Quantitative Aptitude PYQs

Shortcut Trick

Let the total registered voters be 100%.

Voters who did not vote = 8%, so voters who cast votes = 92%.

Winner received 48% of total voters. Runner-up received (92% − 48%) − 120 blank votes = 44% − 120.

Victory Margin = 48% − (44% − 120) = 4% + 120.

Given 4% + 120 = 780 ⇒ 4% = 660 ⇒ 100% = 16,500.

∴ The correct answer is 16,500.

Alternate Method

Given:

⇒ Let Total Voters = x

⇒ Votes Cast = 92% of x = 0.92x

⇒ Winner's Votes = 48% of x = 0.48x

⇒ Runner-up's Votes = (Total Cast − Blank − Winner) = (0.92x − 120 − 0.48x) = 0.44x − 120

Calculations:

⇒ Winner − Runner-up = 780

⇒ 0.48x − (0.44x − 120) = 780

⇒ 0.04x + 120 = 780

⇒ 0.04x = 660

⇒ x = 660 ÷ 0.04

⇒ x = 16,500

∴ The correct answer is 16,500.

Additional Information

Election Base Percentage

Always identify if the percentage is of total registered voters or valid votes cast to set the correct base for calculations.

Valid vs Invalid Votes

Valid Votes = (Total Cast − Invalid/Blank). The sum of votes of all candidates must equal the Total Valid Votes.

Victory Margin Formula

Margin = Votes of Winner − Votes of Runner-up. In percentage terms: (Winner% − Runner-up%) × Total = Margin.

Shortcut Trick

Relative Speed in same direction = 87 − 51 = 36 kmph.

Convert speed to m/s: 36 × (5/18) = 10 m/s.

Total distance (2L) = Speed × Time = 10 × 175 = 1750 m.

Length of one train (L) = 1750 ÷ 2 = 875 m.

∴ The correct answer is 875 m.

Alternate Method

Given: Speed of Rajdhani Express = 87 km/hr, Speed of Garib Rath Express = 51 km/hr, Time taken = 175 seconds, Lengths are equal (L₁ = L₂ = L).

Formula Used: Relative Speed (Same Direction) = S₁ − S₂; Total Distance = L₁ + L₂; Distance = Speed × Time.

⇒ Relative Speed = 87 − 51 = 36 km/hr

⇒ Relative Speed in m/s = 36 × (5/18) = 10 m/s

⇒ Total distance to be covered = L + L = 2L

⇒ 2L = 10 × 175

⇒ 2L = 1750 m

⇒ L = 1750 ÷ 2 = 875 m

∴ The correct answer is 875 m.

Additional Information

Relative Speed Concepts

When moving in the same direction, subtract speeds (S₁ − S₂). When moving in opposite directions, add speeds (S₁ + S₂).

Unit Conversion

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

Train Crossing Rules

The distance covered when one train crosses another is always the sum of their lengths (L₁ + L₂), regardless of the direction.

Shortcut Trick

Any two consecutive integers are always co-prime to each other.

Since P and P + 1 are consecutive integers, their highest common factor (HCF) must be 1.

∴ The correct answer is 1.

Alternate Method

Given values: P is a prime number.

The consecutive number is P + 1.

⇒ Let us assume a prime number, P = 3.

⇒ Then, P + 1 = 3 + 1 = 4.

⇒ Factors of 3 are 1 and 3.

⇒ Factors of 4 are 1, 2, and 4.

⇒ Common factor is 1.

⇒ HCF(3, 4) = 1.

∴ The correct answer is 1.

Additional Information

Co-prime Numbers

Two numbers are said to be co-prime if their highest common factor (HCF) is equal to 1.

HCF of Prime Numbers

The HCF of any two distinct prime numbers is always 1, because prime numbers only have 1 and themselves as factors.

LCM of Co-prime Numbers

The Least Common Multiple (LCM) of any two co-prime numbers, such as P and P + 1, is their product, which is P × (P + 1) = P² + P.

Shortcut Trick

The effective interest rate for 2 years at 14% = 14 + 14 + (14 × 14 ÷ 100) = 29.96%.

Total Interest = 2500 × 29.96 ÷ 100 = ₹749.

Total Amount = Principal + Interest = 2500 + 749 = ₹3249.

∴ The correct answer is ₹3249.

