Quantitative Aptitude PYQs

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 282 + 120 + 496 = 898 km

Total Time = (282 ÷ 47) + (120 ÷ 20) + (496 ÷ 62) = 6 + 6 + 8 = 20 hr

Average Speed = 898 ÷ 20 = 44.9 km/hr

∴ The correct answer is 44.9 km/hr.

Alternate Method

Given:

Distance 1 (D₁) = 282 km, Speed 1 (S₁) = 47 km/hr

Distance 2 (D₂) = 120 km, Speed 2 (S₂) = 20 km/hr

Distance 3 (D₃) = 496 km, Speed 3 (S₃) = 62 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = Total Distance ÷ Total Time

D1 = 282 km
S1 = 47 km/hr
T1 = 6 hr

D2 = 120 km
S2 = 20 km/hr
T2 = 6 hr

D3 = 496 km
S3 = 62 km/hr
T3 = 8 hr

Total Distance = 898 km | Total Time = 20 hr

Calculations:

⇒ Time for 1^st part (T₁) = 282 ÷ 47 = 6 hours

⇒ Time for 2^nd part (T₂) = 120 ÷ 20 = 6 hours

⇒ Time for 3^rd part (T₃) = 496 ÷ 62 = 8 hours

⇒ Total Distance = 282 + 120 + 496 = 898 km

⇒ Total Time = 6 + 6 + 8 = 20 hours

⇒ Average Speed = 898 ÷ 20 = 44.9 km/hr

∴ The correct answer is 44.9 km/hr.

Additional Information

Average Speed for Two Equal Distances

If a person travels two equal distances at speeds x and y, then Average Speed = (2 × x × y) ÷ (x + y).

Average Speed for Three Equal Distances

If a person travels three equal distances at speeds x, y and z, then Average Speed = (3 × x × y × z) ÷ (xy + yz + zx).

Relative Speed

When two bodies move in the same direction, Relative Speed = Difference of speeds. When they move in opposite directions, Relative Speed = Sum of speeds.

Shortcut Trick

Radius of the sphere (r) = Diameter ÷ 2 = 30 ÷ 2 = 15 cm.

Volume of a sphere = 4/3 × π × r³

⇒ Volume = 4/3 × 3.14 × 15³

⇒ Volume = 4/3 × 3.14 × 3375

⇒ Volume = 4 × 3.14 × 1125 = 14,130

∴ The correct answer is 14,130 cm³.

Alternate Method

Sphere Volume Diagram

r = 15 cm

Diameter (d) = 30 cm

Given:

Diameter of the sphere (d) = 30 cm

Value of π ≈ 3.14

Formula Used:

Radius (r) = Diameter ÷ 2

Volume of a Sphere = 4/3 × π × r³

Calculations:

⇒ Radius (r) = 30 ÷ 2 = 15 cm

⇒ Volume = 4/3 × 3.14 × (15)³

⇒ Volume = 4/3 × 3.14 × 3375

⇒ Volume = 4 × 3.14 × (3375 ÷ 3)

⇒ Volume = 4 × 3.14 × 1125

⇒ Volume = 4500 × 3.14

⇒ Volume = 14130 cm³

∴ The correct answer is 14,130 cm³.

Additional Information

Surface Area of a Sphere

The total area covered by the outer surface of a complete 3D sphere. Formula: Area = 4 × π × r².

Volume of a Hemisphere

A hemisphere is exactly half of a complete sphere. Formula: Volume = 2/3 × π × r³.

Total Surface Area of a Hemisphere

It includes both the curved outer surface area and the flat circular base area. Formula: TSA = 3 × π × r².

Shortcut Trick

In ΔABC, ∠BCA = 180° − (54° + 60°) = 66°.

Since BC || AD, alternate interior angle ∠CAD = ∠BCA = 66°.

In isosceles ΔACD (AC = CD), base angles are equal ⇒ ∠CDA = ∠CAD = 66°.

∴ ∠ACD = 180° − (66° + 66°) = 48°.

∴ The correct answer is 48°.

