Quantitative Aptitude PYQs

Shortcut Trick

The relation between Cost Price (CP) and Marked Price (MP) can be determined directly:

CP / MP = (100 − Discount%) / (100 + Profit%)

CP / MP = (100 − 20) / (100 + 28)

CP / MP = 80 / 128 = 5 / 8

⇒ MP / CP = 8 / 5

∴ The correct answer is 8 : 5.

Alternate Method

Given:

Discount on Marked Price = 20%

Profit on Cost Price = 28%

CP: 100x

SP: 128x

MP: 160x

+28% Profit

-20% Discount

MP : CP = 160x : 100x = 8 : 5

Calculations:

Let the Cost Price (CP) of the shoes be 100x.

⇒ Selling Price (SP) = 100x + 28% of 100x = 128x

The retailer offers a 20% discount on the Marked Price (MP).

⇒ SP = MP − 20% of MP = 80% of MP

⇒ 128x = (80 / 100) × MP = 0.8 × MP

⇒ MP = 128x / 0.8 = 160x

Now, finding the ratio of Marked Price to Cost Price:

⇒ MP / CP = 160x / 100x

⇒ MP : CP = 8 : 5

∴ The correct answer is 8 : 5.

Additional Information

Direct Relation Formula

CP / MP = (100 − Discount%) / (100 + Profit%). This direct ratio relation helps solve markup and discount questions instantly without assuming variables.

Successive Discount

When two consecutive discounts D₁% and D₂% are given, the equivalent single discount is calculated as: D₁ + D₂ − (D₁ × D₂) / 100.

Markup Percentage

Markup percentage is always computed on the Cost Price to arrive at the Marked Price. Formula: Markup % = [(MP − CP) / CP] × 100.

Shortcut Trick

The perimeter of a rectangle is simply twice the sum of its adjacent sides.

Sum of sides = 5 + 8 = 13 cm

Perimeter = 2 × 13 = 26 cm

∴ The correct answer is 26 cm.

Alternate Method

8 cm
5 cm
8 cm
5 cm

Given:

Length (L) = 5 cm

Breadth (B) = 8 cm

Formula Used:

Perimeter of a rectangle = 2 × (Length + Breadth)

Calculations:

⇒ Perimeter = 2 × (5 + 8)

⇒ Perimeter = 2 × 13

⇒ Perimeter = 26 cm

∴ The correct answer is 26 cm.

Additional Information

Area of a Rectangle

The area is the total space enclosed by the boundary of the rectangle. Formula: Area = L × B.

Diagonal of a Rectangle

The diagonal divides the rectangle into two identical right-angled triangles. Formula: D = √(L² + B²).

Properties of a Rectangle

Opposite sides are equal and parallel, and all four interior angles are exactly 90°.

Shortcut Trick

Total Work = LCM(20, 22, 14) = 1540 units

Total combined efficiency = (1540 ÷ 20) + (1540 ÷ 22) + (1540 ÷ 14) = 77 + 70 + 110 = 257 units/hour

Time taken together = Total Work ÷ Total efficiency = 1540 ÷ 257 = 5²⁵⁵⁄₂₅₇ hours

∴ The correct answer is 5²⁵⁵⁄₂₅₇ hours.

Alternate Method

Given:

Time taken by pipe A = 20 hours

Time taken by pipe B = 22 hours

Time taken by pipe C = 14 hours

Formula Used:

Total Work = LCM of given times

Efficiency = Total Work ÷ Time taken

Calculations:

⇒ Total efficiency of pipes A, B and C together = 77 + 70 + 110 = 257 units/hour

⇒ Time required when operating together = Total Work ÷ Total efficiency

⇒ Time required = 1540 ÷ 257

⇒ Time required = 5²⁵⁵⁄₂₅₇ hours

∴ The correct answer is 5²⁵⁵⁄₂₅₇ hours.