Alternate Method

Given:

Principal (P) = ₹2,500

Rate (R) = 14% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R ÷ 100)ⁿ

Calculations:

⇒ A = 2500 × (1 + 14 ÷ 100)²

⇒ A = 2500 × (114 ÷ 100) × (114 ÷ 100)

⇒ A = 2500 × (57 ÷ 50) × (57 ÷ 50)

⇒ A = 2500 × (3249 ÷ 2500)

⇒ A = ₹3249

∴ The correct answer is ₹3249.

Additional Information

Compound Interest Formula

The interest is calculated on the principal plus the accumulated interest. CI = A − P.

Effective Rate of Interest

For a two-year period at rate R%, the effective percentage is 2R + (R² ÷ 100)%.

CI vs SI Difference

For 2 years, the difference between CI and SI is given by P × (R ÷ 100)².

Shortcut Trick

Start with the innermost square root: √(11 − 4√7) = √(√7 − 2)² = √7 − 2.

Substitute this value back into the intended expression: √(51 − √7 + √7 − 2).

The irrational terms −√7 and +√7 cancel each other out perfectly.

This simplifies the remaining expression to: √(51 − 2) = √49.

Finally, the square root of 49 is 7.

∴ The correct answer is 7.

Alternate Method

Given:

Expression: √(51 − √7 + √(11 − 4√7))

Formula Used:

(a − b)² = a² + b² − 2ab

Calculation:

⇒ First, evaluate the innermost radical: √(11 − 4√7)

⇒ The term inside can be rewritten as: 11 − 4√7 = 7 + 4 − 2 × 2 × √7

⇒ Applying the algebraic identity, we get: (√7)² + (2)² − 2 × √7 × 2

⇒ This perfectly forms a square: (√7 − 2)²

⇒ Therefore, the square root simplifies to: √(√7 − 2)² = √7 − 2

⇒ Now, substitute this result into the complete expression:

⇒ √(51 − √7 + √7 − 2)

⇒ The −√7 and +√7 terms offset and cancel each other out.

⇒ The expression now becomes: √(51 − 2)

⇒ √49

⇒ 7

∴ The correct answer is 7.

Additional Information

Simplifying Surds of the Form √(a ± √b)

To simplify such surds, we convert the inner expression into a perfect square (x ± y)² = x² + y² ± 2xy, where the rational part equals x² + y² and the irrational part matches 2xy.

Addition and Subtraction of Surds

Only like surds (surds having the same index and same radicand) can be added or subtracted. For example, m√x ± n√x = (m ± n)√x.

Nested Radicals and BODMAS

When solving expressions with nested radicals, always begin simplification from the innermost radical and work your way outward step by step.

Shortcut Trick

Calculate semi-perimeter (s): s = (87 + 108 + 75) ÷ 2 = 135 cm.

Calculate Area using Heron's: Area = √(135 × (135−87) × (135−108) × (135−75)) = √(135 × 48 × 27 × 60).

Area = √(3240²) = 3240 cm².

Altitude (h) on base 108: h = (2 × Area) ÷ base = (2 × 3240) ÷ 108 = 60 cm.

∴ The correct answer is 60 cm.

Alternate Method

Given: Sides of triangle a = 87 cm, b = 108 cm, c = 75 cm.

Formula Used: Heron's Formula for Area = √[s(s − a)(s − b)(s − c)] and Area = (1/2) × base × height.

⇒ Semi-perimeter s = (a + b + c) ÷ 2

⇒ s = (87 + 108 + 75) ÷ 2 = 135 cm

⇒ Area = √[135 × (135 − 87) × (135 − 108) × (135 − 75)]

⇒ Area = √[135 × 48 × 27 × 60]

⇒ Area = √[5 × 27 × 3 × 16 × 27 × 5 × 12]

⇒ Area = √[5² × 27² × 4² × 6²]

⇒ Area = 5 × 27 × 4 × 6 = 3240 cm²

⇒ Area = (1/2) × base × altitude

⇒ 3240 = (1/2) × 108 × altitude

⇒ 3240 = 54 × altitude

⇒ altitude = 3240 ÷ 54 = 60 cm

∴ The correct answer is 60 cm.

Additional Information

Heron's Formula

Used to find the area of a triangle when all three sides (a, b, c) are known: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter.

Altitude of a Triangle

The perpendicular distance from a vertex to the opposite side. The length of the altitude can be found using h = (2 × Area) ÷ Base.