Alternate Method

A
B
C
D

54°
60°
66°
66°
66°
?°

Given:

Trapezium ABCD where BC || AD.

∠ABC = 54°

∠BAC = 60°

AC = CD

Calculation:

⇒ In ΔABC, the sum of all interior angles is 180°.

⇒ ∠BCA = 180° − (∠ABC + ∠BAC)

⇒ ∠BCA = 180° − (54° + 60°) = 180° − 114° = 66°

⇒ Since BC || AD, the alternate interior angles across transversal AC are equal.

⇒ ∠CAD = ∠BCA = 66°

⇒ In ΔACD, it is given that AC = CD.

⇒ In a triangle, angles opposite to equal sides are equal.

⇒ ∠CDA = ∠CAD = 66°

⇒ The sum of angles in ΔACD is 180°.

⇒ ∠ACD = 180° − (∠CAD + ∠CDA)

⇒ ∠ACD = 180° − (66° + 66°) = 180° − 132° = 48°

∴ The correct answer is 48°.

Additional Information

Properties of Trapezium

A convex quadrilateral with at least one pair of parallel sides. The consecutive interior angles between the parallel sides always sum to 180°.

Isosceles Triangle Properties

If two sides of a triangle are equal in length, the angles opposite to them are also equal. The altitude dropped to the base also bisects the base.

Alternate Interior Angles Theorem

When a transversal intersects two parallel lines, the alternate interior angles formed on opposite sides of the transversal are perfectly equal.

Shortcut Trick

Identify the last digit of the given number 740705.

The last digit is 5.

According to the divisibility rule, any number ending in 0 or 5 is completely divisible by 5.

∴ The correct answer is 5.

Alternate Method

Given:

The number is 740705.

Concept Used:

We check the divisibility rules systematically for each of the given options: 12, 4, 5, and 9.

Calculations:

⇒ Divisibility by 4 → The last two digits are 05. Since 05 is not divisible by 4, the entire number is not divisible by 4.

⇒ Divisibility by 12 → For a number to be divisible by 12, it must be divisible by both 3 and 4. Since it fails the divisibility test for 4, it is not divisible by 12.

⇒ Divisibility by 9 → The sum of the digits is 7 + 4 + 0 + 7 + 0 + 5 = 23. Since 23 is not a multiple of 9, the number is not divisible by 9.

⇒ Divisibility by 5 → The unit digit of the number is 5. Since any number with a unit digit of 0 or 5 is exactly divisible by 5, the number is divisible by 5.

∴ The correct answer is 5.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its digits is a multiple of 3.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is perfectly divisible by 8.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is 0 or a multiple of 11.

Shortcut Trick

Since the time of investment is constant, Profit Ratio = Capital Investment Ratio.

Capital Ratio = 1350 : 1470 : 1380 = 45 : 49 : 46

Total ratio parts = 45 + 49 + 46 = 140 parts

140 parts = ₹1330 ⇒ 1 part = 1330 ÷ 140 = ₹9.5

Isha's share = 46 parts = 46 × 9.5 = ₹437

∴ The correct answer is ₹437.

Alternate Method

Given:

Investment by Ritu = ₹1,350

Investment by Sameer = ₹1,470

Investment by Isha = ₹1,380

Total profit at the end of the year = ₹1,330

Formula Used:

Profit Ratio = Capital Ratio × Time Ratio

Since the time period is 1 year for all partners, Profit Ratio = Capital Ratio.

Ritu

₹1350

Sameer

₹1470

Isha

₹1380

Capital Ratio → 45 : 49 : 46

Calculations:

⇒ Profit ratio of Ritu, Sameer and Isha = 1350 : 1470 : 1380

⇒ Simplifying the ratio by dividing by 10 = 135 : 147 : 138

⇒ Further simplifying by dividing by 3 = 45 : 49 : 46

⇒ Sum of the profit ratios = 45 + 49 + 46 = 140

⇒ Isha's profit share = (46 ÷ 140) × 1330

⇒ Isha's profit share = 46 × 9.5

⇒ Isha's profit share = 437

∴ The correct answer is ₹437.