Additional Information

Pipes & Cisterns - Inlet vs Outlet

An inlet pipe fills a tank and its efficiency is considered positive. An outlet pipe empties a tank and its efficiency is taken as negative.

Work & Time Relation

Efficiency and Time are inversely proportional to each other. Total Work = Time taken × Efficiency.

Combined Rate Method

If a pipe can fill a tank in x hours, the part of the tank filled by it in 1 hour is 1/x. For multiple pipes working together, simply add their 1-hour work capacities.

Shortcut Trick

Since the time period is constant (1 year), the profit is distributed in the exact ratio of the partners' investments.

Profit Ratio = 1210 : 1650 : 1300 = 121 : 165 : 130

Total sum of ratio units = 121 + 165 + 130 = 416 units.

Avinash's share = (130 ÷ 416) × 1200 = (5 ÷ 16) × 1200 = 5 × 75 = ₹375.

∴ The correct answer is ₹375.

Alternate Method

Given:

Investment by Neha = ₹1,210

Investment by Rajat = ₹1,650

Investment by Avinash = ₹1,300

Total profit at the end of the year = ₹1,200

Formula Used:

Profit Ratio = (Investment of A × Time of A) : (Investment of B × Time of B) : (Investment of C × Time of C)

₹1210
₹1650
₹1300

Neha
Rajat
Avinash

Ratio → 121 : 165 : 130

Total Profit = ₹1200

Calculations:

⇒ Since the time period is equal (12 months) for everyone, their profit shares are strictly in the ratio of their investment values.

⇒ Profit ratio of Neha : Rajat : Avinash = 1210 : 1650 : 1300

⇒ Simplifying by dividing by 10, we get the Ratio = 121 : 165 : 130

⇒ Total parts of profit = 121 + 165 + 130 = 416

⇒ Avinash's share of profit = (Avinash's Ratio Part ÷ Total Ratio Parts) × Total Profit

⇒ Avinash's share = (130 ÷ 416) × 1200

⇒ Simplifying the fraction (dividing numerator and denominator by 26):

⇒ Avinash's share = (5 ÷ 16) × 1200

⇒ Avinash's share = 5 × 75

⇒ Avinash's share = 375

∴ The correct answer is ₹375.

Additional Information

Simple Partnership

When all partners invest their capitals for the exact same time duration, the final profit or loss is distributed purely in the ratio of their original investments. Formula: Profit Ratio = I₁ : I₂ : I₃

Compound Partnership

When partners invest their respective capitals for varying time periods, the profit is shared proportionally to the product of their investment amounts and time durations. Formula: Profit Ratio = (I₁ × T₁) : (I₂ × T₂) : (I₃ × T₃)

Working vs Sleeping Partner

A working partner is actively involved in business operations and usually receives a pre-decided salary or a certain percentage of the profit outright. A sleeping partner simply acts as an investor. The residual profit is then shared among everyone according to their effective capital investment ratios.

Shortcut Trick

The unit digit of 769501 is 1, which means it is an odd number.

Even numbers like 96 and 90 can never divide an odd number completely.

Any number divisible by 105 (a multiple of 5) must end in 0 or 5, so 105 is also eliminated.

By the process of elimination, the correct divisor has to be 97.

∴ The correct answer is 97.

Alternate Method

Given:

The number is 769501 and the options are 96, 97, 90, 105.

Concept Used:

We can check the divisibility rapidly by observing the unit digit of the given number and the options.

Calculation:

⇒ The given number is 769501, and its unit digit is 1.

⇒ Option 1: 96 is an even number. An even number multiplied by any integer always gives an even unit digit. So, 96 cannot completely divide 769501.

⇒ Option 3: 90 is a multiple of 10. It can only divide numbers ending with 0. So, 90 cannot divide 769501.

⇒ Option 4: 105 is a multiple of 5 (105 = 21 × 5). It can only divide numbers ending with 0 or 5. So, 105 cannot divide 769501.

