentage effectively establishes proportional spatial distribution. It is calculated continuously by dividing the targeted subset area by the overall total area multiplied by 100.

Shortcut Trick

A regular hexagon inscribed in a circle is made up of 6 equilateral triangles with sides equal to the circle's radius.

Side of the hexagon (a) = Radius (r) = 14 cm

Area = 6 × (√3 ÷ 4) × r²

Area = 6 × (√3 ÷ 4) × 196 = 294√3

Area ≈ 294 × 1.732 ≈ 509.21 cm²

∴ The correct answer is 509.21 cm².

Alternate Method

      O r = 14 cm a = 14 cm Regular Hexagon Inside A Circle

Given:

Radius of the circumscribed circle (r) = 14 cm

Formula Used:

Area of a regular hexagon = (3√3 ÷ 2) × a²

Where 'a' is the side length of the regular hexagon.

Calculations:

⇒ For a regular hexagon inscribed in a circle, the side length 'a' is equal to the radius 'r' of the circle.

⇒ Side length (a) = 14 cm

⇒ Area = (3√3 ÷ 2) × (14)²

⇒ Area = (3√3 ÷ 2) × 196

⇒ Area = 3 × 98 × √3

⇒ Area = 294√3 cm²

⇒ Substituting the value of √3 ≈ 1.732:

⇒ Area ≈ 294 × 1.732

⇒ Area ≈ 509.208 cm²

⇒ Rounding off to two decimal places gives 509.21 cm².

∴ The correct answer is 509.21 cm².

Additional Information

Circumradius of a Regular Hexagon

When a regular hexagon is inscribed inside a circle, the radius of the circle (circumradius) is exactly equal to the side length of the hexagon: R = a.

Inradius of a Regular Hexagon

When a circle is inscribed inside a regular hexagon, the radius of the circle (inradius) is equal to the altitude of the equilateral triangles forming the hexagon: r = (√3 ÷ 2) × a.

Composition of a Hexagon

A regular hexagon is formed by 6 identical equilateral triangles. Thus, its total area is exactly 6 times the area of one equilateral triangle: Area = 6 × [(√3 ÷ 4) × a²].

Inscribed vs Circumscribed Areas

For any given circle, the area of the regular hexagon circumscribed around it is exactly 4/3 times the area of the regular hexagon inscribed within it.

Shortcut Trick

Let the first number be 100.

Second number = 150% of 100 = 150.

Third number = 80% of 150 = 120.

Required Ratio = First number : Third number = 100 : 120 = 5 : 6.

∴ The correct answer is 5:6.

Alternate Method

Given:

The second number is 150% of the first number.

The third number is 80% of the second number.

Formula Used:

Value = (Percentage ÷ 100) × Base Value

First: 100x × 1.5 Second: 150x × 0.8 Third: 120x Ratio = 100x : 120x = 5 : 6

Calculation:

⇒ Let the first number be 100x.

⇒ Second number = (150 ÷ 100) × 100x = 150x

⇒ Third number = (80 ÷ 100) × 150x = 120x

⇒ Ratio of first to third = 100x : 120x

⇒ Ratio = 10 : 12 = 5 : 6

∴ The correct answer is 5:6.

Additional Information

Successive Ratio Method (Compound Ratio)

If A:B = x:y and B:C = m:n, then the compound ratio A:C can be found by multiplying the individual ratios: A:C = (x × m) : (y × n).

Fraction Equivalents of Percentages

Converting percentages into their simplest fraction forms speeds up calculation: 150% = 3/2, and 80% = 4/5. Using fractions directly: A : C = 1 : (3/2 × 4/5) = 1 : 6/5 = 5 : 6.

Continuous Percentage Change

When applying percentages sequentially on a base, the final value is found by successive multiplication: Final Value = Base × (1 ± p₁/100) × (1 ± p₂/100).

Shortcut Trick

The given ratio of the legs of the right triangle is 5:12.

This corresponds to the standard Pythagorean triplet (5, 12, 13).

Since the corresponding ratio hypotenuse (13) matches the actual hypotenuse (13 m), the lengths of the legs are exactly 5 m and 12 m.

