Quantitative Aptitude PYQs

Shortcut Trick

ഒരു സംഖ്യ 2 കൊണ്ടും 5 കൊണ്ടും കൃത്യമായി ഹരിക്കണമെങ്കിൽ, അത് 10 കൊണ്ട് ഹരിക്കാൻ കഴിയുന്നതായിരിക്കണം (2 × 5 = 10).

10-ന്റെ ഗുണിതങ്ങളുടെ അവസാന അക്കം എപ്പോഴും 0 ആയിരിക്കും.

നൽകിയിരിക്കുന്ന ഓപ്ഷനുകളിൽ 130 എന്ന സംഖ്യയുടെ അവസാന അക്കം 0 ആണ്.

∴ The correct answer is 130.

Alternate Method

നൽകിയിരിക്കുന്നവ: സംഖ്യകൾ 134, 130, 135, 133.

ഉപയോഗിച്ചിരിക്കുന്ന നിയമങ്ങൾ:

1. 2-ന്റെ ഭാജ്യതാനിയമം: അവസാന അക്കം ഇരട്ടസംഖ്യയോ 0-ഓ ആയിരിക്കണം (0, 2, 4, 6, 8).
2. 5-ന്റെ ഭാജ്യതാനിയമം: അവസാന അക്കം 0 അല്ലെങ്കിൽ 5 ആയിരിക്കണം.

⇒ ഓപ്ഷൻ (1) 134: 2 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 4), എന്നാൽ 5 കൊണ്ട് ഹരിക്കാൻ കഴിയില്ല.

⇒ ഓപ്ഷൻ (2) 130: 2 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 0), 5 കൊണ്ടും ഹരിക്കാം (അവസാന അക്കം 0).

⇒ ഓപ്ഷൻ (3) 135: 5 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 5), എന്നാൽ 2 കൊണ്ട് ഹരിക്കാൻ കഴിയില്ല (ഒറ്റസംഖ്യ).

⇒ ഓപ്ഷൻ (4) 133: 2 കൊണ്ടോ 5 കൊണ്ടോ ഹരിക്കാൻ കഴിയില്ല.

∴ The correct answer is 130.

Additional Information

10-ന്റെ ഭാജ്യതാനിയമം

ഒരു സംഖ്യയുടെ അവസാന അക്കം 0 ആണെങ്കിൽ അത് 10 കൊണ്ട് കൃത്യമായി ഹരിക്കാം. ഇത് 2, 5 എന്നീ സംഖ്യകളുടെ സംയുക്ത നിയമമാണ്.

3-ന്റെ ഭാജ്യതാനിയമം

ഒരു സംഖ്യയിലെ അക്കങ്ങളുടെ തുക 3-ന്റെ ഗുണിതമാണെങ്കിൽ ആ സംഖ്യയെ 3 കൊണ്ട് ഹരിക്കാം.

ഇരട്ട സംഖ്യകളും ഒറ്റ സംഖ്യകളും

2 കൊണ്ട് ഹരിക്കാൻ കഴിയുന്ന സംഖ്യകളെ ഇരട്ട സംഖ്യകൾ എന്നും അല്ലാത്തവയെ ഒറ്റ സംഖ്യകൾ എന്നും വിളിക്കുന്നു.

Shortcut Trick

Ratio of Income (R : S) = 5 : 9 (Difference = 4 units)

Ratio of Expenditure (R : S) = 1 : 3 (Difference = 2 units)

To equate savings, make differences equal: Multiply Expenditure ratio by 2 → 2 : 6

Difference in units (Income − Expenditure) = 5 − 2 = 3 units = ₹39,000

1 unit = ₹13,000 ∴ Rakesh income (5 units) = 5 × 13,000 = ₹65,000

∴ The correct answer is ₹65,000.

Alternate Method

Given: Ratio of incomes = 5 : 9 | Ratio of expenditures = 1 : 3 | Savings = ₹39,000 each.