⇒ Option 2: 97 is a prime number. Let us verify by direct division.

⇒ 769501 ÷ 97 = 7933

⇒ Since the quotient is an integer, 769501 is exactly divisible by 97.

∴ The correct answer is 97.

Additional Information

Divisibility Rule for Even Numbers

An even number always ends in 0, 2, 4, 6, or 8. Only even numbers can be completely divisible by other even numbers. Odd numbers are never divisible by even numbers.

Divisibility Rule for 5

A number is exactly divisible by 5 only if its unit digit is either 0 or 5.

Divisibility Rule for 10

A number is completely divisible by 10 only if its unit digit is 0.

Prime Number Divisibility

For larger prime numbers like 97, there are specific osculator-based rules, but in competitive exams, using the elimination method for the other options is often the fastest strategy.

Shortcut Trick

For a number to be divisible by 22, it must be divisible by both 2 and 11.

Option 3 is odd, so it is instantly eliminated.

Checking divisibility of 11 for 71,32,950 ⇒ (7 + 3 + 9 + 0) − (1 + 2 + 5).

⇒ 19 − 8 = 11 (which is divisible by 11).

Since it is even and perfectly divisible by 11, it is divisible by 22.

∴ The correct answer is 71,32,950.

Alternate Method

Given:

Four numerical options to check for divisibility by 22.

Concept Used:

A number is divisible by 22 if it is completely divisible by its co-prime factors, which are 2 and 11.

Divisibility by 2: The last digit must be an even number (0, 2, 4, 6, 8).

Divisibility by 11: The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

Calculations:

⇒ Checking 65,80,350: Ends in 0 (divisible by 2). For 11 ⇒ (6 + 8 + 3 + 0) − (5 + 0 + 5) = 17 − 10 = 7. Not divisible by 11.

⇒ Checking 64,67,614: Ends in 4 (divisible by 2). For 11 ⇒ (6 + 6 + 6 + 4) − (4 + 7 + 1) = 22 − 12 = 10. Not divisible by 11.

⇒ Checking 61,36,901: Ends in 1 (odd). Not divisible by 2.

⇒ Checking 71,32,950: Ends in 0 (divisible by 2). For 11 ⇒ (7 + 3 + 9 + 0) − (1 + 2 + 5) = 19 − 8 = 11. Divisible by 11.

⇒ Therefore, only 71,32,950 is divisible by both 2 and 11.

∴ The correct answer is 71,32,950.

Additional Information

Divisibility by Composite Numbers

To check divisibility by a composite number, you must express it as the product of two co-prime factors. For example, to check for 12, check divisibility by 3 and 4.

Divisibility Rule of 11

A number is divisible by 11 if the difference of the sum of the alternate digits is either 0 or a multiple of 11.

Divisibility Rule of 7, 11 and 13 combined

If the difference between the number formed by the last three digits and the number formed by the remaining digits is divisible by 7, 11 or 13, then the original number is divisible by them respectively.

Shortcut Trick

Let Anmol's initial time be T.

When his speed doubles, the time taken becomes T/2.

Time saved by Anmol = 6 hours (late initially) + 2 hours (early now) = 8 hours.

Total time saved = T − T/2 = T/2 = 8 hours ⇒ T = 16 hours.

Anmol's speed = Distance ÷ Time = 96 ÷ 16 = 6 km/hr.

∴ The correct answer is 6 km/hr.

Alternate Method

Given:

Total distance = 96 km

Let Anmol's initial time = T_A and Nikhil's time = T_N

Anmol (2x Speed)
Nikhil
Anmol (1x Speed)

TN − 2
TN
TN + 6

2 hrs
6 hrs

Total Time Saved = 8 hrs = TA / 2

Calculations:

⇒ Case 1: T_A = T_N + 6

⇒ Case 2: When Anmol's speed is doubled, his new time = T_A / 2

⇒ T_A / 2 = T_N − 2

⇒ Subtracting Case 2 from Case 1:

⇒ T_A − T_A / 2 = (T_N + 6) − (T_N − 2)

⇒ T_A / 2 = 6 + 2

⇒ T_A / 2 = 8

⇒ T_A = 16 hours

⇒ Speed of Anmol = Distance ÷ Time = 96 ÷ 16 = 6 km/hr

∴ The correct answer is 6 km/hr.

