Quantitative Aptitude PYQs

Shortcut Trick

We can solve this directly using the given options.

Check the sum: All given options sum to 57.

Check the LCM for Option 3: 28 and 29 are consecutive numbers, meaning they are co-prime.

The L.C.M. of co-prime numbers is simply their product: 28 × 29 = 812.

Since this perfectly matches the given condition, we do not need to check further.

∴ The correct answer is 28, 29.

Alternate Method

Given:

Sum of two numbers = 57

L.C.M. of the two numbers = 812

Concept Used:

Let the HCF of the numbers be h. The numbers can be written as hx and hy, where x and y are co-prime.

Sum = h(x + y) and L.C.M. = hxy

Calculations:

⇒ h(x + y) = 57

⇒ hxy = 812

⇒ To find the HCF (h), we check the common factors of 57 (which are 1, 3, 19, 57) and 812.

⇒ Since 812 is not divisible by 3, 19, or 57, the highest common factor h must be 1.

⇒ Therefore, the two numbers are co-prime, meaning x + y = 57 and xy = 812.

⇒ Substituting y = 57 − x into the product equation gives:

⇒ x(57 − x) = 812

⇒ x² − 57x + 812 = 0

⇒ Factorising the quadratic equation: x² − 28x − 29x + 812 = 0

⇒ x(x − 28) − 29(x − 28) = 0

⇒ (x − 28)(x − 29) = 0

⇒ The roots of the quadratic equation are x = 28 and x = 29.

∴ The correct answer is 28, 29.

Additional Information

Product of Numbers Rule

For any two given numbers, the product of the numbers is always equal to the product of their HCF and LCM (First Number × Second Number = HCF × LCM).

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. The LCM of two co-prime numbers is easily found by multiplying them together.

Consecutive Numbers Concept

Any two consecutive natural numbers (like 28 and 29) are always co-prime, meaning they will not share any common factors other than 1.

Shortcut Trick

Subtract the given remainders from the respective numbers: 131 − 8 = 123, 91 − 9 = 82, and 215 − 10 = 205.

To find the HCF quickly, take the difference of the two closest numbers: 123 − 82 = 41.

The greatest common divisor must be a factor of this difference.

Since 41 exactly divides all three numbers (123, 82, 205) and is a prime number, it is the required HCF.

∴ The correct answer is 41.

Alternate Method

Given:

Numbers: 131, 91, and 215

Remainders: 8, 9, and 10 respectively

Method Used:

Required Number = HCF of (Number₁ − Remainder₁), (Number₂ − Remainder₂), and (Number₃ − Remainder₃)

Calculations:

⇒ First number = 131 − 8 = 123

⇒ Second number = 91 − 9 = 82

⇒ Third number = 215 − 10 = 205

⇒ Now, we find the Highest Common Factor (HCF) of 123, 82, and 205.

⇒ Prime factorization of 123 = 3 × 41

⇒ Prime factorization of 82 = 2 × 41

⇒ Prime factorization of 205 = 5 × 41

⇒ The highest common factor present in all three numbers is 41.

⇒ HCF(123, 82, 205) = 41

∴ The correct answer is 41.

Additional Information

HCF with different remainders

If a number divides x, y, and z leaving remainders a, b, and c respectively, the greatest such number is the HCF of (x−a), (y−b), and (z−c).

HCF with same remainder

If a number divides x, y, and z leaving the same remainder 'r' in each case without given value, the greatest such number is the HCF of |x−y|, |y−z|, and |z−x|.

LCM with same remainder

The smallest number which when divided by x, y, and z leaves the same remainder 'r' is given by [LCM of (x, y, z)] + r.

LCM with different remainders

If divided by x, y, z leaving remainders a, b, c respectively where (x−a) = (y−b) = (z−c) = k, the smallest number is [LCM of (x, y, z)] − k.

Shortcut Trick

Let the actual selling price be SP and cost price be CP.

Selling at 2/5 of SP means 0.4 SP, which results in a 26% loss.

0.4 SP = 0.74 CP ⇒ SP = 1.85 CP

New Selling Price = 0.84 SP = 0.84 × 1.85 CP = 1.554 CP

Profit % = (1.554 − 1) × 100 = 55.4%

∴ The correct answer is 55.4%.