Formula Used: Income − Savings = Expenditure

⇒ Let the incomes of Rakesh and Shivam be 5x and 9x respectively.

⇒ According to the question: (5x − 39000) / (9x − 39000) = 1 / 3

⇒ Cross-multiplying: 3(5x − 39000) = 1(9x − 39000)

⇒ 15x − 1,17,000 = 9x − 39,000

⇒ 15x − 9x = 1,17,000 − 39,000

⇒ 6x = 78,000

⇒ x = 13,000

⇒ Rakesh monthly income = 5x = 5 × 13,000 = ₹65,000

∴ The correct answer is ₹65,000.

Additional Information

Income, Expenditure, and Savings Relationship

The basic accounting identity used in such problems is Income = Expenditure + Savings.

Cross-Multiplication Method for Ratios

When the same amount is added/subtracted from both terms of a ratio, the unit difference method is the fastest way to find the initial values.

Ratio Balancing

To balance two ratios with a constant difference, multiply each ratio by the difference of the other ratio.

Shortcut Trick

Use the ratio relationship: CP : MP = (100 − Discount%) : (100 + Profit%).

CP : MP = (100 − 12) : (100 + 10) = 88 : 110.

Simplify the ratio: 88 : 110 = 4 : 5.

Given 4 units (CP) = ₹ 7660, then 1 unit = 7660 ÷ 4 = 1915.

Marked Price (5 units) = 5 × 1915 = ₹ 9575.

∴ The correct answer is ₹ 9575.

Alternate Method

Given: Cost Price (CP) = ₹ 7660, Profit = 10%, Discount = 12%.

Formulas: SP = CP × (100 + P%) ÷ 100 | MP = (SP × 100) ÷ (100 − D%).

⇒ Step 1: Find Selling Price (SP) = 7660 × (110 ÷ 100) = 766 × 11 = ₹ 8426.

⇒ Step 2: Since SP is obtained after a 12% discount on MP, SP = 88% of MP.

⇒ Step 3: Marked Price (MP) = (8426 ÷ 88) × 100.

⇒ Step 4: MP = 95.75 × 100 = ₹ 9575.

∴ The correct answer is ₹ 9575.

Additional Information

Marked Price & Discount

Marked Price is the price printed on the label. Discount is always calculated on the Marked Price unless stated otherwise.

Relation between CP and MP

The direct formula linking CP, MP, Profit, and Discount is: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Apply the BODMAS rule directly: compute 'Of', followed by Division, Multiplication, Addition, and Subtraction.

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − 12 of 4 + 9 × 7

⇒ 21 + 9 × 5 − 48 + 63

⇒ 21 + 45 − 48 + 63

⇒ 129 − 48 = 81

∴ The correct answer is 81.

Alternate Method

Given:

189 ÷ 9 + 117 ÷ 13 × 5 − 12 of 4 + 9 × 7

Concept Used:

BODMAS Rule: First solve Brackets, then Of, Division, Multiplication, Addition, and lastly Subtraction.

Calculation:

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − (12 × 4) + 9 × 7

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − 48 + 9 × 7

⇒ 21 + 9 × 5 − 48 + 9 × 7

⇒ 21 + 45 − 48 + 63

⇒ 66 − 48 + 63

⇒ 18 + 63

⇒ 81

∴ The correct answer is 81.

Additional Information

BODMAS Rule

The fundamental rule that dictates the order of mathematical operations: Brackets first, then Of (multiplication), Division, Multiplication, Addition, and Subtraction.

VBODMAS Extension

'V' stands for Vinculum (or bar). If an expression contains a horizontal bar over some numbers, that specific part must be evaluated first before applying regular BODMAS.

Unit Digit Method

A quick cross-checking tool where you compute only the last digits of each term. In this expression: 1 + 5 − 8 + 3 = 1, matching the unit digit of the correct answer 81.