Additional Information

Profit Ratio in Partnership

Profit is always divided in the ratio of the product of capital invested and the time period of investment (Profit = Capital × Time).

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or a certain percentage of the profit, after which the remaining profit is distributed. A sleeping partner solely provides investment.

Joining Mid-way

If a partner joins or leaves the business mid-way, their respective profit share must be calculated strictly on the exact number of months their capital actually remained invested.

Shortcut Trick

Deviation of the 10 matches from the initial average = 14 − 41 = −27

Total deviation for 10 matches = 10 × (−27) = −270

This total deviation is distributed equally among all 33 matches.

Net change in average = −270 ÷ 33 ≈ −8.18

New average = 41 − 8.18 = 32.82

∴ The correct answer is 32.82 runs.

Alternate Method

Given:

Average runs in the first 23 matches = 41

Average runs in the next 10 matches = 14

Formula Used:

Combined Average = (Sum of all observations) ÷ (Total number of observations)

23 Matches (Avg 41)
10 Matches (Avg 14)
Combined Average ≈ 32.82
Runs = 41
Runs = 14

Calculations:

⇒ Total runs in the first 23 matches = 23 × 41 = 943

⇒ Total runs in the next 10 matches = 10 × 14 = 140

⇒ Total runs in all 33 matches = 943 + 140 = 1083

⇒ Total number of matches = 23 + 10 = 33

⇒ Combined Average = 1083 ÷ 33

⇒ Combined Average = 32.8181...

⇒ Rounding to two decimal places, Combined Average ≈ 32.82

∴ The correct answer is 32.82 runs.

Additional Information

Weighted Average

When groups of different sizes are combined, the overall average is calculated using the formula: (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method

A shortcut technique used to find the combined average quickly by assuming a base average and calculating the net deviation distributed uniformly over the total count.

Arithmetic Mean

It is defined as the sum of a collection of values divided by the total count of values in that collection.

Shortcut Trick

Monthly Interest = 4% of ∞1200 = ₹48

Since interest is simple, it remains constant every month.

Total Interest for 3 months = ₹48 × 3 = ₹144

∴ The correct answer is ₹144.

Alternate Method

Given:

Principal (P) = ₹1,200

Rate (R) = 4% per month

Time (T) = 3 months

Formula Used:

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

P = ₹1200
Month 0

+4%

1
+₹48

+4%

2
+₹48

+4%

3
+₹48

Total Interest = ₹48 × 3 = ₹144

Simple Interest Timeline (Monthly)

Calculations:

⇒ SI = (1200 × 4 × 3) ÷ 100

⇒ SI = 12 × 4 × 3

⇒ SI = 144

∴ The correct answer is ₹144.

Additional Information

Unit Matching Rule

In the simple interest formula, the unit of Time (T) must strictly match the period of the Interest Rate (R). Here, since the rate is per month, the time must also be in months to directly use the formula without conversion.

Annual Rate Conversion

If the given rate was an annual rate, it would need to be divided by 12 to find the monthly interest. E.g., an annual rate of 12% means a monthly rate of 12 ÷ 12 = 1% per month.

Amount Calculation

The final Amount (A) represents the total accrued money after the given time period. Formula: A = P + SI. For this specific case, the Amount would be ₹1200 + ₹144 = ₹1344.

Shortcut Trick

Directly equate the fractions and cross-multiply to find the ratio:

First Number × (39 / 100) = Second Number × (8 / 9)

Ratio = (8 / 9) × (100 / 39)

Ratio = 800 / 351

∴ The correct answer is 800 : 351.

Alternate Method

Given:

39% of the first number is equal to 8/9 of the second number.

Equating Two Values

39% of First Number
(39 / 100) × x

=

8/9 of Second Number
(8 / 9) × y

x : y = 800 : 351

Method Used:

Let the first number be x and the second number be y.

Equate the given expressions and solve for the ratio x / y.