Inradius of a Triangle

The radius of the inscribed circle is given by r = Area ÷ s, where s is the semi-perimeter.

Shortcut Trick

Total amount = ₹10,279

Sum of deductions = 67 + 33 + 13 = ₹113

Remaining amount = 10,279 − 113 = ₹10,166

Ratio after deduction (A : B : C) = 4 : 2 : 20 (Total units = 26)

Value of 1 unit = 10,166 ÷ 26 = ₹391

Difference (C − B) = (20 − 2) units + (13 − 33) = 18 × 391 − 20 = 7,038 − 20 = ₹7,018

∴ The correct answer is ₹7018.

Alternate Method

Given: Total amount = ₹10,279; Deductions: A = ₹67, B = ₹33, C = ₹13; Ratio = 4 : 2 : 20.

Formula: New Share = Original Share − Deduction; Ratio sum × x = Total remaining amount.

⇒ Total remaining = 10,279 − (67 + 33 + 13) = 10,166

⇒ 4x + 2x + 20x = 10,166

⇒ 26x = 10,166

⇒ x = 10,166 ÷ 26 = 391

⇒ B's original share = 2 × 391 + 33 = 782 + 33 = ₹815

⇒ C's original share = 20 × 391 + 13 = 7,820 + 13 = ₹7,833

⇒ Difference = 7,833 − 815 = 7,018

∴ The correct answer is ₹7018.

Additional Information

Ratio and Proportion

Ratio represents the comparison between two or more quantities. For a total S divided in ratio a:b:c, the value of one part is S / (a+b+c).

Deduction Method

When fixed amounts are subtracted from shares, subtract the total deduction from the grand total before applying the ratio distribution.

Shortcut Trick

Mean = (2 ÷ 3) × Mode = (2 ÷ 3) × 3 = 2.

Using the empirical formula: Mode = 3 × Median − 2 × Mean.

3 = 3 × Median − 2(2) ⇒ 3 = 3 × Median − 4 ⇒ Median = 7/3.

Product = 2 × (7/3) = 14/3.

Square = (14/3)² = 196/9 = 21 7/9.

∴ The correct answer is 21 7/9.

Alternate Method

Given:

Mode = 3

Mode = (3/2) × Mean

Formula Used:

Empirical relationship: Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 3 = (3/2) × Mean

⇒ Mean = (3 × 2) ÷ 3 = 2

⇒ Now, substitute values in the empirical formula:

⇒ 3 = 3 × Median − 2(2)

⇒ 3 = 3 × Median − 4

⇒ 7 = 3 × Median

⇒ Median = 7/3

⇒ Required value = (Median × Mean)²

⇒ Result = (7/3 × 2)²

⇒ Result = (14/3)² = 196/9

⇒ Result = 21 7/9

∴ The correct answer is 21 7/9.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is approximately Mode = 3 × Median − 2 × Mean.

Symmetric Distribution

In a perfectly symmetrical distribution, the values of Mean, Median, and Mode are all equal (Mean = Median = Mode).

Positively Skewed Distribution

When the tail of the distribution is on the right side, it is called positively skewed, where Mean > Median > Mode.

Negatively Skewed Distribution

When the tail of the distribution is on the left side, it is called negatively skewed, where Mode > Median > Mean.

Shortcut Trick

Convert times to fractions: Pipe P = 75/2 minutes, Pipe Q = 45 minutes.

Let Total Capacity (LCM of 75 and 45) = 225 units.

Efficiency P = 225 ÷ (75/2) = 6 units/min; Efficiency Q = 225 ÷ 45 = 5 units/min.

Pipe P works for the full 30 minutes: 6 × 30 = 180 units.

Remaining work for Pipe Q = 225 − 180 = 45 units.

Time x = 45 ÷ 5 = 9 minutes.

∴ The correct answer is 9 minutes.

Alternate Method

Given: Time P = 37.5 min = 75/2 min, Time Q = 45 min, Total time = 30 min.

Formula Used: Total Work = Time × Efficiency

⇒ Let Pipe Q be closed after x minutes.

⇒ Work done by P in 30 mins + Work done by Q in x mins = 225

⇒ (6 × 30) + (5 × x) = 225

⇒ 180 + 5x = 225

⇒ 5x = 45

⇒ x = 9 minutes

∴ The correct answer is 9 minutes.