Shortcut Trick
The Empirical relationship between Mean, Median, and Mode is given by:
Median = (Mode + 2 × Mean) ÷ 3
Substituting the values: Median = (14.1 + 2 × 53.1) ÷ 3
Median = (14.1 + 106.2) ÷ 3 = 120.3 ÷ 3 = 40.1
∴ The correct answer is 40.1.

Alternate Method
Given:
⇒ Mean (x̄) = 53.1
⇒ Mode (Z) = 14.1
Formula Used:
⇒ Mode = 3 × Median − 2 × Mean

Calculations:
⇒ 14.1 = 3 × Median − 2 × 53.1
⇒ 14.1 = 3 × Median − 106.2
⇒ 3 × Median = 14.1 + 106.2
⇒ 3 × Median = 120.3
⇒ Median = 120.3 ÷ 3
⇒ Median = 40.1
∴ The correct answer is 40.1.

Additional Information
Empirical Formula Applicability
This relationship, Mode = 3 Median − 2 Mean, is valid for moderately skewed distributions. It is also known as Karl Pearson's formula.
Symmetrical Distribution
In a perfectly symmetrical distribution (like the Normal Distribution), the Mean, Median, and Mode are all equal: Mean = Median = Mode.
Positively Skewed Distribution
When the tail on the right side is longer, the distribution is positively skewed. In this case: Mean > Median > Mode.
Negatively Skewed Distribution
When the tail on the left side is longer, the distribution is negatively skewed. In this case: Mean < Median < Mode.

Shortcut Trick

Total Cost Price (CP) of 9 articles = ₹16.

Total Selling Price (SP) of 9 articles = 9 × 2.8 = ₹25.2.

Profit = SP − CP = 25.2 − 16 = ₹9.2.

Profit% = (Profit ÷ CP) × 100 = (9.2 ÷ 16) × 100.

Profit% = 0.575 × 100 = 57.5%.

∴ The correct answer is 57.5%.

Alternate Method

Given: Total CP of 9 articles = ₹16, SP of 1 article = ₹2.8.

Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ Total Cost Price (CP) = ₹16.

⇒ Total Selling Price (SP) = 9 × 2.8 = ₹25.2.

⇒ Total Profit = 25.2 − 16 = ₹9.2.

⇒ Profit Percentage = (9.2 ÷ 16) × 100 = 57.5%.

∴ The correct answer is 57.5%.

Additional Information

Profit and Loss Basics

Profit occurs when the Selling Price (SP) is greater than the Cost Price (CP). Formula: Profit = SP − CP.

Profit Percentage Calculation

Profit percentage is always calculated on the Cost Price (CP) unless stated otherwise. Formula: Profit% = (Profit ÷ CP) × 100.

Shortcut Trick

In a parallelogram, the area remains constant: Base₁ × Height₁ = Base₂ × Height₂

Let longer side be b₁ = 11.9 cm and shorter side be b₂ = a cm.

⇒ 11.9 × 3.9 = a × 5.1

⇒ a = (11.9 × 3.9) ÷ 5.1 = 9.1 cm

Perimeter = 2 × (Sum of adjacent sides) = 2 × (11.9 + 9.1) = 42 cm

∴ The correct answer is 42 cm.

Alternate Method

Given:

Longer side (L) = 11.9 cm

Shorter side (S) = a cm

Distance between longer sides (h_L) = 3.9 cm

Distance between shorter sides (h_S) = 5.1 cm

Formula Used:

Area of Parallelogram = Base × Perpendicular Height

Perimeter of Parallelogram = 2 × (L + S)

Calculations:

⇒ Area = L × h_L = S × h_S

⇒ 11.9 × 3.9 = a × 5.1

⇒ a = (11.9 × 3.9) / 5.1

⇒ a = (11.9 × 39) / 51

⇒ a = (11.9 × 13) / 17

⇒ a = 0.7 × 13

⇒ a = 9.1 cm

Now, finding the Perimeter:

⇒ Perimeter = 2 × (11.9 + 9.1)

⇒ Perimeter = 2 × 21.0

⇒ Perimeter = 42 cm

∴ The correct answer is 42 cm.