Area = 1/2 × base × height = 1/2 × 12 × 5 = 30 m².

∴ The correct answer is 30 m².

Alternate Method

Given:

The lengths of the legs of a right triangle are in the ratio 5:12.

The hypotenuse of the right triangle = 13 m.

Formula Used:

Area of a right-angled triangle = 1/2 × base × height

Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

5x 12x 13 m A B C

⇒ Let the base and perpendicular of the right triangle be 12x and 5x respectively.

⇒ According to Pythagoras theorem:

⇒ (13)² = (12x)² + (5x)²

⇒ 169 = 144x² + 25x²

⇒ 169 = 169x²

⇒ x² = 1

⇒ x = 1

⇒ The base of the triangle = 12 × 1 = 12 m.

⇒ The height of the triangle = 5 × 1 = 5 m.

⇒ Area of the right triangle = 1/2 × 12 × 5

⇒ Area = 6 × 5 = 30 m²

∴ The correct answer is 30 m².

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy the condition a² + b² = c² is known as a Pythagorean triplet. Common and frequently tested triplets include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Heron's Formula

For any given triangle with side lengths a, b, and c, the area can be calculated using Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter calculated as s = (a+b+c) ÷ 2.

Equilateral Triangle Area

For an equilateral triangle where all three sides are equal, the area can be directly calculated using the formula: Area = (√3 ÷ 4) × side².

Shortcut Trick

Let the original number be 100.

Mistaken value (increased by 25%) = 125.

Correct value (decreased by 20%) = 80.

Required percentage = [(125 − 80) ÷ 80] × 100%

Required percentage = (45 ÷ 80) × 100% = 56.25%

∴ The correct answer is 56.25%.

Alternate Method

Given:

A number is mistakenly increased by 25% instead of being decreased by 20%.

            Correct (80) Original (100) Mistaken (125)  −20%  +25%  Difference = +45 Percentage Change Visualization

Calculations:

⇒ Let the original number be 100.

⇒ Correct value = 100 − (20% of 100) = 100 − 20 = 80

⇒ Mistaken value = 100 + (25% of 100) = 100 + 25 = 125

⇒ Difference between mistaken value and correct value = 125 − 80 = 45

⇒ Required percentage = (Difference ÷ Correct value) × 100%

⇒ Required percentage = (45 ÷ 80) × 100%

⇒ Required percentage = (9 ÷ 16) × 100%

⇒ Required percentage = 900 ÷ 16 = 56.25%

∴ The correct answer is 56.25%.

Additional Information

Percentage Error

Percentage Error = (|Mistaken Value − Correct Value| ÷ Correct Value) × 100%. It measures the deviation of an incorrect operation from the true or expected value.

Base of Percentage

When asked 'by what percent is A more or less than B', the value B is considered the base. The formula used is: [(A − B) ÷ B] × 100%.

Successive Percentage Change

When a number is changed successively by x% and y%, the net percentage change is given by: Net Change = x + y + (x × y) ÷ 100.

Shortcut Trick

Let the cost price for the wholesaler be CP.

Effective price multiplier = 1.20 (profit) × 1.50 (markup) × 0.80 (discount) = 1.44

Customer SP = CP × 1.44 = 480 ⇒ CP = 480 ÷ 1.44 = 333.33

∴ The correct answer is Rs. 333.33.

Alternate Method

Given:

Profit% = 20%, Markup% = 50%, Discount% = 20%

Final customer selling price = Rs. 480

 100u 120u 180u 144u Wholesaler CP Retailer CP Marked Price Customer = Rs. 480   +20%   +50%   −20% 144u = Rs. 480 ⇒ 100u = Rs. 333.33

Calculations:

⇒ Let the cost price for the wholesaler be 100u.

⇒ Retailer's CP = 100u + 20% of 100u = 120u

⇒ Marked Price = 120u + 50% of 120u = 120u + 60u = 180u

⇒ Customer's SP = 180u − 20% of 180u = 180u − 36u = 144u

⇒ According to the given data, 144u = 480

⇒ 1u = 480 ÷ 144 = 10 ÷ 3

⇒ Wholesaler CP (100u) = 100 × (10 ÷ 3) = 1000 ÷ 3 = 333.33

∴ The correct answer is Rs. 333.33.