Additional Information

Pipes and Cisterns Efficiency

Efficiency is inversely proportional to time. If a pipe fills a tank in T hours, its efficiency is 1/T per hour.

Inlet and Outlet Pipes

Inlet pipes have positive efficiency (filling), while outlet/leakage pipes have negative efficiency (emptying).

Combined Work Formula

If pipe A fills in x and pipe B fills in y, time to fill together is (x × y) ÷ (x + y).

Shortcut Trick

In a 30°−60°−90° triangle, the sides are in the ratio 1 : √3 : 2.

The side adjacent to 60° (base) is 1 unit, and the hypotenuse (ladder) is 2 units.

Given: 1 unit = 5.6 m

⇒ 2 units = 5.6 × 2 = 11.2 m

∴ The correct answer is 11.2 m.

Alternate Method

Given:

Angle of elevation (θ) = 60°

Distance from wall (Base) = 5.6 m

Formula Used:

cos(θ) = Base ÷ Hypotenuse

Calculations:

⇒ cos(60°) = 5.6 ÷ Length of ladder

⇒ 1/2 = 5.6 ÷ Length of ladder

⇒ Length of ladder = 5.6 × 2

⇒ Length of ladder = 11.2 m

∴ The correct answer is 11.2 m.

Additional Information

Trigonometric Ratios

For a right-angled triangle, sin θ = Perpendicular/Hypotenuse, cos θ = Base/Hypotenuse, and tan θ = Perpendicular/Base.

Standard Values

Key values to remember: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Pythagoras Theorem

In any right-angled triangle, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Total Perimeter = 2 × Straight Length + 2 × π × Radius

400 = 2(100) + 2 × (22/7) × r

200 = (44/7) × r

r = 1400 ÷ 44 = 31.818...

∴ The correct answer is 31.82 metres.

Alternate Method

Given:

Length of straight portions (L) = 100 m

Total Perimeter (P) = 400 m

Value of π = 22/7

Formula Used:

Perimeter of Track = (2 × Length of straight part) + (Circumference of a full circle formed by two semicircles)

Perimeter = 2L + 2πr

Calculations:

⇒ 400 = (2 × 100) + (2 × 22/7 × r)

⇒ 400 = 200 + (44/7) × r

⇒ 400 − 200 = (44/7) × r

⇒ 200 = (44/7) × r

⇒ r = (200 × 7) ÷ 44

⇒ r = 1400 ÷ 44

⇒ r = 31.818181...

⇒ r ≈ 31.82 m

∴ The correct answer is 31.82 metres.

Additional Information

Area of the Athletic Track

The total area is the sum of the rectangular area and the area of the full circle: Area = (L × 2r) + πr².

Circumference of a Semicircle

The distance around the curved part of a semicircle is πr. The total perimeter of a closed semicircle is πr + 2r.

Standard Running Track

A standard 400m outdoor track usually has a radius of approximately 36.5 metres for the innermost lane.

Shortcut Trick

Total distance for one complete round = 4 × Side = 4 × 25 = 100 m.

Speed in m/s = 4 × (5/18) = 10/9 m/s.

Time = Distance ÷ Speed = 100 ÷ (10/9) = 10 × 9 = 90 seconds.

∴ The correct answer is 90 seconds.

Alternate Method

Given:

Side of the square field = 25 m

Speed of the boy = 4 km/hr

Formula Used:

Perimeter of square = 4 × side

Time = Distance ÷ Speed

Conversion: 1 km/hr = 5/18 m/s

Calculations:

⇒ Total Distance = Perimeter = 4 × 25 = 100 m

⇒ Speed in m/s = 4 × (5/18) = 20/18 = 10/9 m/s

⇒ Time taken = 100 ÷ (10/9)

⇒ Time taken = 100 × (9/10)

⇒ Time taken = 10 × 9 = 90 seconds

∴ The correct answer is 90 seconds.

Additional Information

Unit Conversion

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

Square Properties

The distance covered in one complete round is the Perimeter (4a). The area is side² (a²).

Basic Speed Formula

Speed = Distance ÷ Time | Distance = Speed × Time | Time = Distance ÷ Speed.

Shortcut Trick

Mean proportional (MP) formula: MP² = Product of numbers

⇒ 84² = X × 147

⇒ X = (84 × 84) ÷ 147

⇒ X = 48

∴ The correct answer is 48.