Additional Information

Properties of a Parallelogram

Opposite sides are equal and parallel. Opposite angles are equal. Adjacent angles are supplementary (sum = 180°).

Area and Height Relationship

Area = b₁h₁ = b₂h₂. This implies that the ratio of sides is inversely proportional to the ratio of their corresponding heights: b₁/b₂ = h₂/h₁.

Diagonals of a Parallelogram

The diagonals bisect each other but are not necessarily equal in length or perpendicular. The sum of the squares of the diagonals equals the sum of the squares of the four sides: d₁² + d₂² = 2(a² + b²).

Shortcut Trick

If a tap fills a cistern in x hours and a leak empties it in y hours, the time taken to fill it is (xy) ÷ (y − x).

Given x = 7 hours and y = 8 hours.

Time = (7 × 8) ÷ (8 − 7) = 56 ÷ 1 = 56 hours.

∴ The correct answer is 56 hr.

Alternate Method

Given:

Filling Tap time = 7 hours

Emptying Hole time = 8 hours

Calculations:

⇒ Total Work (Capacity of Cistern) = LCM of (7, 8) = 56 units

⇒ Efficiency of Filling Tap = 56 ÷ 7 = +8 units/hr

⇒ Efficiency of Emptying Hole = 56 ÷ 8 = −7 units/hr

⇒ Combined Efficiency (Net flow) = 8 − 7 = +1 unit/hr

⇒ Time taken to fill = Total Work ÷ Combined Efficiency

⇒ Time = 56 ÷ 1 = 56 hours

∴ The correct answer is 56 hr.

Additional Information

Net Efficiency Rule

When pipes/taps work together, their efficiencies are added (positive for filling, negative for emptying) to find the total work per unit time.

Pipes and Cisterns Concept

This topic is an application of Time and Work where filling pipes represent positive work and leaks/outlets represent negative work.

Shortcut Trick

In right-angled ΔBDC, the angle at C is 90° − 71° = 19°.

For ΔAEC, ∠AEB is the exterior angle at vertex E.

By the exterior angle property, ∠AEB = ∠CAE + ∠ACE.

∠AEB = 55° + 19° = 74°.

∴ The correct answer is 74°.

Alternate Method

Given: BD ⊥ AC at D, ∠DBC = 71°, and ∠CAE = 55°.

Formula Used: Angle Sum Property of a triangle and the Exterior Angle Theorem.

⇒ In ΔBDC, ∠BDC = 90° (since BD ⊥ AC).

⇒ ∠BCD = 180° − (∠BDC + ∠DBC)

⇒ ∠BCD = 180° − (90° + 71°) = 180° − 161° = 19°.

⇒ In ΔAEC, ∠AEB is the exterior angle at vertex E because B, E, C are collinear.

⇒ ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 55° + 19° = 74°.

∴ The correct answer is 74°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of all interior angles of any triangle is always 180°.

Complementary Angles

In a right-angled triangle, the two acute angles are complementary, meaning their sum is exactly 90°.

Shortcut Trick

Relation between Volume (V) and Curved Surface Area (CSA): V = (CSA × r) ÷ 2

First find Radius (r): 2 × (22/7) × r × 6 = 264 ⇒ r = 7 cm

Direct Calculation: Volume = (264 × 7) ÷ 2 = 132 × 7 = 924 cm³

∴ The correct answer is 924 cm³.

Alternate Method

Given:

Height (h) of the cylinder = 6 cm

Curved Surface Area (CSA) = 264 cm²

Formula Used:

Curved Surface Area (CSA) = 2πrh

Volume (V) = πr²h

Calculations:

⇒ 2 × (22/7) × r × 6 = 264

⇒ (264/7) × r = 264

⇒ r = 7 cm

Now, substitute the value of Radius (r) in the Volume formula:

⇒ Volume = (22/7) × 7 × 7 × 6

⇒ Volume = 22 × 7 × 6

⇒ Volume = 154 × 6

⇒ Volume = 924 cm³

∴ The correct answer is 924 cm³.