Calculations:

⇒ 39% of x = (8 / 9) of y

⇒ (39 / 100) × x = (8 / 9) × y

⇒ x / y = (8 / 9) ÷ (39 / 100)

⇒ x / y = (8 / 9) × (100 / 39)

⇒ x / y = (8 × 100) / (9 × 39)

⇒ x / y = 800 / 351

∴ The correct answer is 800 : 351.

Additional Information

Ratio & Proportion Basics

A ratio compares two quantities, indicating how many times the first number contains the second. It is commonly expressed as a : b or as a fraction a / b.

Percentage to Fraction Conversion

To convert any percentage to a fraction, divide the percentage value by 100. For example, x% equals x / 100.

Cross-Multiplication Rule

When solving equations involving fractions, if a / b = c / d, then multiplying the extremes and the means gives ad = bc. This simplifies proportional relationships directly.

Shortcut Trick

For an object placed on a building, observed from distance d with elevation angles α (roof) and β (top):

Height of flagpole = d × (tan β − tan α)

⇒ Height = 30 × (tan 60° − tan 30°)

⇒ Height = 30 × (√3 − 1/√3)

⇒ Height = 30 × (2/√3) = 20√3 m

∴ The correct answer is 20√3 m.

Alternate Method

House

30°

60°

A

B

C

D

30 m

h&sub2;

Given:

Distance from the base of the house, AB = 30 m

Angle of elevation of the house roof, ∠CAB = 30°

Angle of elevation of the top of the flagpole, ∠DAB = 60°

Formula Used:

tan θ = Perpendicular / Base

Calculations:

Let the height of the house be BC = h₁

Let the height of the flagpole be CD = h₂

In right-angled ▵ABC:

⇒ tan 30° = BC / AB

⇒ 1/√3 = h₁ / 30

⇒ h₁ = 30 / √3 = 10√3 m

In right-angled ▵ABD:

⇒ tan 60° = BD / AB

⇒ √3 = (h₁ + h₂) / 30

⇒ 30√3 = 10√3 + h₂

⇒ h₂ = 30√3 − 10√3 = 20√3 m

∴ The correct answer is 20√3 m.

Additional Information

Angle of Elevation vs Depression

The angle of elevation is measured upwards from the horizontal, while the angle of depression is measured downwards. Both are geometrically equal due to alternate interior angles formed by parallel lines.

Important Trigonometric Ratios

For standard height and distance problems, memorize the basic tangent values: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Complementary Angles Trick

If two points on the ground are in a straight line with the base, and their angles of elevation to the top are complementary (α + β = 90°), the height of the object can be directly calculated as h = √(d₁ × d₂).

Shortcut Trick

Apply the BODMAS rule directly to evaluate the given expression.

⇒ 16 + 12 ÷ 6 − 2 × 4

⇒ 16 + 2 − 8

⇒ 10

∴ The correct answer is 10.

Alternate Method

Given:

Expression: 16 + 12 ÷ 6 − 2 × 4

Concept Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ 16 + (12 ÷ 6) − (2 × 4)

First, evaluate the division part:

⇒ 16 + 2 − 2 × 4

Next, perform the multiplication:

⇒ 16 + 2 − 8

Then, perform the addition:

⇒ 18 − 8

Finally, complete the subtraction:

⇒ 10

∴ The correct answer is 10.

Additional Information

BODMAS Rule

It dictates the order of mathematical operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Same Precedence Operations

Division and Multiplication share the same precedence, as do Addition and Subtraction. They are evaluated sequentially from left to right.

Types of Brackets (VBODMAS)

V stands for Vinculum (bar over numbers). The priority order to solve brackets is: Vinculum, Simple brackets (), Curly brackets {}, and Square brackets [].

Shortcut Trick

Selling Price = (100% − Discount%) of Marked Price

SP = (100% − 25%) of 800

SP = 75% of 800 = (3 ÷ 4) × 800

SP = 600

∴ The correct answer is ₹600.

Alternate Method

Given:

Printed price (MP) = ₹800

Discount = 25%

Formula Used:

Selling Price = MP × (100 − Discount%) ÷ 100

Marked Price
₹800

− 25% Discount
(− ₹200)

Selling Price
₹600

Calculations:

⇒ SP = 800 × (100 − 25) ÷ 100

⇒ SP = 800 × 75 ÷ 100

⇒ SP = 8 × 75

⇒ SP = 600

∴ The correct answer is ₹600.