Alternate Method

Given:

First term = X

Third term = 147

Mean Proportional = 84

Formula Used:

If b is the mean proportional between a and c, then b² = ac or b = √(ac).

Calculations:

⇒ 84 = √(X × 147)

⇒ 84 × 84 = X × 147

⇒ 7056 = X × 147

⇒ X = 7056 ÷ 147

⇒ X = 48

∴ The correct answer is 48.

Additional Information

Third Proportional

If a and b are two numbers, then the third proportional c is given by the relation a:b :: b:c, which means c = b²/a.

Fourth Proportional

If a, b, and c are three numbers, the fourth proportional d is such that a:b :: c:d, which means d = (b × c)/a.

Continued Proportion

Three numbers a, b, c are in continued proportion if a/b = b/c, meaning b is the geometric mean of a and c.

Shortcut Trick

Profit per unit = Selling Price − Cost Price

Profit per kg = 100 − 85 = ₹15

Total Profit = Quantity × Profit per unit

Total Profit = 10 × 15 = ₹150

∴ The correct answer is ₹150.

Alternate Method

Given: Quantity = 10 kg, Cost Price (CP) = ₹85/kg, Selling Price (SP) = ₹100/kg.

Formulas: Total Profit = Total SP − Total CP | Total CP = Quantity × CP per unit | Total SP = Quantity × SP per unit

⇒ Total Cost Price = 10 × 85 = ₹850

⇒ Total Selling Price = 10 × 100 = ₹1000

⇒ Total Profit = 1000 − 850 = ₹150

∴ The correct answer is ₹150.

Additional Information

Profit Percentage

It is the ratio of profit to cost price expressed as a percentage: Profit% = (Profit ÷ CP) × 100.

Net Gain and Loss

Profit occurs when SP > CP, while loss occurs when CP > SP. Loss = CP − SP.

Shortcut Trick

Net discount in 'Buy x get y free' schemes is calculated as y ÷ (x + y) × 100.

Here, x = 30 and y = 11.

Discount % = 11 ÷ (30 + 11) × 100 = 11 ÷ 41 × 100.

Discount % = 1100 ÷ 41 = 26.829...

Rounding to two decimal places, we get 26.83%.

∴ The correct answer is 26.83%.

Alternate Method

Given: Buy 30 items, get 11 items free.

Formulas: Discount % = (Discount ÷ MP) × 100 | Total Items = Paid Items + Free Items.

⇒ Let the Marked Price (MP) of 1 item be 1.

⇒ Total items given to customer = 30 + 11 = 41.

⇒ Total MP of 41 items = 41.

⇒ Customer pays for only 30 items, so Selling Price (SP) = 30.

⇒ Discount = MP − SP = 41 − 30 = 11.

⇒ Discount % = (11 ÷ 41) × 100 = 26.829...%.

∴ The correct answer is 26.83%.

Additional Information

Marked Price & Discount

The price listed on the label is Marked Price. Discount is always calculated on the MP unless stated otherwise. SP = MP × (100 − D%) ÷ 100.

Free Goods Schemes

In "Buy A get B free", the effective discount rate is B ÷ (A + B) × 100. The shopkeeper bears the cost of (A+B) items but receives revenue for only A items.

Shortcut Trick

In right-angled △BDC, ∠C = 90° − ∠DBC = 90° − 39° = 51°.

For △AEC, ∠AEB is an exterior angle at vertex E, formed by the line segment BC.

By Exterior Angle Theorem: ∠AEB = ∠CAE + ∠C

⇒ ∠AEB = 16° + 51° = 67°.

∴ The correct answer is 67°.

Alternate Method

Given:

BD ⊥ AC at D, so ∠BDC = 90°

∠DBC = 39° and ∠CAE = 16°

Formula Used:

Sum of angles in a triangle = 180°

Angles on a straight line add up to 180° (Linear Pair).

39°

16°

?
A
B
C
D
E
∠AEB = 67°

Calculations:

⇒ In right-angled △BDC:

⇒ ∠BDC + ∠DBC + ∠C = 180°

⇒ 90° + 39° + ∠C = 180°

⇒ 129° + ∠C = 180°

⇒ ∠C = 180° − 129° = 51°

⇒ Now, considering △AEC:

⇒ ∠AEC + ∠CAE + ∠C = 180°

⇒ ∠AEC + 16° + 51° = 180°

⇒ ∠AEC + 67° = 180°

⇒ ∠AEC = 180° − 67° = 113°

⇒ Since points B, E, and C are collinear, they form a straight line:

⇒ ∠AEB + ∠AEC = 180° (Linear Pair)

⇒ ∠AEB + 113° = 180°

⇒ ∠AEB = 180° − 113° = 67°

∴ The correct answer is 67°.

