Quantitative Aptitude PYQs

Shortcut Trick

Let total work be the LCM of 17, 21, and 19 = 6783 units.

Combined efficiency = (6783 ÷ 17) + (6783 ÷ 21) + (6783 ÷ 19) = 399 + 323 + 357 = 1079 units/hour.

Time required when working together = Total Work ÷ Combined Efficiency.

⇒ Time = 6783 ÷ 1079 = 6 ³⁰⁹/₁₀₇₉ hours.

∴ The correct answer is 6 ³⁰⁹/₁₀₇₉ hours.

Alternate Method

Given:

Time taken by pipe A = 17 hours

Time taken by pipe B = 21 hours

Time taken by pipe C = 19 hours

Formula Used:

Total Work = LCM of times taken by all individual pipes.

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Total Efficiency of pipes A, B, and C together = 399 + 323 + 357 = 1079 units/hour

⇒ Total Time taken together = Total Work ÷ Total Efficiency

⇒ Time = 6783 ÷ 1079

⇒ Time = 6 ³⁰⁹/₁₀₇₉ hours

∴ The correct answer is 6 ³⁰⁹/₁₀₇₉ hours.

Additional Information

Pipes and Cisterns

Inlet pipes fill the tank (assigned positive efficiency), while outlet pipes empty it (assigned negative efficiency). Net efficiency is the algebraic sum of all actively working pipes.

Concept of Efficiency

Efficiency is inversely proportional to the time taken. For any given setup, Work = Efficiency × Time.

Work Equivalence

If multiple independent units contribute to a shared objective, their per unit outputs (1/T_A + 1/T_B + 1/T_C) directly sum up to equal the net output rate (1/T_Total).

Shortcut Trick

Let Cost Price (CP) = 100. Since the loss is 30%, initial SP = 70.

Actual SP = 70 × (14 ÷ 4) = 245.

New SP = 54% of Actual SP = 0.54 × 245 = 132.3.

Profit = 132.3 − 100 = 32.3% 

∴ The correct answer is 32.3%.

Alternate Method

Given:

Selling price 1 (SP₁) = 4/14 of Actual SP.

Loss at SP₁ = 30%.

Selling price 2 (SP₂) = 54% of Actual SP.

         Actual SP = 245  SP₁ = 70  SP₂ = 132.3  CP = 100  × 4/14  × 54%  -30% Loss  +32.3% Profit

Calculations:

⇒ Let the Cost Price (CP) be 100 units.

⇒ Since the loss is 30% at SP₁, the value of SP₁ = 100 − 30 = 70.

⇒ According to the question: SP₁ = (4 ÷ 14) × Actual SP.

⇒ Actual SP = 70 × (14 ÷ 4) = 980 ÷ 4 = 245.

⇒ The new selling price (SP₂) is 54% of Actual SP.

⇒ SP₂ = (54 ÷ 100) × 245 = 0.54 × 245 = 132.3.

⇒ Profit = SP₂ − CP = 132.3 − 100 = 32.3 units.

⇒ Profit Percentage = (32.3 ÷ 100) × 100% = 32.3%.

∴ The correct answer is 32.3%.

Additional Information

Cost Price & Selling Price

Cost price (CP) is the base value on which profit or loss is calculated. Selling price (SP) = CP × (100 ± Profit/Loss)%.

Marked Price & Discount

Marked price (MP) is the listed price. Discount is always calculated on MP. SP = MP × (100 − Discount)%.

Successive Percentage Change

When a value is changed successively by a% and b%, the net change is given by the formula: (a + b + a × b ÷ 100)%.

Shortcut Trick

Convert the floor dimensions to cm: 1215 cm and 1134 cm.

Side of the largest square tile = HCF(1215, 1134) = 81 cm.

Total tiles = (1215 × 1134) ÷ (81 × 81).

Total tiles = 15 × 14 = 210.

∴ The correct answer is 210.