Additional Information

Marked Price (MP)

It is the printed price or tag price of an article. Discounts are always calculated on the Marked Price unless stated otherwise.

Successive Discount

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

Relationship between CP and MP

The ratio of Marked Price to Cost Price is given by: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Total work = LCM(13, 24) = 312 units.

Combined efficiency of A and B = 24 + 13 = 37 units/day.

Work done in 5 days = 37 × 5 = 185 units.

Work remaining = 312 − 185 = 127 units.

Fraction of work left = 127/312.

∴ The correct answer is 127/312.

Alternate Method

Given:

Time taken by A to complete the work = 13 days

Time taken by B to complete the work = 24 days

Number of days they worked together = 5 days

Calculations:

⇒ Total efficiency of A and B together = 24 + 13 = 37 units/day

⇒ Work done by A and B in 5 days = 37 × 5 = 185 units

⇒ Remaining work = Total work − Work done

⇒ Remaining work = 312 − 185 = 127 units

⇒ Fraction of work left = Remaining work ÷ Total work

⇒ Fraction of work left = 127/312

∴ The correct answer is 127/312.

Additional Information

Total Work Concept

Assuming total work as the Least Common Multiple (LCM) of individual completion times makes calculations integer-based and much easier than fractional addition.

Work & Efficiency

Efficiency is inversely proportional to time. A person taking more days has lower efficiency. Efficiency = Total Work ÷ Total Days.

Work Left Fraction

Fraction of work left = 1 − (Days worked × Combined 1-day work). It simply represents the ratio of the uncompleted units to the total units.

Pipes and Cisterns

Pipes and Cisterns problems use the exact same principles as Time & Work, with leaks or emptying pipes treated as having a negative efficiency.

Shortcut Trick

The final price after successive discounts can be found directly by multiplying the initial price with the respective discount multiplying factors.

Final Price = MP × (100 − D₁)% × (100 − D₂)% × (100 − D₃)%

Final Price = 10000 × 0.80 × 0.90 × 0.95

Final Price = 10000 × 0.684 = 6840

∴ The correct answer is ₹6,840.

Alternate Method

Given:

Marked Price of the product = ₹10,000

Trade discount (D₁) = 20%

Scheme discount (D₂) = 10%

Cash discount (D₃) = 5%

Formula Used:

Selling Price = MP × [(100 − D₁)/100] × [(100 − D₂)/100] × [(100 − D₃)/100]

₹10,000
₹8,000
₹7,200
₹6,840
(Final Price)

−20%
−10%
−5%

Trade Discount
Scheme Discount
Cash Discount

Calculations:

⇒ Final Selling Price = 10000 × (100 − 20)/100 × (100 − 10)/100 × (100 − 5)/100

⇒ Final Selling Price = 10000 × (80/100) × (90/100) × (95/100)

⇒ Final Selling Price = 10000 × 4/5 × 9/10 × 19/20

⇒ Final Selling Price = 10000 × (4 × 9 × 19) / (5 × 10 × 20)

⇒ Final Selling Price = 10000 × 684 / 1000

⇒ Final Selling Price = 10 × 684 = 6840

∴ The correct answer is ₹6,840.

Additional Information

Successive Discounts

When two discounts D₁% and D₂% are applied successively, the effective discount is given by: D_eff = (D₁ + D₂ − (D₁ × D₂) / 100)%.

Marked Price and Selling Price

Selling Price (SP) = Marked Price (MP) × (1 − Discount/100). The discount is always calculated on the Marked Price, not on the Cost Price.

Types of Discounts

A trade discount is offered directly on the catalog price to arrive at the invoice price. A cash discount is further applied on this invoice price to encourage prompt payment.

Shortcut Trick

The overall literacy rate is given as 22%, and the male literacy rate is also 22%.

According to the weighted average concept, if a subgroup has an average identical to the overall group average, the remaining subgroup must also share the exact same average.