Additional Information

Total Surface Area (TSA) of Cylinder

The total surface area includes the curved surface and the area of two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Hollow Cylinder

If the cylinder has an outer radius R and inner radius r, the volume is V = π(R² − r²)h.

Relation between Cone and Cylinder

For the same radius and height, the volume of a cylinder is exactly three times the volume of a cone (V_cylinder = 3 × V_cone).

Shortcut Trick

Net Profit/Loss% = a + b + (a × b) ÷ 100, where a is Mark-up% and b is Discount% (negative).

Given: a = +37 and b = −16.

Net change = 37 − 16 + (37 × (−16)) ÷ 100.

Net change = 21 − (592 ÷ 100) = 21 − 5.92 = 15.08%.

Positive result indicates a gain.

∴ The correct answer is 15.08% gain.

Alternate Method

Given → Mark-up = 37% above CP, Discount = 16% on MP.

Formulas → MP = CP × (100 + Mark-up%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Gain% = (SP − CP) ÷ CP × 100.

⇒ Let Cost Price (CP) = 100.

⇒ Marked Price (MP) = 100 + 37% of 100 = 137.

⇒ Selling Price (SP) = 137 − 16% of 137.

⇒ SP = 137 − (16 × 137 ÷ 100) = 137 − 21.92 = 115.08.

⇒ Gain = SP − CP = 115.08 − 100 = 15.08.

⇒ Gain% = (15.08 ÷ 100) × 100 = 15.08%.

∴ The correct answer is 15.08% gain.

Additional Information

Relation between CP and MP

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

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net effect is (x − y − xy ÷ 100)%.

Shortcut Trick

Calculate the total number of days: Feb (25) + March (31) + April (17) = 73 days.

Convert days to years: T = 73 ÷ 365 = 1/5 year.

Apply direct SI formula: SI = (2000 × 7 × 1) ÷ (100 × 5).

SI = 20 × 7 ÷ 5 = 4 × 7 = 28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate of Interest (R) = 7% per annum

Time period = 3 February 2023 to 17 April 2023

Formula Used:

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

Calculations:

⇒ Days in February (Non-leap year) = 28 − 3 = 25 days

⇒ Days in March = 31 days

⇒ Days in April = 17 days

⇒ Total days (T) = 25 + 31 + 17 = 73 days

⇒ Time in years = 73 ÷ 365 = 1/5 year

⇒ SI = (2000 × 7 × (1/5)) ÷ 100

⇒ SI = (2000 × 7) ÷ 500

⇒ SI = 14000 ÷ 500 = 28

∴ The correct answer is ₹28.

Additional Information

Time Calculation Rule

In interest calculation, the day on which money is deposited is generally excluded, and the day on which money is withdrawn is included.

Standard Day Fractions

73 days is a very common period in exams as it equals exactly 1/5 of a 365-day year. Similarly, 146 days = 2/5 year, and 219 days = 3/5 year.

Leap Year Check

Always check if the year is a leap year (divisible by 4). If it is, February has 29 days and the total year has 366 days.

Shortcut Trick

Net rate of filling = 1/6 − 1/15 = (5 − 2)/30 = 3/30 = 1/10 per minute.

Time to fill full tank = 10 minutes.

Time to fill one-half of the tank = 10 ÷ 2 = 5 minutes.

∴ The correct answer is 5 minutes.

Alternate Method

Given: Filling time = 6 min, Emptying time = 15 min.

Formula Used: Total Work = LCM of times; Efficiency = Total Work ÷ Time.

⇒ Net Efficiency (A + B) = 5 − 2 = 3 units/min.

⇒ Target Work = 1/2 × Total Capacity = 1/2 × 30 = 15 units.

⇒ Time required = Target Work ÷ Net Efficiency = 15 ÷ 3 = 5 minutes.