Additional Information

Speed and Time Relation

For a constant distance, speed is inversely proportional to time. If speed becomes n times, the time taken becomes 1/n times.

Relative Time Shift

When a journey takes t₁ time more initially and later takes t₂ time less than a reference point, the net change in time is exactly t₁ + t₂.

Average Speed

When equal distances are covered at different speeds a and b, the overall average speed is calculated using (2 × a × b) ÷ (a + b).

Shortcut Trick

Given 6A = 9B = 2C.

Ratio of A : B : C = 1/6 : 1/9 : 1/2 = 3 : 2 : 9.

Total parts = 3 + 2 + 9 = 14.

Share of A = (3 ÷ 14) × 73,500 = 15,750.

∴ The correct answer is ₹15,750.

Alternate Method

Given:

Total amount to be divided = ₹73,500

Equality condition: 6 × A = 9 × B = 2 × C

Formula Used:

If pA = qB = rC, then A : B : C = 1/p : 1/q : 1/r.

Total = 14 parts = ₹73,500

A (3)
B (2)
C (9)

₹15,750
₹10,500
₹47,250

Calculations:

⇒ Let 6A = 9B = 2C = LCM(6, 9, 2) = 18 units

⇒ Ratio of A : B : C = 18/6 : 18/9 : 18/2 = 3 : 2 : 9

⇒ Total ratio parts = 3 + 2 + 9 = 14 parts

⇒ 14 parts = ₹73,500

⇒ 1 part = 73,500 ÷ 14 = ₹5,250

⇒ Share of A = 3 parts = 3 × 5,250 = ₹15,750

∴ The correct answer is ₹15,750.

Additional Information

Inverse Proportion Ratio

If pA = qB = rC, then the ratio of A : B : C is calculated as 1/p : 1/q : 1/r. Multiply by the LCM of p, q, and r to convert these into simplest integer ratios.

Proportional Division

To divide a total amount N into a ratio x : y : z, the share of the first part is calculated directly as (x ÷ (x + y + z)) × N.

Combined Ratio

If A : B = a : b and B : C = x : y, then the combined ratio A : B : C can be found by making the common term B equal, resulting in the compound ratio ax : bx : by.

Shortcut Trick

Number of cubes = Total Volume ÷ Volume of 1 small cube

Number of cubes = (24 × 16 × 10) ÷ (4 × 4 × 4)

Number of cubes = 3840 ÷ 64 = 60

∴ The correct answer is 60 cubes.

Alternate Method

Given:

Length of cuboid (L) = 24 cm, Breadth (B) = 16 cm, Height (H) = 10 cm

Side of small cube (a) = 4 cm

Formula Used:

Number of cubes = Volume of cuboid ÷ Volume of one cube

24 cm
10 cm
16 cm
Solid Cuboid

Melted

4 cm
Small Cubes

Calculations:

⇒ Volume of cuboid = L × B × H

⇒ Volume of cuboid = 24 × 16 × 10 = 3840 cm³

⇒ Volume of one small cube = a³

⇒ Volume of one small cube = 4 × 4 × 4 = 64 cm³

⇒ Number of cubes formed = 3840 ÷ 64

⇒ Number of cubes formed = 60

∴ The correct answer is 60 cubes.

Additional Information

Melting and Recasting

When a solid 3D object is melted to form multiple smaller objects, the overall volume remains perfectly constant (Total Volume = n × Volume of small object).

Volume of Cuboid

Calculated using the formula L × B × H. The longest diagonal of a cuboid is given by √(L² + B² + H²).