Alternate Method

Given:

Length of the floor = 12.15 m = 1215 cm

Breadth of the floor = 11.34 m = 1134 cm

Length = 12.15 m (1215 cm)
11.34 m

81 cm
Square Tile

Total Tiles = Area of Floor ÷ Area of 1 Tile

Formula Used:

Number of tiles = Total Area of Floor ÷ Area of one Tile

Maximum side of tile = HCF(Length, Breadth)

Calculations:

⇒ Side of the square tile = HCF of 1215 and 1134

⇒ Prime factorization of 1215 = 3⁵ × 5

⇒ Prime factorization of 1134 = 2 × 3⁴ × 7

⇒ HCF(1215, 1134) = 3⁴ = 81 cm

⇒ Area of the floor = 1215 × 1134 cm²

⇒ Area of one square tile = 81 × 81 cm²

⇒ Total tiles required = (1215 × 1134) ÷ (81 × 81)

⇒ Total tiles required = (1215 ÷ 81) × (1134 ÷ 81)

⇒ Total tiles required = 15 × 14 = 210

∴ The correct answer is 210.

Additional Information

Application of HCF in Mensuration

When completely covering a rectangular surface with maximum-sized square pieces without any cuts, the side length of the square piece must be the Highest Common Factor (HCF) of the dimensions of the surface.

Prime Factorization for HCF

To find the HCF of two numbers, factorize them into prime numbers and multiply the lowest powers of all common prime factors present in both numbers.

Shortcut Trick

For equal simple interests over the same time, the ratio of Principals is the inverse of the ratio of Rates.

P₁ : P₂ = R₂ : R₁ = 3 : 13

Total parts = 3 + 13 = 16 parts ≡ ₹22,400 ⇒ 1 part = ₹1,400

Sum invested at 13% (P₁) = 3 × 1400 = ₹4,200

∴ The correct answer is ₹4,200.

Alternate Method

Given:

Total Investment = ₹22,400

Rate for Part 1 (R₁) = 13% per annum

Rate for Part 2 (R₂) = 3% per annum

Time (T) = 7 years for both

Formula Used:

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

Total: ₹22,400

Part 1 (P1)
13% for 7 years

Part 2 (P2)
3% for 7 years

Simple Interest 1 = Simple Interest 2

Calculation:

Let the sum invested at 13% be P₁ and at 3% be P₂.

⇒ P₁ + P₂ = 22400

Given that the interests from both investments are equal after 7 years:

⇒ (P₁ × 13 × 7) ÷ 100 = (P₂ × 3 × 7) ÷ 100

⇒ P₁ × 13 = P₂ × 3

⇒ P₁ ÷ P₂ = 3 ÷ 13

The total ratio parts = 3 + 13 = 16 parts

⇒ 16 parts = 22400

⇒ 1 part = 22400 ÷ 16 = 1400

The sum invested at 13% corresponds to 3 parts:

⇒ P₁ = 3 × 1400 = 4200

∴ The correct answer is ₹4,200.

Additional Information

When Final Amounts are Equal

If the total amounts (Principal + SI) from two parts are equal, the ratio of their principals is P₁ : P₂ = (100 + R₂T₂) : (100 + R₁T₁).

Simple Interest Rule

Simple Interest is calculated only on the initial principal amount. SI = (P × R × T) ÷ 100.

Difference Between SI and CI (2 Years)

The difference between Compound Interest and Simple Interest for 2 years is calculated as D = P(R ÷ 100)².

Shortcut Trick

Count the total number of given values, which is 11 (an odd number).

The median position is exactly the middle term: (11 + 1) ÷ 2 = 6^th position.

To save time, we do not need to sort all numbers. Just identify the 6 smallest values mentally.

The 6 smallest terms in ascending order are: 16, 50, 55, 57, 60, and 69.

The 6^th term is 69, which is our required median.

∴ The correct answer is 69.

Alternate Method

Given:

Data set: 57, 96, 60, 91, 69, 50, 72, 16, 78, 73, 55

Total number of terms (n) = 11

Formula Used:

Median (for an odd number of terms) = Value of the ((n + 1) ÷ 2)^th term in the sorted data set.