∴ The correct answer is 5 minutes.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency) while outlet pipes remove water (negative efficiency) from the tank.

Net Rate Formula

If an inlet fills in 'x' and an outlet empties in 'y', net work done in 1 hour = (1/x − 1/y).

Pipes and Cisterns vs Time and Work

The concept is identical to Time and Work, where emptying a tank is equivalent to doing negative work.

Shortcut Trick

Let Tower Height be h = 50 m.

Using standard 30°-60°-90° triangle ratios: Distance to car = h × √3 = 50√3 m.

Relative height of balloon from tower top = Distance × tan(60°) = 50√3 × √3 = 150 m.

Total height of balloon = Height of tower + Relative height = 50 + 150 = 200 m.

∴ The correct answer is 200 metres.

Alternate Method

Given: Height of Tower (AB) = 50 m, Angle of depression = 30°, Angle of elevation = 60°.

Formula Used: tan(θ) = Perpendicular / Base

⇒ Let distance from tower to car be x.

⇒ In ΔABC: tan(30°) = 50 / x

⇒ 1/√3 = 50 / x ⇒ x = 50√3 m.

⇒ Let height of balloon above the tower top level be H.

⇒ In the upper triangle: tan(60°) = H / x

⇒ √3 = H / 50√3

⇒ H = 50√3 × √3 = 150 m.

⇒ Total Height of Balloon from ground = Tower Height + H = 50 + 150 = 200 m.

∴ The correct answer is 200 metres.

Additional Information

Angle of Depression and Elevation

The Angle of Depression from an observer to an object below is equal to the Angle of Elevation from the object to the observer due to alternate interior angles.

Trigonometric Ratios for Standard Angles

For heights and distances, remember: tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) = √3.

Shortcut Trick

Let original price = 100

Net multiplier = (1 − 0.45) × (1 − 0.55) × (1 − 0.50) × (1 + 0.40)

Net multiplier = 0.55 × 0.45 × 0.50 × 1.4 = 0.17325

Net % change = (0.17325 − 1) × 100 = −82.675%

∴ The correct answer is Decreases by 82.68%.

Alternate Method

Given: Successive decreases of 45%, 55%, 50% and an increase of 40%.

Step-by-step Calculation:

⇒ Let the initial price of the fuel be 100 units.

⇒ Price after 1^st month decrease of 45% = 100 × (1 − 0.45) = 55

⇒ Price after 2^nd month decrease of 55% = 55 × (1 − 0.55) = 55 × 0.45 = 24.75

⇒ Price after 3^rd month decrease of 50% = 24.75 × (1 − 0.50) = 12.375

⇒ Price after 4^th month increase of 40% = 12.375 × (1 + 0.40) = 12.375 × 1.4 = 17.325

⇒ Net Change = Final Price − Original Price = 17.325 − 100 = −82.675

⇒ Net Percentage Decrease = (82.675 / 100) × 100 = 82.675% ≈ 82.68%

∴ The correct answer is Decreases by 82.68%.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the net change is (a + b + ab/100)%. Negative values are used for decrease.

Multiplying Factor Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ... This is the most efficient method for multiple successive changes.

Base Change Property

Successive percentage changes are cumulative, meaning each subsequent percentage is calculated on the updated value, not the original base.

Shortcut Trick

Convert time directly into hours to find the distance in km, then multiply by 1000 for meters.

Time = 1.2 min = 1.2 ÷ 60 hr = 1/50 hr.

Distance = Speed × Time = 99 × (1/50) = 1.98 km.

Length of bridge = 1.98 × 1000 = 1980 m.

∴ The correct answer is 1980 m.

Alternate Method

Given:

Speed of the car = 99 km/hr

Time taken to cross the bridge = 1.2 minutes

Speed = 99 km/hr | Time = 1.2 min

Bridge

Length = D meters

Formula Used:

Distance = Speed × Time

Calculations:

⇒ First, convert the speed from km/hr to m/s.