Volume of Cube

Calculated as a³, where 'a' is the length of its edge. The total surface area of a cube is 6a².

Cutting vs Melting Rules

Cutting a solid into smaller blocks requires finding whole pieces along each physical dimension (ignoring leftover parts), whereas melting ignores dimensions and relies purely on total volume.

Shortcut Trick

Let the average of the two consecutive even numbers be 'a' (which will be an odd number).

The numbers can be written as (a − 1) and (a + 1), and their required sum is simply 2a.

Sum of their squares = (a − 1)² + (a + 1)² = 2740

⇒ 2(a² + 1) = 2740 ⇒ a² + 1 = 1370 ⇒ a² = 1369 ⇒ a = 37

The sum of the numbers = 2 × 37 = 74.

∴ The correct answer is 74.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 2740.

Method Used:

Use the algebraic identity: (x + y)² = x² + 2xy + y²

Calculations:

Let the two consecutive even natural numbers be x and x + 2.

⇒ x² + (x + 2)² = 2740

⇒ x² + x² + 4x + 4 = 2740

⇒ 2x² + 4x − 2736 = 0

Dividing the entire equation by 2, we get:

⇒ x² + 2x − 1368 = 0

Now, split the middle term to factorize the quadratic equation:

⇒ x² + 38x − 36x − 1368 = 0

⇒ x(x + 38) − 36(x + 38) = 0

⇒ (x − 36)(x + 38) = 0

Since natural numbers cannot be negative, x cannot be −38. Therefore, x = 36.

The two consecutive even numbers are 36 and 38.

⇒ Sum of the numbers = 36 + 38 = 74

∴ The correct answer is 74.

Additional Information

Sum of First n Natural Numbers

The sum of the first n natural numbers is given by the formula: S = n(n + 1) ÷ 2.

Sum of Squares of First n Natural Numbers

The sum of the squares of the first n natural numbers is calculated using: S = n(n + 1)(2n + 1) ÷ 6.

Properties of Even Numbers

The sum or difference of any two even numbers is always even. The square of any even number is always a multiple of 4.

Quadratic Roots Formula

For a standard quadratic equation ax² + bx + c = 0, the roots can be directly found using: x = (−b ± √(b² − 4ac)) ÷ 2a.

Shortcut Trick

Youngest + Eldest = 2 × Middle sibling

⇒ x + (2x + 8) = 2 × 25

⇒ 3x + 8 = 50 ⇒ 3x = 42 ⇒ x = 14

Eldest age = 2x + 8 = 2(14) + 8 = 36 years

∴ The correct answer is 36 years.

Alternate Method

Given: Three ages in ascending order are x, 25, and 2x + 8.

Condition: Sum of Youngest and Eldest = 2 × Middle.

⇒ x + (2x + 8) = 2 × 25

⇒ 3x + 8 = 50

⇒ 3x = 50 − 8

⇒ 3x = 42

⇒ x = 14

Eldest sibling age:

⇒ 2x + 8 = 2(14) + 8

⇒ 28 + 8 = 36 years

∴ The correct answer is 36 years.

Additional Information

Arithmetic Progression (AP)

When the sum of the first and third terms is double the second term, the terms are in AP. 2b = a + c.

Average Property

For three numbers in AP, the middle number is the Arithmetic Mean (Average) of the other two.

Linear Equations

Solving equations involves isolating the variable by performing inverse operations on both sides.

Shortcut Trick

Total distance = Sum of lengths of two trains = 136 + 144 = 280 m.

Relative speed (opposite direction) = Sum of speeds = 65 + 79 = 144 km/hr.

Relative speed in m/s = 144 × (5/18) = 8 × 5 = 40 m/s.

Time taken = Total distance ÷ Relative speed = 280 ÷ 40 = 7 seconds.

∴ The correct answer is 7 sec.