Finding the Median

16

50

55

57

60

69

72

73

78

91

96

Median = 69
(6^th term out of 11)

Calculations:

⇒ First, arrange the data set in ascending order:

⇒ 16, 50, 55, 57, 60, 69, 72, 73, 78, 91, 96

⇒ Number of terms (n) = 11 (which is odd)

⇒ Median position = (11 + 1) ÷ 2 = 12 ÷ 2 = 6^th position

⇒ The 6^th term in the sorted sequence is 69.

∴ The correct answer is 69.

Additional Information

Mean

Mean is the simple average of a data set, calculated by dividing the sum of all observations by the total number of observations.

Median (Even Number of Terms)

When the number of observations (n) is even, the median is the simple average of the (n ÷ 2)^th and ((n ÷ 2) + 1)^th terms.

Mode

Mode is the value that appears most frequently in a given data set. A dataset may have one mode, multiple modes, or no mode at all.

Range

The range of a data set is the mathematical difference between the highest and lowest values (Range = Maximum value − Minimum value).

Shortcut Trick

The third proportional to any two numbers 'a' and 'b' is calculated using the formula: (b × b) ÷ a.

Here, the given numbers are a = 9 and b = 57.

⇒ Third proportional = (57 × 57) ÷ 9

⇒ Third proportional = 3249 ÷ 9 = 361

∴ The correct answer is 361.

Alternate Method

Given:

The first two numbers are 9 and 57.

Let the required third proportional be x.

Formula Used:

If x is the third proportional to a and b, then the ratio holds true as a : b :: b :

x. Thus, Product of extremes = Product of means.

9

:

57

::

57

:

x

Product of extremes

Product of means

9 × x = 57 × 57

Calculations:

⇒ 9 × x = 57 × 57

⇒ 9x = 3249

⇒ x = 3249 ÷ 9

⇒ x = 361

∴ The correct answer is 361.

Additional Information

First Proportional

If x is the first proportional to a, b, and c, then x : a :: b : c, which gives x = (a × b) ÷ c.

Mean Proportional

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

Fourth Proportional

If x is the fourth proportional to a, b, and c, then a : b :: c : x, which evaluates to x = (b × c) ÷ a.

Invertendo Property

If a ratio a : b = c : d holds true, then by the invertendo property, its inverse is also true: b : a = d : c.

Shortcut Trick

Equivalent single discount for successive discounts a% and b% = [a + b − (a × b) ÷ 100]%

⇒ Equivalent Discount = 10 + 15 − (10 × 15) ÷ 100

⇒ Equivalent Discount = 25 − 1.5 = 23.5%

∴ The correct answer is 23.5%.

Alternate Method

Given:

Marked Price (MP) = ₹8,000

Successive discounts = 10% and 15%

Formula Used:

Selling Price (SP) = MP × (100 − D₁)/100 × (100 − D₂)/100

Equivalent Discount % = (Total Discount ÷ MP) × 100

MP: ₹8000

₹7200

SP: ₹6120

− 10%

− 15%

Overall Discount: 23.5%

Total Discount = ₹8000 − ₹6120 = ₹1880

Calculations:

⇒ SP = 8000 × (100 − 10)/100 × (100 − 15)/100

⇒ SP = 8000 × 90/100 × 85/100

⇒ SP = 8000 × 0.9 × 0.85 = ₹6,120

⇒ Total Discount = MP − SP = 8000 − 6120 = ₹1,880

⇒ Equivalent Discount % = (1880 ÷ 8000) × 100 = 23.5%

∴ The correct answer is 23.5%.

Additional Information

Successive Discount for Three Items

For three successive discounts a%, b% and c%, first find the equivalent of a% and b%, then apply the formula again with c%, or directly use SP = MP × (1 − a/100) × (1 − b/100) × (1 − c/100).