Since the male literacy matches the overall literacy, the female literacy rate must naturally be equal to the overall average without any calculation.

∴ The correct answer is 22%.

Alternate Method

Given:

Total population = 359,000

Total males = 179,000

⇒ Total females = Total population − Total males = 359,000 − 179,000 = 180,000

Overall literacy rate = 22%

Male literacy rate = 22%

Formula Used:

Total literate people = Literate males + Literate females

Males: 22%

Females: x%

Overall: 22%

x − 22
0

Population Ratio → 179 : 180

Calculations:

Let the literacy rate of females be x%.

⇒ 22% of 359,000 = (22% of 179,000) + (x% of 180,000)

⇒ (22 ÷ 100) × 359000 = (22 ÷ 100) × 179000 + (x ÷ 100) × 180000

Multiplying the entire equation by 100 to remove fractions:

⇒ 22 × 359000 = 22 × 179000 + x × 180000

⇒ 22 × 359000 − 22 × 179000 = x × 180000

⇒ 22 × (359000 − 179000) = x × 180000

⇒ 22 × 180000 = x × 180000

⇒ x = (22 × 180000) ÷ 180000

⇒ x = 22

∴ The correct answer is 22%.

Additional Information

Weighted Average Property

When a group is divided into two or more parts and one part has the same average as the overall average, the remaining parts must also have an average identical to the overall average.

Rule of Alligation

It is used to find the ratio in which two groups with different averages are mixed to obtain an overall average. Formula: (Quantity 1 ÷ Quantity 2) = (Average 2 − Overall Average) ÷ (Overall Average − Average 1).

Percentage Basics

Percentage literally means "per hundred". It is a fraction with a denominator of 100. To find a percentage of any quantity: Value = (Percentage × Total Base) ÷ 100.

Shortcut Trick

The largest circle inscribed inside a rectangle always has its diameter equal to the smaller side (breadth) of the rectangle.

Diameter of the largest circle (d) = 217 m.

Perimeter of circle = π × d

Perimeter = (22 ÷ 7) × 217 = 22 × 31 = 682 m.

∴ The correct answer is 682 m.

Alternate Method

Given:

Length of the rectangle (L) = 249 m

Breadth of the rectangle (B) = 217 m

Value of π = 22 ÷ 7

d = 217 m
L = 249 m
B = 217 m

Formula Used:

When a circle is drawn inside a rectangle such that it covers the maximum possible area, its diameter is strictly restricted by the smaller side of the rectangle.

Diameter of largest circle (d) = Breadth of rectangle

Perimeter (Circumference) of a circle = π × d

Calculations:

⇒ Diameter of the inscribed circle = 217 m

⇒ Perimeter of the circle = (22 ÷ 7) × 217

⇒ Perimeter = 22 × (217 ÷ 7)

⇒ Perimeter = 22 × 31

⇒ Perimeter = 682 m

∴ The correct answer is 682 m.

Additional Information

Area of Largest Inscribed Circle

The diameter of the largest circle inscribed in a rectangle is equal to its smaller side (breadth). Area = π × (B ÷ 2)².

Circumcircle of a Rectangle

The diameter of the circle circumscribing a rectangle equals the diagonal of the rectangle. Diameter = √(L² + B²).

Largest Square Inside a Circle

The diagonal of the largest inscribed square is equal to the diameter of the circle. Side of square = d ÷ √2.

Shortcut Trick

Time for P to fill tank = 15 × 4/3 = 20 h, and Q = 25 × 6/5 = 30 h.

Let Total Work = LCM(20, 30) = 60 units.

Efficiency: P = 3 units/h, Q = 2 units/h. Filled in 2 hrs = (3 + 2) × 2 = 10 units.

Pipe R empties 10 units in 5 hrs ⇒ Efficiency of R = −2 units/h.

Combined efficiency of P, Q, and R = 3 + 2 − 2 = 3 units/h.

∴ The correct answer is 20 hours.