Alternate Method

Given: Length of Train 1 (L₁) = 136 m, Length of Train 2 (L₂) = 144 m, Speed of Train 1 (S₁) = 65 km/hr, Speed of Train 2 (S₂) = 79 km/hr.

Formula Used: Time = (L₁ + L₂) ÷ (S₁ + S₂)

⇒ Total length = 136 + 144 = 280 m

⇒ Relative speed = 65 + 79 = 144 km/hr

⇒ Conversion to m/s = 144 × (5/18) m/s

⇒ Relative speed = 40 m/s

⇒ Time = 280 ÷ 40 = 7 sec

∴ The correct answer is 7 sec.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, relative speed is the difference of their 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.

Crossing Stationary Objects

When a train crosses a platform/bridge of length P, total distance covered is Length of Train + P.

Shortcut Trick

Pipe P fills full tank in 30 × (8/3) = 80 hr and Q in 48 × (5/3) = 80 hr.

Combined efficiency of P + Q = 1/80 + 1/80 = 2/80 = 1/40 units/hr.

In 4 hours, P + Q fill 4/40 = 1/10 part of the tank.

Pipe R empties this 1/10 part in 6 hours → R empties full tank in 60 hr.

Net efficiency (P + Q − R) = 1/40 − 1/60 = (3 − 2)/120 = 1/120.

∴ The correct answer is 120 hr.

Alternate Method

Given: P fills 3/8 in 30 hr; Q fills 3/5 in 48 hr; R empties filled water in 6 hr.

Formula Used: Time = Total Work ÷ Net Efficiency

⇒ Total work (LCM of 80, 80) = 240 units.

⇒ Water filled by P and Q in 4 hours = (3 + 3) × 4 = 24 units.

⇒ Pipe R empties 24 units in 6 hours → Efficiency of R = 24 ÷ 6 = 4 units/hr (negative).

⇒ Combined efficiency of P, Q and R = 3 + 3 − 4 = 2 units/hr.

⇒ Time taken by all three to fill the tank = 240 ÷ 2 = 120 hours.

∴ The correct answer is 120 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes have positive efficiency as they add water, while outlet/leak pipes have negative efficiency as they remove water.

Time and Efficiency Relation

Time taken is inversely proportional to efficiency. If a pipe fills 1/n part in 1 hour, it takes n hours to fill the full tank.

Pipes & Cisterns Net Work

Net part filled in 1 hour = (1/A + 1/B − 1/C), where A and B are inlets and C is an outlet.

Shortcut Trick

For two numbers a and b, the third proportional x is calculated as x = b² ÷ a.

Given a = 9 and b = 51.

⇒ x = (51 × 51) ÷ 9

⇒ x = 2601 ÷ 9 = 289

∴ The correct answer is 289.

Alternate Method

Given: First number (a) = 9, Second number (b) = 51, Third proportional = x.

Formula Used: In a continued proportion a : b :: b : x, the product of means equals the product of extremes.

⇒ 9 : 51 = 51 : x

⇒ 9 × x = 51 × 51

⇒ 9x = 2601

⇒ x = 2601 ÷ 9

⇒ x = 289

∴ The correct answer is 289.

Additional Information

Mean Proportional

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

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a/b = c/d, implying d = (b × c) ÷ a.

Duplicate Ratio

The duplicate ratio of a : b is a² : b².

Shortcut Trick

Combined efficiency of A and B = 1/2 work/day.

Efficiency of B alone = 1/4 work/day.

Efficiency of A = 1/2 − 1/4 = 1/4 work/day.

Time for A to complete single work = 4 days.

Time for A to complete double the work = 4 × 2 = 8 days.

∴ The correct answer is 8 days.

Alternate Method

Given: Time taken by (A + B) = 2 days, Time taken by B = 4 days.