Alternate Method

Given:

Loss = 26% when SP₁ = 2/5 × Actual SP

New Selling Price SP₂ = 84% × Actual SP

SP₁
0.4 SP
(74% CP)

Cost Price
1.0 CP
(100% CP)

SP₂
0.84 SP
(155.4% CP)

Actual SP
1.0 SP
(185% CP)

← 26% Loss

55.4% Profit →

Calculations:

⇒ Let the Actual SP be 100x.

⇒ SP₁ = (2/5) × 100x = 40x

⇒ Since selling at 40x incurs a 26% loss, the Cost Price (CP) will be calculated as:

⇒ CP × (100 − 26)% = 40x

⇒ CP × 0.74 = 40x

⇒ CP = 40x / 0.74 = 4000x / 74 = 2000x / 37

⇒ Now, the new selling price SP₂ = 84% of Actual SP = 84x

⇒ Profit = SP₂ − CP = 84x − (2000x / 37)

⇒ Profit = (3108x − 2000x) / 37 = 1108x / 37

⇒ Profit % = (Profit / CP) × 100

⇒ Profit % = [(1108x / 37) / (2000x / 37)] × 100

⇒ Profit % = (1108 / 2000) × 100

⇒ Profit % = 1108 / 20 = 55.4%

∴ The correct answer is 55.4%.

Additional Information

Profit & Loss Percentage

Profit % = (Profit / CP) × 100 and Loss % = (Loss / CP) × 100. Both are always calculated on the Cost Price (CP) unless stated otherwise.

Marked Price & Discount

Discount is always offered on the Marked Price (MP). SP = MP × (100 − Discount%) / 100.

Successive Discounts

If two successive discounts of a% and b% are given, the equivalent single discount is calculated as (a + b − ab / 100)%.

Shortcut Trick

When the same distance is covered at speeds S₁ and S₂, the average speed formula is:

⇒ Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

⇒ Average Speed = (2 × 55 × 66) ÷ (55 + 66)

⇒ Average Speed = (7260) ÷ 121

⇒ Average Speed = 60 km/hr

∴ The correct answer is 60 km/hr.

Alternate Method

Given:

⇒ Speed of onward journey (S₁) = 55 km/hr

⇒ Speed of return journey (S₂) = 66 km/hr

⇒ Let the uniform distance between Visakhapatnam and Nagpur be D km.

Visakhapatnam
Nagpur

Distance = D

Speed = 55 km/hr

Speed = 66 km/hr

Formula Used:

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = D + D = 2D km

⇒ Time taken for onward journey = D ÷ 55 hours

⇒ Time taken for return journey = D ÷ 66 hours

⇒ Total Time = (D ÷ 55) + (D ÷ 66)

⇒ Total Time = (6D + 5D) ÷ 330 = 11D ÷ 330 = D ÷ 30 hours

⇒ Average Speed = 2D ÷ (D ÷ 30)

⇒ Average Speed = 2 × 30 = 60 km/hr

∴ The correct answer is 60 km/hr.

Additional Information

Average Speed for 3 Equal Distances

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

Conversion of Units

To convert km/hr to m/s, multiply the value by (5 ÷ 18). Conversely, to convert m/s to km/hr, multiply the value by (18 ÷ 5).

Relative Speed

If two objects move in the same direction at speeds S₁ and S₂ (where S₁ > S₂), their Relative Speed = S₁ − S₂. If they move in opposite directions, their Relative Speed = S₁ + S₂.

Shortcut Trick

Sum of the remaining two numbers = Total sum − (Sum of first two + Third number)

Sum of the remaining two = 655 − (2 × 80 + 101) = 655 − 261 = 394

Average of remaining two numbers = 394 ÷ 2 = 197

∴ The correct answer is 197.

Alternate Method

Given:

Total sum of 5 numbers = 655

Average of first 2 numbers = 80

Third number = 101

Sum = 160
101
Remaining Sum = 394

First 2 numbers
3^rd num
Remaining 2 numbers

Total Sum = 655

Avg = 394 ÷ 2
Avg = 197

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Sum of first 2 numbers = 80 × 2 = 160

⇒ Sum of first 3 numbers = 160 + 101 = 261

⇒ Sum of remaining 2 numbers = Total sum − Sum of first 3 numbers

⇒ Sum of remaining 2 numbers = 655 − 261 = 394

⇒ Average of remaining 2 numbers = 394 ÷ 2

⇒ Average = 197

∴ The correct answer is 197.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, the overall average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Sum and Average Relation

The sum of a dataset is always directly proportional to its average. If a number is removed, the new average depends on whether the removed number is greater or smaller than the old average.