Additional Information

Marked Price and Discount

Discount is always computed on the Marked Price (MP), not the Cost Price (CP). The final selling price is given by SP = MP − Discount.

Successive Percentage Changes

If an initial value experiences consecutive changes of a% and b%, the combined net percentage change is calculated using: Net Change = (a + b + a×b ÷ 100)%.

Chain Sale Rule

When an item passes consecutively through multiple sellers (A to B to C) at corresponding markups or profit percentages p% and q%, the final price equals Initial CP × [(100 + p) ÷ 100] × [(100 + q) ÷ 100].

Shortcut Trick

For a regular right prism, Base Area = Volume ÷ Height.

Base Area = 1024 ÷ 16 = 64 cm².

Since the base is a square, Side = √Base Area.

Side = √64 = 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

Given:

Volume of the prism (V) = 1024 cm³

Height of the prism (h) = 16 cm

The base of the prism is a square.

Formula Used:

Volume of a prism = Base Area × Height

Area of a square = Side²

       h = 16 cm  Side = a a V = 1024 cm³

Calculations:

Let the side of the square base be a cm.

⇒ Base Area = a²

⇒ Volume = Base Area × Height

⇒ 1024 = a² × 16

⇒ a² = 1024 ÷ 16

⇒ a² = 64

⇒ a = √64

⇒ a = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Right Prism

A prism is a 3D solid with two identical, parallel bases. Volume = Base Area × Height. Lateral Surface Area = Base Perimeter × Height.

Right Circular Cylinder

A cylinder is a special prism with circular bases. Volume = πr²h, where r is the base radius and h is the height.

Pyramid

A pyramid has a polygon base and triangular faces that meet at a common apex. Volume = (1 ÷ 3) × Base Area × Height.

Total Surface Area of Prism

Total Surface Area (TSA) = Lateral Surface Area + 2 × Base Area.

Shortcut Trick

Time required for 1 tap to fill half the tank = 14 ÷ 2 = 7 hours.

Total taps working on the remaining half = 1 (initial) + 7 (new) = 8 taps.

Time for 8 taps to fill the remaining half = 7 ÷ 8 hours = 52.5 minutes = 52 minutes 30 seconds.

Total time = 7 hours + 52 minutes 30 seconds = 7 hr 52 min 30 sec.

∴ The correct answer is 7 hr 52 min 30 sec.

Alternate Method

Given:

Time taken by 1 tap to fill the complete tank = 14 hours.

After half the tank is filled, 7 more similar taps are opened.

Calculations:

⇒ Total capacity of the tank = 14 units.

⇒ Time taken by 1 tap to fill half the tank (7 units) = 7 units ÷ 1 unit/hour = 7 hours.

⇒ Remaining tank capacity to be filled = 14 − 7 = 7 units.

⇒ Total number of taps working now = 1 + 7 = 8 taps.

⇒ Combined efficiency of 8 taps = 8 × 1 = 8 units/hour.

⇒ Time taken to fill the remaining 7 units = 7 units ÷ 8 units/hour = 7/8 hours.

⇒ Convert 7/8 hours to minutes: (7 ÷ 8) × 60 = 105 ÷ 2 = 52.5 minutes.

⇒ 52.5 minutes is exactly 52 minutes and 30 seconds.

⇒ Total time taken to completely fill the tank = 7 hours + 52 minutes 30 seconds = 7 hr 52 min 30 sec.

∴ The correct answer is 7 hr 52 min 30 sec.

Additional Information

Pipes & Cisterns Concept

If a pipe can fill a tank in x hours, then the part filled in 1 hour is 1/x. Efficiency is inversely proportional to the time taken to complete the total work.

Combined Work Rule

When n identical entities work together, their combined efficiency becomes n times the individual efficiency, and the total time taken to do the same work reduces to 1/n of the individual time.

Time Conversion

To convert a fraction of an hour into minutes, simply multiply the fractional value by 60. To further convert a fraction of a minute into seconds, multiply that decimal or fraction by 60 again.