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

⇒ Efficiency of A = Efficiency of (A + B) − Efficiency of B

⇒ Efficiency of A = 2 − 1 = 1 unit/day

⇒ New Total Work = 2 × 4 units = 8 units

⇒ Time taken by A = New Total Work ÷ Efficiency of A

⇒ Time taken by A = 8 ÷ 1 = 8 days

∴ The correct answer is 8 days.

Additional Information

Work-Efficiency Relationship

Time taken is inversely proportional to efficiency. If the amount of work is doubled, the time taken at the same efficiency also doubles.

Combined Work Formula

If A takes 'a' days and B takes 'b' days, together they take (a × b) ÷ (a + b) days to complete the work.

Pipes and Cisterns

Similar logic applies where efficiency of an outlet pipe is taken as negative, while the inlet pipe is positive.

Shortcut Trick

Ratio of CP : MP = (100 − Discount%) : (100 + Profit%).

CP : MP = (100 − 25) : (100 + 50) = 75 : 150 = 1 : 2.

Profit = 50% of CP ⇒ 0.5 × CP = 1290 ∴ CP = 2580.

Since MP = 2 × CP, Advertised Price = 2 × 2580 = 5,160.

∴ The correct answer is ₹5,160.

Alternate Method

Given: Profit% = 50%, Discount% = 25%, Profit Amount = ₹1,290.
Formulas: CP = (Profit ÷ Profit%) × 100 | SP = CP + Profit | MP = (SP × 100) ÷ (100 − Discount%)

⇒ Cost Price (CP): (1290 ÷ 50) × 100 = ₹2,580.

⇒ Selling Price (SP): 2580 + 1290 = ₹3,870.

⇒ Marked Price (MP): MP × (100 − 25)% = 3870.

⇒ MP × 0.75 = 3870 ⇒ MP = 3870 ÷ 0.75 = ₹5,160.

∴ The correct answer is ₹5,160.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price can be directly found using CP/MP = (100 − D%)/(100 + P%).

Advertised Price

Advertised Price is another term for Marked Price (MP) or List Price, on which discounts are calculated.

Shortcut Trick

Product of percentages and fraction: 0.60 × 0.30 × (1/3) = 0.06

Value given: 0.06 × Number = 204

Number = 204 ÷ 0.06 = 3400

∴ The correct answer is 3400.

Alternate Method

Given:

Percentage 1 = 60%

Percentage 2 = 30%

Fraction = \( \frac{1}{3} \)

Resulting Value = 204

Calculations:

⇒ (60/100) × (30/100) × (1/3) × Number = 204

⇒ (6/10) × (3/10) × (1/3) × Number = 204

⇒ (18/300) × Number = 204

⇒ (6/100) × Number = 204

⇒ Number = (204 × 100) ÷ 6

⇒ Number = 34 × 100

⇒ Number = 3400

∴ The correct answer is 3400.

Additional Information

Percentage to Fraction Conversion

To convert x% to a fraction, divide by 100: x/100. Example: 60% = 60/100 = 3/5.

Successive Percentage Calculation

When applying percentages successively (a% of b% of c), the net multiplier is (a/100) × (b/100) × (c/100).

Base Change Rule

If the result R is known for a series of operations on X, then X = R ÷ (Product of all multipliers).

Shortcut Trick

In a rectangle, the sum of length and width (semi-perimeter) is 64 ÷ 2 = 32 cm.

Let the width be x, then length is x + 8.

x + (x + 8) = 32 → 2x = 24 → x = 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given: Perimeter of the rectangle = 64 cm, and Length = Width + 8 cm.

Formula Used: Perimeter = 2 × (Length + Width)

Let the width be W cm.

Then, the length (L) = W + 8 cm.

⇒ 2 × (L + W) = 64

⇒ (W + 8) + W = 64 ÷ 2

⇒ 2W + 8 = 32

⇒ 2W = 32 − 8

⇒ 2W = 24

⇒ W = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Area of a Rectangle

The total space occupied by the rectangle is calculated as Area = Length × Width.