Marked Price and Cost Price Relation

When profit % and discount % are given, the ratio of Cost Price to Marked Price is determined by the formula: CP/MP = (100 − Discount %) ÷ (100 + Profit %).

Buy X Get Y Free

When y items are given free on the purchase of x items, the discount percentage is calculated as [y ÷ (x + y)] × 100.

Shortcut Trick

Curved Surface Area (CSA) of a cylinder = 2πrh

⇒ Outer radius (r) = CSA ÷ 2πh

⇒ r = 1500 ÷ (2 × 22/7 × 68)

⇒ r = (1500 × 7) ÷ (44 × 68) = 10500 ÷ 2992

⇒ r ≈ 3.51 cm

∴ The correct answer is 3.51 cm.

Alternate Method

Given:

Outer curved surface area (CSA) = 1500 cm²

Length of the rod (h) = 68 cm

r = ?

h = 68 cm

CSA = 1500 sq.cm

Formula Used:

Curved Surface Area of a cylinder = 2πrh

Where 'r' is the base radius and 'h' is the height or length.

Calculations:

⇒ 2 × (22/7) × r × 68 = 1500

⇒ (44/7) × 68 × r = 1500

⇒ 2992 × r = 1500 × 7

⇒ 2992 × r = 10500

⇒ r = 10500 ÷ 2992

⇒ r ≈ 3.5093 cm

Rounding to two places of decimal, we get 3.51 cm.

∴ The correct answer is 3.51 cm.

Additional Information

Total Surface Area (TSA) of a Cylinder

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

Volume of a Cylinder

The capacity of a cylindrical shape is calculated by multiplying the area of its base by its height. Formula: Volume = πr²h

Curved Surface Area of a Hollow Cylinder

For a hollow cylindrical tube with an outer radius R and an inner radius r, the total curved surface area considers both surfaces. Formula: CSA = 2πRh + 2πrh

Shortcut Trick

Ratio of Amount to Principal for 2 years = 841 / 625

For 1 year, we take the square root of the ratio: √(841 / 625) = 29 / 25

This means a principal of ₹25 amounts to ₹29 in 1 year.

Interest for 1 year = 29 − 25 = 4

Rate of interest = (4 / 25) × 100 = 16%

∴ The correct answer is 16%.

Alternate Method

Given:

Principal (P) = ₹625

Amount (A) = ₹841

Time (n) = 2 years

Formula Used:

Amount = P × (1 + R/100)ⁿ

Principal
₹625

Amount
₹841
Time = 2 years
Rate = R% p.a. (CI)

Calculations:

⇒ 841 = 625 × (1 + R/100)²

⇒ 841 / 625 = (1 + R/100)²

⇒ (29 / 25)² = (1 + R/100)²

⇒ 29 / 25 = 1 + R/100

⇒ R/100 = (29 / 25) − 1

⇒ R/100 = (29 − 25) / 25

⇒ R/100 = 4 / 25

⇒ R = (4 × 100) / 25

⇒ R = 16

∴ The correct answer is 16%.

Additional Information

Effective Rate of Compound Interest

For 2 years, the effective rate of CI can be calculated directly using the successive percentage formula: R_eff = 2R + (R² / 100)%.

Half-Yearly & Quarterly Compounding

If compounded half-yearly, Rate = R/2 and Time = 2n. If compounded quarterly, Rate = R/4 and Time = 4n.

Difference Between CI and SI for 2 Years

The difference between Compound Interest and Simple Interest for 2 years is given by the formula: Difference = P × (R / 100)².

Shortcut Trick

Convert the speed directly into km/min for a faster direct calculation.

Speed = 84 km/hr = 84 ÷ 60 km/min = 1.4 km/min

Distance = Speed × Time

Distance = 1.4 × 1.6 = 2.24 km

∴ The correct answer is 2.24 km.

Alternate Method

Given:

Speed of the car = 84 km/hr

Time taken to cross the bridge = 1.6 minutes

Formula Used:

Distance = Speed × Time

Speed = 84 km/hr
Time = 1.6 mins
Bridge Length = ?