Alternate Method

Given:

Time taken by P to fill 3/4 part = 15 hours

⇒ Time for P to fill the full tank = 15 × 4/3 = 20 hours

Time taken by Q to fill 5/6 part = 25 hours

⇒ Time for Q to fill the full tank = 25 × 6/5 = 30 hours

Part of tank filled by P and Q in 2 hours = 2 × (1/20 + 1/30) = 2 × (3 + 2)/60 = 1/6

Pipe R empties this 1/6 part in 5 hours.

⇒ Time for R to empty the full tank = 5 × 6 = 30 hours

Calculations:

⇒ Combined efficiency of P, Q, and R = 3 + 2 − 2 = 3 units/hr

⇒ Time taken by P, Q, and R together = Total Work ÷ Combined Efficiency

⇒ Time taken = 60 ÷ 3 = 20 hours

∴ The correct answer is 20 hours.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank and its work is considered positive (+). An outlet or emptying pipe drains the tank, so its work is considered negative (−).

Total Work and Efficiency

Efficiency is the amount of work done in a unit time. It is inversely proportional to the time taken. Total Work = Time Taken × Combined Efficiency.

Work Cycle Principle

If pipes operate on alternate hours, the total work is divided by the work done in one complete cycle to find the number of cycles needed.

Shortcut Trick

Let Father's age be F and Son's age be S.

Given: F + 5S = 79 and 2.5F + S = 94

Multiply the second equation by 5 to equalise S: 12.5F + 5S = 470

Subtract the first equation: (12.5F + 5S) − (F + 5S) = 470 − 79

⇒ 11.5F = 391 ⇒ F = 34

∴ The correct answer is 34 years.

Alternate Method

Given:

F + 5S = 79   --- (Equation 1)

2.5F + S = 94   --- (Equation 2)

Father (F)

5 × Son (5S)

= 79 years

2.5 × Father (2.5F)

Son (S)

= 94 years

Calculations:

⇒ From Equation 2, express S in terms of F:

⇒ S = 94 − 2.5F

⇒ Substitute the value of S in Equation 1:

⇒ F + 5(94 − 2.5F) = 79

⇒ F + 470 − 12.5F = 79

⇒ 470 − 79 = 12.5F − F

⇒ 391 = 11.5F

⇒ F = 391 ÷ 11.5

⇒ F = 34

∴ The correct answer is 34 years.

Additional Information

Linear Equations in Two Variables

A system of two linear equations can be solved using substitution, where one variable is isolated and substituted into the other equation.

Elimination Method

Multiply both equations by suitable numbers to make the coefficients of one variable equal, then subtract to eliminate it.

Consistency of Equations

Unique solutions exist when the ratio of coefficients a₁/a₂ is not equal to b₁/b₂.

Shortcut Trick

The ratio of the prices is 3 : 5 : 7, which forms an Arithmetic Progression (AP).

For an odd number of terms in an AP, the average is simply the middle term, which is 5 units.

We are given that the average price is ₹15,990.

⇒ 5 units = ₹15,990 → 1 unit = ₹15,990 ÷ 5 = ₹3,198

The price of the cheapest piece corresponds to 3 units.

⇒ Cheapest piece = 3 × ₹3,198 = ₹9,594

∴ The correct answer is ₹9,594.

Alternate Method

Given:

Average price of 3 pieces of furniture = ₹15,990

Ratio of their prices = 3 : 5 : 7

Formula Used:

Total sum of observations = Average × Number of observations

Average = (3x + 5x + 7x) ÷ 3 = 5x = ₹15,990

3x
5x
7x
Cheapest

Total = 15x = 3 × ₹15,990 = ₹47,970

Calculations:

Let the prices of the three pieces of furniture be 3x, 5x, and 7x respectively.

⇒ Total price = 3x + 5x + 7x = 15x

⇒ We know that, Total price = Average × Number of items

⇒ 15x = 15,990 × 3

⇒ 15x = 47,970

⇒ x = 47,970 ÷ 15

⇒ x = 3,198

The price of the cheapest piece of furniture corresponds to the smallest ratio part, which is 3x.

⇒ Cheapest price = 3 × 3,198 = ₹9,594

∴ The correct answer is ₹9,594.