Net Deviation Method

Instead of calculating total sums, one can assume an average and find the net sum of deviations. If the net deviation is zero, the assumed average is the true average.

Shortcut Trick
Express x and y as powers of the same base and substitute them directly into the equation.
⇒ (580.5)z = (580.79)10
⇒ 580.5z = 587.9
⇒ 0.5z = 7.9 ⇒ z = 15.8
∴ The correct answer is 15.8.

Alternate Method
Given:
x = 580.5
y = 580.79
xz = y10
Formula Used:
(am)n = amn
If am = an, then m = n (for a > 0, a ≠ 1)
Calculations:
Substitute the given values of x and y into the equation xz = y10:
⇒ (580.5)z = (580.79)10
Apply the power of a power rule to both sides:
⇒ 580.5 × z = 580.79 × 10
⇒ 580.5z = 587.9
Since the bases are equal on both sides, we can equate the exponents:
⇒ 0.5z = 7.9
⇒ z = 7.9 ÷ 0.5
⇒ z = 7.9 × 2
⇒ z = 15.8
∴ The correct answer is 15.8.

Additional Information
Power of a Power Rule
When raising a base with a power to another power, you multiply the exponents: (am)n = amn.
Equating Exponents
If two powers with the same positive base (other than 1) are equal, then their exponents must be equal. If am = an, then m = n.
Product Rule of Exponents
When multiplying two expressions with the same base, you add the exponents: am × an = am+n.
Division Rule of Exponents
When dividing two expressions with the same base, you subtract the exponents: am ÷ an = am−n.

Shortcut Trick

Total CP = 740 + 16 = ₹756 and SP = ₹558

Ratio of SP to CP = 558 : 756 = 31 : 42

Loss in ratio units = 42 − 31 = 11 units.

Loss % = (11 ÷ 42) × 100 = 550/21 %.

Since SP is less than CP, there is a loss of 550/21%.

∴ The correct answer is 550/21% loss.

Alternate Method

Given:

Cost price of the toy = ₹740

Decoration cost (Overhead expenses) = ₹16

Selling price (SP) of the toy = ₹558

Formula Used:

Total CP = Initial CP + Overhead expenses

Loss = Total CP − SP

Loss % = (Loss ÷ Total CP) × 100

CP: ₹740

• ₹16

Total CP: ₹756

Sold

SP: ₹558

Loss = ₹198

Calculations:

⇒ Total CP = 740 + 16 = ₹756

⇒ Loss = 756 − 558 = ₹198

⇒ Loss % = (198 ÷ 756) × 100

⇒ Loss % = (11 ÷ 42) × 100

⇒ Loss % = 1100 ÷ 42 = 550/21 %

∴ The correct answer is 550/21% loss.

Additional Information

Overhead Expenses

Costs such as repairs, transportation, and decoration incurred after purchasing an item are always added to the initial cost price to arrive at the Total Cost Price.

Profit & Loss Base

Profit and loss percentage calculations are always evaluated against the Total Cost Price (CP) as the base, not the Selling Price (SP), unless specifically mentioned.

Shortcut Trick

Fraction value of 80% = 4/5

Original number = 5 units, Increase = 4 units

New number = 5 + 4 = 9 units

9 units = 2790 ⇒ 1 unit = 310

Original number = 5 units = 5 × 310 = 1550

∴ The correct answer is 1550.

Alternate Method

Given:

A number when increased by 80% gives 2790.

Original Number (100%)

• 80% Increase

180% = 2790

Formula Used:

Final Value = Initial Value × (1 + Percentage Increase ÷ 100)

Calculations:

⇒ Let the original number be x.

⇒ x + 80% of x = 2790

⇒ x + 0.8x = 2790

⇒ 1.8x = 2790

⇒ x = 2790 ÷ 1.8

⇒ x = 27900 ÷ 18

⇒ x = 1550

∴ The correct answer is 1550.

Additional Information

Percentage Increase

The formula to calculate percentage increase is: % Increase = (Final Value − Initial Value) ÷ Initial Value × 100.