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 (opposite) interior angles.

Linear Pair Axiom

If a ray stands on a line, then the sum of two adjacent angles so formed is 180°. Conversely, if the sum of two adjacent angles is 180°, then their non-common arms form a straight line.

Altitudes of a Triangle

An altitude is a perpendicular line segment drawn from a vertex to the opposite side. The point where all three altitudes of a triangle intersect is called the Orthocenter.

Shortcut Trick

For 3 equal parts of a journey, Average Speed = 3xyz ÷ (xy + yz + zx)

Here x = 2, y = 87, z = 53.

xy + yz + zx = (2 × 87) + (87 × 53) + (53 × 2) = 174 + 4611 + 106 = 4891

Average Speed = (3 × 2 × 87 × 53) ÷ 4891 = 27666 ÷ 4891 ≈ 5.656 km/hr

Rounding off to 1 decimal place, we get 5.7 km/hr.

∴ The correct answer is 5.7 km/hr.

Alternate Method

Total Distance = 3d

Distance = d
Distance = d
Distance = d

2 km/hr
87 km/hr
53 km/hr

Given:

Speeds for each one-third distance are 2 km/hr, 87 km/hr, and 53 km/hr respectively.

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculation:

⇒ Let the total distance be 3d. Thus, each one-third distance is d.

⇒ Total Time = (d ÷ 2) + (d ÷ 87) + (d ÷ 53)

⇒ Total Time = d × [(87 × 53) + (2 × 53) + (2 × 87)] ÷ (2 × 87 × 53)

⇒ Total Time = d × [4611 + 106 + 174] ÷ 9222

⇒ Total Time = d × (4891 ÷ 9222)

⇒ Average Speed = 3d ÷ [d × (4891 ÷ 9222)]

⇒ Average Speed = (3 × 9222) ÷ 4891 = 27666 ÷ 4891

⇒ Average Speed ≈ 5.656 km/hr

⇒ Rounded to 1 decimal place = 5.7 km/hr.

∴ The correct answer is 5.7 km/hr.

Additional Information

Average Speed for Two Equal Distances

If a journey is divided into two equal halves covered at speeds x and y, Average Speed = 2xy ÷ (x + y).

Average Speed General Formula

When distances are unequal, strictly use Average Speed = Total Distance ÷ Total Time.

Relative Speed

For objects moving in opposite directions: Relative Speed = S₁ + S₂. For same direction: |S₁ − S₂|.

Shortcut Trick

For terms with the same exponent, we can multiply the bases directly:

a^x × b^x × c^x = (a × b × c)^x

⇒ 5⁻²³ × 16⁻²³ × 21⁻²³ = (5 × 16 × 21)⁻²³

⇒ (80 × 21)⁻²³ = 1680⁻²³

⇒ 1 / 1680²³

∴ The correct answer is 1 / 1680²³.

Alternate Method

Given:

Expression = 5⁻²³ × 16⁻²³ × 21⁻²³

Formula Used:

1. a^m × b^m = (a × b)^m

2. a^−m = 1 / a^m

Calculations:

⇒ 5⁻²³ × 16⁻²³ × 21⁻²³

⇒ (5 × 16)⁻²³ × 21⁻²³

⇒ 80⁻²³ × 21⁻²³

⇒ (80 × 21)⁻²³

⇒ 1680⁻²³

⇒ 1 / 1680²³

∴ The correct answer is 1 / 1680²³.

Additional Information

Power of a Product Rule

When multiplying bases with the same exponent, you can multiply the bases first and keep the exponent intact: a^m × b^m = (a × b)^m.

Negative Exponent Rule

A negative exponent indicates the reciprocal of the base raised to a positive power: a⁻ⁿ = 1 / aⁿ.

Product Rule with Same Base

When multiplying terms with the same base, simply add their exponents: a^m × aⁿ = a^m+n.

Zero Exponent Rule

Any non-zero base raised to the power of zero is always equal to one: a⁰ = 1, for all a ≠ 0.