Calculation:

⇒ First, convert the given time into hours to match the speed unit.

⇒ Time = 1.6 ÷ 60 hours

⇒ Distance = Speed × Time

⇒ Distance = 84 × (1.6 ÷ 60)

⇒ Distance = (84 ÷ 60) × 1.6

⇒ Distance = 1.4 × 1.6

⇒ Distance = 2.24 km

Since the length of the car is not provided, we consider it negligible compared to the massive length of the bridge.

∴ The correct answer is 2.24 km.

Additional Information

Unit Conversions

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

Crossing Objects with Length

When a moving object of length L₁ crosses a stationary object of length L₂ (like a bridge or platform), the total distance covered equals L₁ + L₂.

Average Speed

When an object travels equal distances at speeds S₁ and S₂, the average speed is given by (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

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

Checking the last digit of the given options: 3, 9, 4, and 0.

Only 70,48,030 has a unit digit of 0.

∴ The correct answer is 70,48,030.

Alternate Method

Given:

Four numbers: 61,75,753 ; 65,67,359 ; 71,32,954 ; 70,48,030

Concept Used:

Divisibility Rule of 5: The unit digit of the number must be 0 or 5.

Calculations:

⇒ Option 1: 61,75,753 → Unit digit is 3 (Not divisible by 5)

⇒ Option 2: 65,67,359 → Unit digit is 9 (Not divisible by 5)

⇒ Option 3: 71,32,954 → Unit digit is 4 (Not divisible by 5)

⇒ Option 4: 70,48,030 → Unit digit is 0 (Divisible by 5)

⇒ Thus, 70,48,030 is perfectly divisible by 5.

∴ The correct answer is 70,48,030.

Additional Information

Divisibility Rule of 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

Divisibility Rule of 3

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

Divisibility Rule of 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Divisibility Rule of 10

A number is divisible by 10 if its last digit is exactly 0.

Shortcut Trick

Let F and S be their ages 2 years ago. Given: F − 2S = 22.

Present ages are (F + 2) and (S + 2). The new relation is (F + 2) − 2(S + 2) = F − 2S − 2 = 20.

Let after y years, the father be twice his son's age. Then (Present Father + y) = 2 × (Present Son + y).

This simplifies to: Present Father − 2 × Present Son = y.

Since we already found this difference is 20, we directly get y = 20 years without solving for individual ages.

∴ The correct answer is 20 years.

Alternate Method

Given:

Let son's age 2 years ago = x

Father's age 2 years ago = 2x + 22

2 Years Ago
Son: x
Father: 2x+22

Present
Son: x+2
Father: 2x+24

After y Years
Son: x+2+y
Father: 2x+24+y

+2
+y

Condition: 2x + 24 + y = 2 × (x + 2 + y)

Calculations:

⇒ Present age of son = x + 2

⇒ Present age of father = 2x + 22 + 2 = 2x + 24

⇒ Let after y years, father's age be twice son's age.

⇒ Father's age after y years = 2x + 24 + y

⇒ Son's age after y years = x + 2 + y

⇒ 2x + 24 + y = 2 × (x + 2 + y)

⇒ 2x + 24 + y = 2x + 4 + 2y

⇒ 24 − 4 = 2y − y

⇒ y = 20

∴ The correct answer is 20 years.

Additional Information

Linear Equations in Ages

Age problems often involve setting up linear equations based on past, present, and future timelines. Always relate expressions back to a single reference point (usually present age) to maintain consistency in calculations.

Age Differences over Time

While the absolute difference between two persons' ages remains constant over time, any scaled mathematical difference (like Father's age − 2 × Son's age) will change linearly as the years progress.

Ratio Method for Ages

If two age ratios at different time periods are provided, comparing the difference in ratio terms to the actual time gap is a fast way to find current ages without solving multiple simultaneous equations.