Base Value Concept

In percentage problems, the initial or original quantity is always considered as the base value or 100% unless otherwise specified.

Successive Percentage

When a number is increased by a% and then by b%, the net percentage increase is given by (a + b + (a × b) ÷ 100)%.

Shortcut Trick

The net price is the remaining value after deducting the discount percentage from the listed price.

Net Price = Listed Price × (100% − Discount%)

Net Price = 900 × (100% − 30%)

Net Price = 900 × 70% = 630

∴ The correct answer is ₹630.

Alternate Method

Given:

Listed Price (MP) = ₹900

Trade Discount = 30%

Formula Used:

Discount Amount = MP × (Discount% ÷ 100)

Net Price = MP − Discount Amount

MP = ₹900

−30% Discount

Net Price = ?

Net Price = 900 − 270 = ₹630

Calculations:

⇒ Discount Amount = 900 × (30 ÷ 100)

⇒ Discount Amount = 900 × 0.30

⇒ Discount Amount = ₹270

⇒ Net Price = 900 − 270

⇒ Net Price = ₹630

∴ The correct answer is ₹630.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP) or Listed Price of an article. Net Price or Selling Price (SP) = MP − Discount.

Successive Discounts

If two successive discounts of a% and b% are given on an item, the equivalent single discount is given by the formula: (a + b − ab ÷ 100)%.

Profit and Loss Calculation

Unlike discounts which are evaluated on the Marked Price, Profit or Loss is always calculated on the Cost Price (CP). Profit% = (Profit ÷ CP) × 100.

Shortcut Trick

For a cylinder, Radius (r) = CSA ÷ (2 × π × h)

⇒ r = (3300 × 7) ÷ (2 × 22 × 72)

⇒ r = 23100 ÷ 3168

⇒ r = 175 ÷ 24 = 7.2916... cm

∴ The correct answer is 7.29 cm.

Alternate Method

Given:

Curved Surface Area (CSA) = 3300 cm²

Length or Height (h) = 72 cm

Formula Used:

CSA of a Cylinder = 2 × π × r × h

r = ?

h = 72 cm
CSA = 3300 sq. cm

Calculations:

⇒ 3300 = 2 × (22 ÷ 7) × r × 72

⇒ r = (3300 × 7) ÷ (2 × 22 × 72)

⇒ r = (150 × 7) ÷ (2 × 72)

⇒ r = (75 × 7) ÷ 72

⇒ r = 525 ÷ 72

⇒ r = 175 ÷ 24

⇒ r = 7.2916...

⇒ r ≈ 7.29 cm

∴ The correct answer is 7.29 cm.

Additional Information

Total Surface Area of a Cylinder

The TSA includes the curved surface and both circular bases. Formula: TSA = 2 × π × r × (r + h).

Volume of a Cylinder

The capacity or three-dimensional space enclosed by a cylinder. Formula: V = π × r² × h.

Curved Surface Area of a Cone

The outer surface area excluding the circular base. Formula: CSA = π × r × l, where l is the slant height (√(r² + h²)).

Shortcut Trick

The given number is 481355.

By simply observing the unit digit, which is 5, we can determine its divisibility instantly.

Any number ending in 0 or 5 is exactly divisible by 5.

∴ The correct answer is 5.

Alternate Method

Given:

Number = 481355

Method Used:

Application of standard divisibility rules for 3, 5, 6, and 15.

Calculations:

⇒ Divisibility by 3: Sum of digits = 4 + 8 + 1 + 3 + 5 + 5 = 26.

⇒ Since 26 is not perfectly divisible by 3, the number 481355 is not divisible by 3.

⇒ Divisibility by 6: The number must be divisible by both 2 and 3.

⇒ Since the number ends in 5 (which makes it an odd number), it is not divisible by 2, and hence cannot be divisible by 6.

⇒ Divisibility by 15: The number must be divisible by both 3 and 5.

⇒ As proven above, it is not divisible by 3, so it is not divisible by 15.

⇒ Divisibility by 5: The unit digit of the number is 5.

⇒ Therefore, the number is perfectly 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 individual digits is exactly divisible by 3.

Divisibility Rule of 5

A number is divisible by 5 if its last digit (unit digit) is either 0 or 5.