⇒ Speed = 99 × (5 / 18) m/s

⇒ Speed = 55 / 2 m/s = 27.5 m/s

⇒ Now, convert the time from minutes to seconds.

⇒ Time = 1.2 minutes = 1.2 × 60 seconds = 72 seconds

⇒ Length of the bridge = Distance covered by the car

⇒ Distance = 27.5 m/s × 72 s

⇒ Distance = 1980 m

∴ The correct answer is 1980 m.

Additional Information

Unit Conversion

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

Relative Speed

When two bodies move in the same direction, their relative speed is the difference of their speeds (S₁ − S₂). When they move in opposite directions, it is the sum of their speeds (S₁ + S₂).

Crossing Objects

When a vehicle (like a train) crosses a stationary object with significant length (bridge, platform), the total distance covered = length of vehicle + length of object. For a point object (man, pole), distance = length of vehicle.

Shortcut Trick

When interests earned are equal and the time period is the same, the ratio of investments is inversely proportional to their rates.

Ratio of investments = P₁ : P₂ = R₂ : R₁ = 13 : 9

Total parts = 13 + 9 = 22 parts → ₹20,900

1 part → 20,900 ÷ 22 = ₹950

Sum invested at 9% (P₁) = 13 × 950 = ₹12,350

∴ The correct answer is ₹12,350.

Alternate Method

Given:

Total investment = ₹20,900

Rate for Part 1 (R₁) = 9% p.a., Rate for Part 2 (R₂) = 13% p.a.

Time period (T) = 3 years for both parts

Formula Used:

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

Total Sum: ₹20,900

Part 1 (P1)
9% p.a. for 3 yrs

Part 2 (P2)
13% p.a. for 3 yrs

Interest 1 = Interest 2

Calculations:

⇒ Let the principal invested at 9% be P₁ and at 13% be P₂.

⇒ P₁ + P₂ = 20,900

⇒ According to the question, SI on Part 1 = SI on Part 2

⇒ (P₁ × 9 × 3) ÷ 100 = (P₂ × 13 × 3) ÷ 100

⇒ P₁ × 27 = P₂ × 39

⇒ P₁ / P₂ = 39 / 27 = 13 / 9

⇒ Let P₁ = 13x and P₂ = 9x.

⇒ 13x + 9x = 20,900

⇒ 22x = 20,900

⇒ x = 20,900 ÷ 22 = 950

⇒ Sum invested at 9% (P₁) = 13 × 950 = 12,350

∴ The correct answer is ₹12,350.

Additional Information

Equal Simple Interest Rule

When a sum is divided into multiple parts such that their Simple Interests are equal, the ratio of the principal amounts is P₁ : P₂ : P₃ = 1 / (R₁T₁) : 1 / (R₂T₂) : 1 / (R₃T₃).

Equal Amounts Rule

If a sum is divided such that the final amounts (Principal + SI) after a certain time are equal, the investment ratio is P₁ : P₂ = 1 / (100 + R₁T₁) : 1 / (100 + R₂T₂).

Simple Interest vs Compound Interest

Simple Interest is always calculated consistently on the initial principal. In Compound Interest, the interest of the previous periods is accumulated with the principal to calculate the new interest.

Shortcut Trick

Since the time period is the same for all partners, the profit ratio equals their investment ratio.

Profit ratio = 1310 : 1430 : 1040 = 131 : 143 : 104

Total profit units = 131 + 143 + 104 = 378 units

Given 378 units = ₹1890 ⇒ 1 unit = ₹5

Yuvraj's share = 104 units = 104 × 5 = ₹520

∴ The correct answer is ₹520.