Shortcut Trick

For two consecutive even numbers, Sum of Squares = 2 × (Average² + 1)

⇒ Average² + 1 = 2180 ÷ 2 = 1090

⇒ Average² = 1089 ⇒ Average = 33

Sum of the two numbers = 2 × Average = 2 × 33 = 66

∴ The correct answer is 66.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 2180.

Formula Used:

(a + b)² = a² + 2ab + b²

Calculations:

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

⇒ x² + (x + 2)² = 2180

⇒ x² + (x² + 4x + 4) = 2180

⇒ 2x² + 4x + 4 = 2180

⇒ 2x² + 4x − 2176 = 0

Dividing the equation by 2:

⇒ x² + 2x − 1088 = 0

⇒ x² + 34x − 32x − 1088 = 0

⇒ x(x + 34) − 32(x + 34) = 0

⇒ (x − 32)(x + 34) = 0

⇒ Roots are x = 32 and x = −34

⇒ Since the given values are natural numbers, x cannot be negative.

⇒ x = 32

⇒ The second consecutive even number is x + 2 = 34

⇒ Sum of the numbers = 32 + 34 = 66

∴ The correct answer is 66.

Additional Information

Consecutive Even and Odd Numbers

The difference between any two consecutive even or odd numbers is always 2. If the first even/odd number is x, the next consecutive number will be x + 2.

Sum of First n Natural Numbers

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

Sum of First n Even Natural Numbers

The sum of the first n even natural numbers can be found directly using the formula S = n(n + 1).

Shortcut Trick

M's Profit = (1860 + 372) ÷ 2 = ₹1116, and N's Profit = (1860 − 372) ÷ 2 = ₹744.

Profit Ratio (M : N) = 1116 : 744 = 3 : 2.

Investment Ratio = (Profit M ÷ Time M) : (Profit N ÷ Time N) = (3 ÷ 7) : (2 ÷ 5) = 15 : 14.

Difference in investment ratio parts = 15 − 14 = 1 unit.

Given difference in capital = ₹35,000 ⇒ 1 unit = ₹35,000.

Capital of M = 15 units = 15 × 35,000 = ₹5,25,000.

∴ The correct answer is ₹5,25,000.

Alternate Method

Given:

Total profit = ₹1,860

M's profit is ₹372 more than N's profit.

Time period of M = 7 months, Time period of N = 5 months.

M's capital is ₹35,000 more than N's capital.

Formula Used:

Profit Ratio = (Capital_M × Time_M) : (Capital_N × Time_N)

Partner M
Time = 7 months
Profit = ₹1116 (3 parts)
Capital = 15 units

Partner N
Time = 5 months
Profit = ₹744 (2 parts)
Capital = 14 units

Difference = 15 − 14 = 1 unit = ₹35,000

M's Capital = 15 × 35,000 = ₹5,25,000

Calculations:

⇒ Let N's profit be P_N and M's profit be P_M.

⇒ P_M + P_N = 1860 and P_M − P_N = 372

⇒ Adding both equations: 2 × P_M = 2232 ⇒ P_M = 1116

⇒ P_N = 1860 − 1116 = 744

⇒ Profit Ratio (M : N) = 1116 : 744 = 3 : 2

⇒ Let N's capital be x. Then M's capital is (x + 35000).

⇒ [(x + 35000) × 7] ÷ [x × 5] = 3 ÷ 2

⇒ 14 × (x + 35000) = 15x

⇒ 14x + 490000 = 15x

⇒ x = 490000

⇒ N's capital = ₹4,90,000

⇒ M's capital = 490000 + 35000 = ₹5,25,000

∴ The correct answer is ₹5,25,000.

Additional Information

Simple vs Compound Partnership

In simple partnership, capitals are invested for the same time period. In compound partnership, the time periods differ and profit is shared based on the product of Capital × Time.

Working vs Sleeping Partner

A working partner receives a fixed salary or a certain percentage of the profit for managing the business, while the remaining profit is distributed in the ratio of their effective investments.