Divisibility Rule of 6

A number is divisible by 6 if it is divisible by both 2 (meaning it is an even number) and 3.

Divisibility Rule of 15

A number is divisible by 15 if it is perfectly divisible by both of its coprime factors, which are 3 and 5.

Shortcut Trick

For any regular polygon, the exterior angle is directly related to the number of sides by a simple ratio.

⇒ Exterior Angle = 360° ÷ n

⇒ Exterior Angle = 360° ÷ 30 = 12°

Since the interior and exterior angles always form a straight line (180°):

⇒ Interior Angle = 180° − 12° = 168°

∴ The correct answer is 168°.

Alternate Method

Given:

Number of sides of the regular polygon (n) = 30

Formula Used:

Sum of interior angles of a regular polygon = (n − 2) × 180°

Measure of one interior angle = [(n − 2) × 180°] ÷ n

168°
12°
Interior Angle
Exterior Angle
Polygon Side
Extended Line

Interior Angle + Exterior Angle = 180°

Calculations:

⇒ Sum of interior angles = (30 − 2) × 180°

⇒ Sum of interior angles = 28 × 180° = 5040°

⇒ Measure of each interior angle = 5040° ÷ 30

⇒ Measure of each interior angle = 168°

∴ The correct answer is 168°.

Additional Information

Sum of Exterior Angles

For any convex polygon, regardless of the number of sides, the sum of all its exterior angles is always exactly 360°.

Number of Diagonals

The total number of diagonals in a polygon with n sides can be quickly calculated using the formula: n(n − 3) ÷ 2.

Regular vs Irregular Polygons

A regular polygon is both equilateral (all sides equal) and equiangular (all angles equal). An irregular polygon lacks one or both of these properties.

Shortcut Trick

Selling Price after 1^st discount = 1400 × (100 − 12) ÷ 100 = ₹1232

Second discount amount = 1232 − 1133.44 = ₹98.56

Second discount % = (98.56 ÷ 1232) × 100 = 8%

∴ The correct answer is 8%.

Alternate Method

Given:

Marked Price (MP) = ₹1400

Final Selling Price (SP) = ₹1133.44

First discount (D₁) = 12%

Let the second discount be D₂%.

Formula Used:

SP = MP × (1 − D₁/100) × (1 − D₂/100)

₹1400Marked Price−12%₹1232After 1^st Discount−8%₹1133.44Final PriceDiscount = ₹98.56

Calculation:

⇒ Price after 1^st discount = 1400 − (12% of 1400)

⇒ Price after 1^st discount = 1400 − 168 = ₹1232

⇒ Final SP after 2^nd discount = ₹1133.44

⇒ Second discount amount = 1232 − 1133.44 = ₹98.56

⇒ D₂ = (98.56 ÷ 1232) × 100

⇒ D₂ = 8%

∴ The correct answer is 8%.

Additional Information

Successive Discounts

When two discounts D₁% and D₂% are given successively, the effective discount percentage is equivalent to (D₁ + D₂ − (D₁ × D₂) ÷ 100)%.

Marked Price & Selling Price Relationship

Selling Price (SP) is obtained after deducting the total discount from the Marked Price (MP). The direct multiplier formula is SP = MP × (100 − Discount%) ÷ 100.

Markup and Discount

The relationship between Cost Price, Markup Percentage (M%), and Discount Percentage (D%) to find Profit Percentage (P%) is given by: P% = M% − D% − (M% × D%) ÷ 100.

Shortcut Trick

Put x = 90° and y = 0° in the given expression:

⇒ sin(90° − 90°) cos(180° − (90° − 0°)) + cos(90° − 90°) sin(180° − (0° − 90°))

⇒ sin(0°) cos(90°) + cos(0°) sin(270°)

⇒ 0 × 0 + 1 × (−1) = −1

Substituting y = 0° in the options, only Option 3 (−cos y) gives −1.

∴ The correct answer is −cos y.