Shortcut Trick

The given number is 814242541, which ends with an odd digit (1).

Odd numbers can never be exactly divided by even numbers, which eliminates 8 and 12.

Any number perfectly divisible by 15 must end in 0 or 5, which eliminates 15.

By eliminating the incorrect options, the given number must be divisible by 7.

∴ The correct answer is 7.

Alternate Method

Given:

Number = 814242541

Method Used:

Divisibility rule of 7: Subtract the sum of even-placed 3-digit blocks from the sum of odd-placed 3-digit blocks (counted from right to left). If the final difference is a multiple of 7, the original number is divisible by 7.

Calculations:

⇒ Divide the number into 3-digit blocks from right to left: 541, 242, and 814.

⇒ Odd-placed blocks (1^st and 3^rd) = 541 and 814

⇒ Even-placed block (2^nd) = 242

⇒ Sum of odd-placed blocks = 541 + 814 = 1355

⇒ Sum of even-placed blocks = 242

⇒ Difference = 1355 − 242 = 1113

⇒ Now, divide 1113 by 7 to check for divisibility.

⇒ 1113 ÷ 7 = 159 (Remainder = 0)

⇒ Since 1113 is perfectly divisible by 7, the given number 814242541 is also perfectly divisible by 7.

∴ The correct answer is 7.

Additional Information

Divisibility Rule of 7, 11, and 13 (Block Method)

Make groups of 3 digits starting from the rightmost digit. The original number is divisible by 7, 11, or 13 if the difference between the alternating sums of these blocks is perfectly divisible by 7, 11, or 13 respectively.

Divisibility Rule of 8 and 16

A number is divisible by 8 if the number formed by its last 3 digits is divisible by 8. It is divisible by 16 if the number formed by its last 4 digits is divisible by 16.

Divisibility Rule for Composite Numbers

To test if a number is perfectly divisible by a composite number like 12 or 15, check if it is divisible by all of its co-prime factors independently (e.g., check 3 and 4 for 12; check 3 and 5 for 15).

Shortcut Trick

By the rule of alligation, if the overall transaction results in neither loss nor gain (0%), the ratio of their cost prices is inversely proportional to their respective percentage changes.

Ratio of CP₁ : CP₂ = Gain% : Loss% = 22 : 8 = 11 : 4.

Total ratio parts = 11 + 4 = 15 parts.

Value of 15 parts = ₹1200 ⇒ 1 part = ₹80.

CP of the first speaker (11 parts) = 11 × 80 = ₹880.

∴ The correct answer is ₹880.

Alternate Method

Given:

Total Cost Price (CP) of two speakers = ₹1,200

Loss on the first speaker = 8%

Gain on the second speaker = 22%

Overall transaction result = No profit, no loss (0%)

Formula Used:

In a case of no profit and no loss, Absolute Loss on one item = Absolute Gain on the other item.

-8%
+22%
0%
22
8
1^st Speaker
2^nd Speaker
Overall
Ratio of CP = 22 : 8 = 11 : 4

Calculations:

⇒ Let the Cost Price of the first speaker be CP₁ and the second be CP₂.

⇒ According to the condition: 8% of CP₁ = 22% of CP₂

⇒ (CP₁ ÷ CP₂) = 22 ÷ 8

⇒ Ratio of CP₁ to CP₂ = 11 : 4

⇒ Total ratio parts = 11 + 4 = 15 parts

⇒ 15 parts = ₹1,200

⇒ 1 part = 1200 ÷ 15 = ₹80

⇒ Cost Price of the first speaker (CP₁) = 11 parts = 11 × 80 = ₹880

∴ The correct answer is ₹880.

Additional Information

No Profit No Loss Condition

When a transaction results in neither a gain nor a loss, the monetary value of the gain on one item perfectly balances the monetary value of the loss on the other item. Thus, Profit% × CP₁ = Loss% × CP₂.

Alligation Method for CP Ratio

Alligation is a powerful tool to find the ratio of quantities. In profit and loss problems, taking the alligation of profit/loss percentages of two individual items with their overall net percentage always yields the ratio of their Cost Prices (CP).

Successive Profit/Loss

When the price of an article undergoes successive changes of a% and b%, the net change is computed using the formula: (a + b + ab ÷ 100)%. Note that losses are substituted as negative values in the formula.