Finding Unknown Capital

When profit ratio and time ratio are known, the capital ratio can be directly computed using: Capital Ratio = (Profit Ratio) ÷ (Time Ratio).

Shortcut Trick

The area of a triangle can be directly calculated when the base and height are known using the direct formula.

⇒ Area = 1/2 × base × altitude

⇒ Area = 1/2 × 28 × 15

⇒ Area = 14 × 15 = 210 cm²

∴ The correct answer is 210 cm².

Alternate Method

Base = 28 cm
Altitude = 15 cm
A
B
C
Area = 1/2 × Base × Altitude

Given:

⇒ Base of the triangle (b) = 28 cm

⇒ Corresponding altitude (h) = 15 cm

Formula Used:

⇒ Area of a Triangle = 1/2 × b × h

Calculations:

⇒ Area = 1/2 × 28 × 15

⇒ Area = 14 × 15

⇒ Area = 210 cm²

∴ The correct answer is 210 cm².

Additional Information

Heron's Formula

Used when all three side lengths (a, b, c) are known: Area = √[s(s − a)(s − b)(s − c)], where semi-perimeter s = (a + b + c) ÷ 2.

Area of an Equilateral Triangle

For a triangle with all three equal sides of length a: Area = (√3 ÷ 4) × a².

Area of a Right-Angled Triangle

When the base and perpendicular are explicitly given forming a right angle: Area = 1/2 × Base × Perpendicular.

Shortcut Trick

Solve the expression strategically by grouping similar denominators.

Subtract the similar fractions first: 5/4 − 3/4 = 2/4 = 1/2

Simplify the complex fraction: 1 ÷ (1 + 4/7) = 1 ÷ (11/7) = 7/11

Add the final simplified values: 1/2 + 7/11 = (11 + 14) / 22 = 25/22

∴ The correct answer is 25/22.

Alternate Method

Given:

Expression: 5/4 + 1 / (1 + 4/7) − 3/4

Concept Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and basic fraction arithmetic.

Calculations:

⇒ 5/4 + 1 / ((7 + 4) / 7) − 3/4

⇒ 5/4 + 1 / (11/7) − 3/4

⇒ 5/4 + 7/11 − 3/4

Rearranging the terms to combine the fractions with a common denominator first:

⇒ (5/4 − 3/4) + 7/11

⇒ 2/4 + 7/11

⇒ 1/2 + 7/11

Taking the Least Common Multiple (LCM) of 2 and 11, which is 22:

⇒ (1 × 11) / (2 × 11) + (7 × 2) / (11 × 2)

⇒ 11/22 + 14/22

⇒ (11 + 14) / 22

⇒ 25/22

∴ The correct answer is 25/22.

Additional Information

BODMAS Rule

BODMAS dictates the order of mathematical operations to ensure a single, correct evaluation. It stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Complex Fractions

A fraction where the numerator, denominator, or both contain another fraction. Always simplify by resolving the inner expressions into single proper or improper fractions before executing the outer division.

Reciprocal of a Fraction

To find the reciprocal of any given fraction (a/b), simply flip it upside down to form (b/a). Multiplying a fraction by its reciprocal always results in 1.

Shortcut Trick

When a number is increased by 100%, it becomes 200% of its original value (or simply doubles).

⇒ 200% of the Number = 2790

⇒ 100% of the Number = 2790 ÷ 2 = 1395

∴ The correct answer is 1395.

Alternate Method

Given:

A number, when increased by 100%, gives 2790.

Original (100%)

• Increase (100%)

Total (200%) = 2790

Formula Used:

Final Value = Original Value + (Percentage Increase × Original Value)

Calculation:

Let the required number be x.

⇒ x + (100% of x) = 2790

⇒ x + (100/100 × x) = 2790

⇒ x + x = 2790

⇒ 2x = 2790

⇒ x = 2790 ÷ 2

⇒ x = 1395

∴ The correct answer is 1395.

Additional Information

Percentage Increase / Decrease

New Value = Original Value × (1 ± R/100), where R is the percentage change.