Alternate Method

Given:

sin(90° − x) cos(180° − (x − y)) + cos(90° − x) sin(180° − (y − x))

Formula Used:

sin(90° − θ) = cos θ

cos(90° − θ) = sin θ

cos(180° − θ) = −cos θ

sin(180° − θ) = sin θ

sin(−θ) = −sin θ

cos(A − B) = cos A cos B + sin A sin B

Calculations:

⇒ sin(90° − x) = cos x

⇒ cos(180° − (x − y)) = −cos(x − y)

⇒ cos(90° − x) = sin x

⇒ sin(180° − (y − x)) = sin(y − x) = −sin(x − y)

Substituting these values into the given expression:

⇒ (cos x) × (−cos(x − y)) + (sin x) × (−sin(x − y))

⇒ −cos x cos(x − y) − sin x sin(x − y)

⇒ −[cos x cos(x − y) + sin x sin(x − y)]

Using the identity cos(A − B) = cos A cos B + sin A sin B:

⇒ −[cos(x − (x − y))]

⇒ −cos(x − x + y)

⇒ −cos y

∴ The correct answer is −cos y.

Additional Information

Trigonometric Signs by Quadrant (ASTC Rule)

1^st Quadrant: All positive | 2^nd Quadrant: Sine & Cosec positive | 3^rd Quadrant: Tan & Cot positive | 4^th Quadrant: Cos & Sec positive.

Compound Angle Identities

sin(A + B) = sin A cos B + cos A sin B

cos(A + B) = cos A cos B − sin A sin B

Complementary & Supplementary Angles

Angles that sum to 90° are complementary (sin(90°−θ) = cos θ). Angles summing to 180° are supplementary (cos(180°−θ) = −cos θ).

Shortcut Trick

The third proportional of two numbers a and b is given by b² ÷ a.

Here, a = 81 and b = 27.

⇒ Third proportional = (27 × 27) ÷ 81

⇒ Third proportional = 729 ÷ 81 = 9

∴ The correct answer is 9.

Alternate Method

Given:

First number (a) = 81

Second number (b) = 27

Let the third proportional be x.

Formula Used:

If three numbers a, b, and c are in continued proportion, then a : b :: b : c

⇒ Product of Extremes = Product of Means

⇒ a × c = b × b

81
:

27
::

27
:

x

Product of Means

Product of Extremes

Calculations:

⇒ 81 × x = 27 × 27

⇒ 81x = 729

⇒ x = 729 ÷ 81

⇒ x = 9

∴ The correct answer is 9.

Additional Information

Continued Proportion

If three quantities a, b, and c are such that a : b = b : c, they are said to be in continued proportion where b is the mean proportional.

Mean Proportional

The mean proportional between two numbers a and c is represented by b = √(a × c).

Fourth Proportional

If four terms a, b, c, and d are in proportion (a : b = c : d), then d is called the fourth proportional. It is evaluated as d = (b × c) ÷ a.

Compounded Ratio

The compounded ratio of two distinct ratios a : b and c : d is the product of their corresponding terms, yielding ac : bd.

Shortcut Trick

Convert percentage changes into fraction multipliers:

⇒ +25% = +1/4 → Multiplier = 5/4

⇒ +60% = +3/5 → Multiplier = 8/5

⇒ −50% = −1/2 → Multiplier = 1/2

⇒ Net effect = 5/4 × 8/5 × 1/2 = 40/40 = 1

∴ The correct answer is 12,014.

Alternate Method

Given:

Initial population of the town = 12,014

Successive changes in 3 years = +25%, +60%, and −50%

Formula Used:

Final Value = Initial Value × (1 ± R₁/100) × (1 ± R₂/100) × (1 ± R₃/100)

12,014
Initial

+25%
×5/4

15,017.5

+60%
×8/5

24,028

−50%
×1/2

12,014
Final

Net Multiplier = 5/4 × 8/5 × 1/2 = 1

Calculations:

⇒ Multiplier for Year 1 = (1 + 25/100) = 125/100 = 5/4

⇒ Multiplier for Year 2 = (1 + 60/100) = 160/100 = 8/5

⇒ Multiplier for Year 3 = (1 − 50/100) = 50/100 = 1/2

⇒ Final population = 12,014 × (5/4) × (8/5) × (1/2)

⇒ Final population = 12,014 × (40/40)

⇒ Final population = 12,014 × 1 = 12,014

∴ The correct answer is 12,014.

Additional Information

Successive Percentage Change

When a value is successively changed by x% and y%, the net percentage change can also be found using the formula: Net % Change = x + y + (xy / 100). This method is useful for consecutive years but can become complex for more than two variables.