Additional Information

Average of Terms in Arithmetic Progression (AP)

When a sequence of numbers is in Arithmetic Progression (meaning the difference between consecutive terms is constant), their average is exactly equal to the middle term (for an odd number of terms) or the average of the two middle terms (for an even number of terms).

Ratio and Proportion

When a total quantity is divided in the ratio a : b : c, the share of the first part is calculated directly using the formula: Share = [a ÷ (a + b + c)] × Total Quantity.

Weighted Average

Used when computing the overall average of sets with different numbers of items. The formula is: A_w = (n₁A₁ + n₂A₂ + ... + n_kA_k) ÷ (n₁ + n₂ + ... + n_k).

Shortcut Trick

Effective % for Case 1 = 7.7 × 20/7 = 22%

22% of Principal = ₹1,188 ⇒ Principal = ₹5,400

Effective % for Case 2 = 8.5 × 41/9 = 348.5/9%

New Interest = (348.5/9)% of 5400 = 348.5 × 6 = ₹2,091

Total Amount = Principal + New Interest = 5400 + 2091 = ₹7,491

∴ The correct answer is ₹7,491.

Alternate Method

Given:

SI₁ = ₹1,188

R₁ = 7.7% and T₁ = 20/7 years

R₂ = 8.5% and T₂ = 41/9 years

Formula Used:

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

Amount = Principal (P) + SI

Principal (P) = ?

R = 7.7%, T = 20/7 yrs
SI = 1188
Find P = 5400

R = 8.5%, T = 41/9 yrs
Find SI = 2091
Amount = 7491

Calculations:

⇒ SI₁ = (P × R₁ × T₁) ÷ 100

⇒ 1188 = [P × 7.7 × (20/7)] ÷ 100

⇒ 1188 = (P × 1.1 × 20) ÷ 100

⇒ 1188 = P × (22 ÷ 100)

⇒ P = (1188 × 100) ÷ 22 = ₹5,400

Now, finding the interest for the second case:

⇒ SI₂ = (P × R₂ × T₂) ÷ 100

⇒ SI₂ = [5400 × 8.5 × (41/9)] ÷ 100

⇒ SI₂ = 54 × 8.5 × (41/9)

⇒ SI₂ = 6 × 8.5 × 41

⇒ SI₂ = 51 × 41 = ₹2,091

Total Amount = Principal + SI₂

⇒ Amount = 5400 + 2091 = ₹7,491

∴ The correct answer is ₹7,491.

Additional Information

Effective Interest Rate

The total simple interest over a time period T is directly proportional to the product of Rate and Time. Effective Interest % = R × T.

Amount in Simple Interest

The final accumulated sum is the original Principal plus the generated Interest. Formula: A = P + (P × R × T) ÷ 100.

Principal Invariance

Unlike compound interest, the principal in simple interest remains constant throughout the entire term, making the interest linearly dependent on time.

Shortcut Trick

Convert time directly into hours to match the speed unit.

Time in hours = 5.4 ÷ 60 = 0.09 hours

Distance = Speed × Time

Distance = 42 × 0.09 = 3.78 km

∴ The correct answer is 3.78 km.

Alternate Method

Given:

Speed of the person = 42 km/h

Time taken to cross the bridge = 5.4 minutes

Speed = 42 km/h
Time = 5.4 min

Length of Bridge = ?

Formula Used:

Distance = Speed × Time

1 hour = 60 minutes

Calculations:

⇒ First, convert the time from minutes to hours.

⇒ Time = 5.4 ÷ 60 hours

⇒ Time = 0.09 hours

⇒ Now, use the distance formula:

⇒ Length of the bridge = Distance covered

⇒ Length of the bridge = 42 × 0.09

⇒ Length of the bridge = 3.78 km

∴ The correct answer is 3.78 km.

Additional Information

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.

Relative Speed

If two bodies move in the same direction, relative speed = |S₁ − S₂|. In opposite directions, relative speed = S₁ + S₂.

Average Speed

If equal distances are covered at speeds S₁ and S₂, Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).