Diagonal of a Rectangle

The distance between opposite vertices is given by the Pythagorean theorem: d = √(L² + W²).

Perimeter of a Square

Since all sides are equal in a square, the perimeter is simply 4 × side.

Shortcut Trick

Check the ratio of the given sides: 35 : 21 : 28 ⇒ 5 : 3 : 4.

Since 3, 4, 5 is a Pythagorean Triplet, the triangle is right-angled.

The sides 21 cm and 28 cm are the perpendicular legs, and 35 cm is the hypotenuse.

In a right-angled triangle, the altitude to one leg is the other leg itself.

∴ The correct answer is 28 cm.

Alternate Method

Given: Sides of the triangle are 35 cm, 21 cm, and 28 cm.

Formula Used: Area of Triangle = 1/2 × Base × Height (Altitude)

Calculations:

⇒ First, verify the type of triangle using the Pythagoras Theorem:

⇒ 21² + 28² = 441 + 784 = 1225

⇒ 35² = 1225

⇒ Since 21² + 28² = 35², the triangle is right-angled at vertex B.

⇒ For a right-angled triangle, if we take the side 21 cm (BC) as the base, then the side 28 cm (AB) becomes the altitude.

⇒ Area = 1/2 × 21 × 28 = 294 cm².

⇒ Let altitude to side 21 cm be h.

⇒ 1/2 × 21 × h = 294

⇒ h = (294 × 2) ÷ 21 = 28 cm.

∴ The correct answer is 28 cm.

Additional Information

Pythagorean Triplets

Sets of three integers (a, b, c) that satisfy a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17). Any multiple like (21, 28, 35) is also a triplet.

Heron's Formula

Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. This is used for finding the area of any scalene triangle.

Altitude to Hypotenuse

In a right triangle with legs a, b and hypotenuse c, the altitude to the hypotenuse h = (a × b) ÷ c.

Shortcut Trick

Area of Right-angled Triangle = 1/2 × 36 × 64 = 1152 cm²

Area of Rhombus = Area of Triangle ÷ (3/4) = 1152 × 4/3 = 1536 cm²

Other diagonal (d₂) = (2 × 1536) ÷ 48 = 64 cm

Side of Rhombus = √((48/2)² + (64/2)²) = √(24² + 32²) = 40 cm

Perimeter = 4 × 40 = 160 cm

∴ The correct answer is 160 cm.

Alternate Method

Given:

Right-angled triangle sides = 36 cm and 64 cm.

One diagonal of rhombus (d₁) = 48 cm.

Formula Used:

Area of Triangle = 1/2 × base × height

Area of Rhombus = 1/2 × d₁ × d₂

Side of Rhombus (s) = √((d₁/2)² + (d₂/2)²)

Calculations:

⇒ Area of Right-angled Triangle = 1/2 × 36 × 64

⇒ Area of Triangle = 18 × 64 = 1152 cm²

⇒ According to question, 1152 = 3/4 × Area of Rhombus

⇒ Area of Rhombus = 1152 × 4/3 = 1536 cm²

⇒ 1536 = 1/2 × 48 × d₂

⇒ 1536 = 24 × d₂

⇒ d₂ = 1536 ÷ 24 = 64 cm

⇒ Side of Rhombus (s) = √((48/2)² + (64/2)²)

⇒ s = √(24² + 32²) = √(576 + 1024)

⇒ s = √1600 = 40 cm

⇒ Perimeter of Rhombus = 4 × side = 4 × 40 = 160 cm

∴ The correct answer is 160 cm.

Additional Information

Properties of a Rhombus

All sides are equal, and diagonals bisect each other at right angles (90°). Opposite angles are equal.

Relationship between Side and Diagonals

For a rhombus with diagonals d₁ and d₂, the side s satisfies: 4s² = d₁² + d₂².

Area of Triangle

Area = 1/2 × base × height. For a right-angled triangle, the base and height are the sides forming the 90° angle.