Fraction Equivalents

Converting standard percentages to fractions significantly boosts calculation speed. Standard fractions include 25% = 1/4, 60% = 3/5, and 50% = 1/2. The new value multiplier is simply (1 + fraction) for an increase or (1 − fraction) for a decrease.

Base Independence

In a chain of successive percentage changes, the final result is independent of the order in which the changes occur due to the commutative property of multiplication. An increase followed by a decrease produces the exact same outcome if reversed.

Shortcut Trick

Apply the BODMAS rule directly to the expression (Division → Multiplication → Subtraction):

⇒ (−9) − (−60) ÷ (−10) + (−3) × 6

⇒ −9 − (6) − 18

⇒ −15 − 18 = −33

∴ The correct answer is −33.

Alternate Method

Given:

(−9) − (−60) ÷ (−10) + (−3) × 6

Formula Used:

BODMAS Rule: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Calculation:

⇒ (−9) − [(−60) ÷ (−10)] + [(−3) × 6]

First, perform the division part:

⇒ (−60) ÷ (−10) = 6

Substitute this value back into the original expression:

⇒ (−9) − 6 + (−3) × 6

Next, perform the multiplication part:

⇒ (−3) × 6 = −18

Substitute this value back into the expression:

⇒ (−9) − 6 + (−18)

Finally, perform the addition and subtraction from left to right:

⇒ −15 − 18

⇒ −33

∴ The correct answer is −33.

Additional Information

BODMAS Rule

It defines the correct order of operations to solve an expression: Brackets first, then Orders (powers/roots), followed by Division and Multiplication (evaluated left to right), and finally Addition and Subtraction (evaluated left to right).

Sign Rules for Multiplication and Division

Multiplying or dividing two numbers with the same sign always gives a positive result (− × − = +). Conversely, multiplying or dividing numbers with different signs always gives a negative result (+ × − = −).

Sign Rules for Addition and Subtraction

When combining two negative numbers, you add their absolute values and retain the negative sign (e.g., −9 − 6 = −15). When signs are different, subtract the smaller number from the larger and keep the sign of the larger absolute number.

Shortcut Trick

Profit ratio of Ritik to Sanjay = (16500 − 3600) : 3600 = 12900 : 3600 = 43 : 12

Investment ratio is the same as the profit ratio = 43 : 12 (since time is constant).

Difference in ratio parts = 43 − 12 = 31 parts, and total = 55 parts.

Investment difference = (31 ÷ 55) × 79200 = 31 × 1440 = 44640

∴ The correct answer is ₹44,640.

Alternate Method

Given:

⇒ Total investment = ₹79,200

⇒ Total profit = ₹16,500

⇒ Sanjay's profit = ₹3,600

⇒ Ritik's profit = ₹16,500 − ₹3,600 = ₹12,900

Formula Used:

⇒ Profit Ratio = Investment Ratio (when the time period is constant)

⇒ Required Value = (Ratio Difference ÷ Total Ratio) × Total Value

Total Investment = ₹79,200 (55 Ratio Parts)

Ritik: ₹12,900 Profit (43 parts)

S: 12 parts

Investment Difference = 31 Ratio Parts
31 × (79200 ÷ 55) = ₹44,640

Calculations:

⇒ Profit ratio (Ritik : Sanjay) = 12900 : 3600

⇒ Profit ratio simplified = 129 : 36 = 43 : 12

⇒ Sum of ratio parts = 43 + 12 = 55

⇒ Value of 55 parts = ₹79,200

⇒ Value of 1 part = 79200 ÷ 55 = ₹1,440

⇒ Difference in parts = 43 − 12 = 31

⇒ Difference in investment = 31 × 1440

⇒ Difference in investment = ₹44,640

∴ The correct answer is ₹44,640.

Additional Information

Partnership for Same Time Period

When investments are made for the exact same duration, the total profit is distributed exactly in the ratio of their capital investments. Profit Ratio = Investment Ratio.

Partnership for Different Time Periods

When time periods differ, profit is distributed based on the product of investment and time. Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Active and Sleeping Partners

An active partner is paid a fixed salary or commission from the total profit for managing the business. The remaining profit is then divided according to their equivalent investment ratios.