Alternate Method

Given:

Rahul's investment = ₹1,310

Monika's investment = ₹1,430

Yuvraj's investment = ₹1,040

Total profit = ₹1,890

Formula Used:

Profit Share = (Individual Investment Ratio ÷ Total Investment Ratio) × Total Profit

₹1310
Rahul

₹1430
Monika

₹1040
Yuvraj

Calculations:

⇒ Ratio of investments of Rahul, Monika and Yuvraj = 1310 : 1430 : 1040

⇒ Simplified Ratio = 131 : 143 : 104

⇒ Sum of the profit ratios = 131 + 143 + 104 = 378

⇒ Share of Yuvraj in the profit = (104 ÷ 378) × 1890

⇒ Share of Yuvraj = 104 × 5 = ₹520

∴ The correct answer is ₹520.

Additional Information

Simple Partnership

When all partners invest their capital for the exact same duration, their profit or loss is distributed purely in the ratio of their invested amounts.

Compound Partnership

If partners invest capital for different periods, the profit sharing ratio is derived from the product of the capital and time (Profit Ratio = C₁×T₁ : C₂×T₂).

Working and Sleeping Partners

A working partner manages the firm and may receive a fixed salary out of the profits before distribution. A sleeping partner simply invests money and only gets a proportionate share of the remaining profit.

Shortcut Trick

Side of the square = Perimeter ÷ 4

Side = 336 ÷ 4 = 84 m

Area of the square = side²

Area = 84 × 84 = 7056 m²

∴ The correct answer is 7056 m².

Alternate Method

a = 84 m
a = 84 m
a = 84 m
a = 84 m
Area = 7056 sq. m
Perimeter = 336 m

Given:

Perimeter of the square = 336 m

Formula Used:

Perimeter of a square = 4 × side

Area of a square = side²

Calculation:

⇒ 4 × side = 336

⇒ side = 336 ÷ 4

⇒ side = 84 m

Now, calculate the area of the square:

⇒ Area = 84 × 84

⇒ Area = 7056 m²

∴ The correct answer is 7056 m².

Additional Information

Relation between Area and Perimeter

The area of a square can be directly related to its perimeter (P) using the formula: Area = P² ÷ 16.

Diagonal of a Square

The length of the diagonal of a square with side 'a' is given by the formula: d = a√2.

Area using Diagonal

If the length of the diagonal 'd' is given, the area can be easily calculated as: Area = d² ÷ 2.

Properties of a Square

All four sides of a square are equal. The diagonals are equal in length and bisect each other exactly at 90°.

Shortcut Trick

The age difference between two people remains constant forever.

Let Son's age 3 years ago be x, then Father's age was 2x + 21, making the difference x + 21.

When Father becomes twice the Son's age, the Son's age must exactly equal this difference (x + 21).

Presently, Son's age is x + 3. Hence, years required from now = (x + 21) − (x + 3) = 18.

∴ The correct answer is 18 years.

Alternate Method

3 Years Ago
Present
Future (+n yrs)

Son = x
Dad = 2x+21

Son = x+3
Dad = 2x+24

Son = S
Dad = 2S

+3

+n

Age Difference = (2x + 21) − x = x + 21 (Constant)
When Dad is 2 × Son, Son's Age = Difference = x + 21
Years from now (n) = (x + 21) − (x + 3) = 18

Given:

3 years ago, Father's age was 21 years more than twice the Son's age.

We need to find the number of years from now when the Father's age will be twice the Son's age.

Calculations:

Let the present age of the Father be F and the Son be S.

According to the condition 3 years ago:

⇒ (F − 3) = 2 × (S − 3) + 21

⇒ F − 3 = 2S − 6 + 21

⇒ F − 2S = 18     ... (Equation 1)

Let after 'n' years from now, the Father's age be twice the Son's age:

⇒ F + n = 2 × (S + n)

⇒ F + n = 2S + 2n

⇒ n = F − 2S

Substituting the value of (F − 2S) from Equation 1:

⇒ n = 18

∴ The correct answer is 18 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals is always constant. If person A is older than person B by x years, A will always remain older by x years in the past or future.

Double Age Concept

When one person is exactly twice the age of another, the younger person's age is exactly equal to the age difference between them.