Shortcut Trick

Weight of Mohan = (Sum of first two + Sum of last two) − Sum of all three

Weight of Mohan = (2 × 43 + 2 × 50) − (3 × 45)

Weight of Mohan = (86 + 100) − 135

Weight of Mohan = 186 − 135 = 51

∴ The correct answer is 51 kg.

Alternate Method

Given:

Average weight of Vipul, Mohan, and Mohammad = 45 kg

Average weight of Vipul and Mohan = 43 kg

Average weight of Mohan and Mohammad = 50 kg

Formula Used:

Total Weight = Average Weight × Number of Persons

Calculation:

Total = 135 kg

Vipul + Mohan = 86 kg

Mohan + Mohammad = 100 kg

Overlap = Mohan
Mohan = 86 + 100 − 135 = 51 kg

⇒ Total weight of Vipul, Mohan, and Mohammad = 45 × 3 = 135 kg

⇒ Total weight of Vipul and Mohan = 43 × 2 = 86 kg

⇒ Total weight of Mohan and Mohammad = 50 × 2 = 100 kg

⇒ Total of the two pairs = (Vipul + Mohan) + (Mohan + Mohammad) = 86 + 100 = 186 kg

⇒ The sum of the pairs includes Mohan's weight exactly twice.

⇒ Weight of Mohan = Sum of the two pairs − Total weight of all three

⇒ Weight of Mohan = 186 − 135 = 51 kg

∴ The correct answer is 51 kg.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂ respectively, the combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method in Averages

The sum of deviations of all observations from their respective average is always zero. This property can be used as a shortcut to quickly find missing variables without large multiplications.

Replacement Method

If a new item replaces an old item and the average increases, the value of the new item is equal to the value of the old item plus the product of total members and the net increase in the average.

Shortcut Trick

Relative speed of Train 1 w.r.t man = 195 ÷ 13 = 15 m/s = (15 × 18/5) = 54 km/hr.

Since both are moving in the same direction, actual speed of Train 1 = 54 + 5 = 59 km/hr.

Relative speed of the two trains (moving in opposite directions) = 59 + 76 = 135 km/hr.

Converting relative speed to m/s = 135 × (5/18) = 37.5 m/s.

Time required to completely cross = (195 + 255) ÷ 37.5 = 450 ÷ 37.5 = 12 seconds.

∴ The correct answer is 12 seconds.

Alternate Method

Given:

Length of Train 1 (L₁) = 195 m

Speed of the man (v_m) = 5 km/hr

Time taken to overtake the man (t₁) = 13 seconds

Length of Train 2 (L₂) = 255 m

Speed of Train 2 (v₂) = 76 km/hr

Formula Used:

Time = Total Distance ÷ Relative Speed

Relative Speed (same direction) = v₁ − v₂

Relative Speed (opposite direction) = v₁ + v₂

Train 1 : 195 m

v1 = 59 km/hr

Train 2 : 255 m

v2 = 76 km/hr

Relative Speed = 59 + 76 = 135 km/hr = 37.5 m/s
Total Distance = 195 + 255 = 450 m
Time = 450 ÷ 37.5 = 12 seconds

Calculations:

⇒ Relative speed of Train 1 w.r.t the man = L₁ ÷ t₁ = 195 ÷ 13 = 15 m/s

⇒ Convert relative speed into km/hr = 15 × (18/5) = 54 km/hr

⇒ Since the man is moving in the same direction, relative speed = v₁ − 5

⇒ v₁ − 5 = 54

⇒ Speed of Train 1 (v₁) = 59 km/hr

⇒ When crossing Train 2, total distance to cover = L₁ + L₂ = 195 + 255 = 450 m

⇒ Relative speed (opposite direction) = v₁ + v₂ = 59 + 76 = 135 km/hr

⇒ Convert this relative speed into m/s = 135 × (5/18) = 37.5 m/s

⇒ Time required to cross Train 2 = Total Distance ÷ Relative Speed

⇒ Time = 450 ÷ 37.5 = 12 seconds

∴ The correct answer is 12 seconds.

Additional Information

Relative Speed Concepts

When two objects move in the same direction, relative speed = |S₁ − S₂|. When moving in opposite directions, relative speed = (S₁ + S₂).

Trains Crossing Point Objects

When a train crosses a stationary man, pole, or tree, the distance covered is exactly equal to the length of the train itself.

Unit Conversion Rule

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