Successive Percentage Change

Net % change = a + b + (a × b)/100, where a and b are successive percentage changes.

Multiplier Method

Increasing a value by 100% doubles the original quantity (Multiplier = 2). Increasing it by 200% triples it, and so on.

Shortcut Trick

Since the investment time period is the same, the profit is divided in the ratio of their investments.

Investment Ratio (Aarti : Anil : Naveen) = 1110 : 1760 : 1230 = 111 : 176 : 123

Sum of ratio parts = 111 + 176 + 123 = 410

Naveen's share = [123 / 410] × Total Profit

Naveen's share = (123 / 410) × 1620 = (3 / 10) × 1620 = 3 × 162 = 486

∴ The correct answer is ₹486.

Alternate Method

Given:

Aarti's investment = ₹1,110

Anil's investment = ₹1,760

Naveen's investment = ₹1,230

Total annual profit = ₹1,620

Formula Used:

Profit Ratio = Investment Ratio (when the time duration is identical for all partners)

₹1110
Aarti
1 Year

₹1760
Anil
1 Year

₹1230
Naveen
1 Year

Ratio → 111 : 176 : 123

Calculations:

⇒ Ratio of their profits = 1110 : 1760 : 1230

⇒ Simplifying by dividing by 10 = 111 : 176 : 123

⇒ Sum of the ratio parts = 111 + 176 + 123 = 410

⇒ Naveen's ratio share = 123 parts out of 410 total parts

⇒ Naveen's profit share = (123 / 410) × 1620

⇒ Naveen's profit share = 3 / 10 × 1620

⇒ Naveen's profit share = 3 × 162 = 486

∴ The correct answer is ₹486.

Additional Information

Simple Partnership

When all partners invest their money for the exact same duration, the profits are distributed strictly in the ratio of their capital investments.

Compound Partnership

When the time periods of investment are different, the profit ratio is calculated as the ratio of the product of their capital and time (C₁ × T₁ : C₂ × T₂).

Sleeping Partner

A partner who merely provides capital but does not actively manage the business is called a sleeping partner. Their share of profit depends solely on their investment and time duration.

Shortcut Trick

Price after first 12% discount = ₹2860 × (100 − 12)/100 = ₹2860 × 0.88 = ₹2516.8

Value of second discount = Price after 1^st discount − Final Selling Price

Second discount = ₹2516.8 − ₹2390.96 = ₹125.84

Rate of second discount = (125.84 ÷ 2516.8) × 100 = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Marked Price (MP) = ₹2860

Selling Price (SP) = ₹2390.96

First discount (d₁) = 12%

Let the second discount be d₂.

Formula Used:

SP = MP × (1 − d₁/100) × (1 − d₂/100)

−12%

−d₂%

Marked Price
₹2860

After 1^st Disc.
₹2516.8

Selling Price
₹2390.96

Successive Discount Flow

Calculations:

⇒ 2390.96 = 2860 × (1 − 12/100) × (1 − d₂/100)

⇒ 2390.96 = 2860 × 88/100 × (1 − d₂/100)

⇒ 2390.96 = 2516.8 × (1 − d₂/100)

⇒ 1 − d₂/100 = 2390.96 ÷ 2516.8

⇒ 1 − d₂/100 = 0.95

⇒ d₂/100 = 1 − 0.95

⇒ d₂/100 = 0.05

⇒ d₂ = 5%

∴ The correct answer is 5%.

Additional Information

Successive Discounts

When two consecutive discounts d₁% and d₂% are offered, the single equivalent discount is given by: D = d₁ + d₂ − (d₁ × d₂) / 100.

Relation Between MP and CP

The direct ratio between Marked Price and Cost Price can be calculated using the profit and discount percentages: MP / CP = (100 + Profit%) / (100 − Discount%).

Selling Price Formula

The selling price after a standard discount is evaluated over the marked price directly: SP = MP × (100 − Discount%) / 100.

Shortcut Trick

Winning margin = (60% − 40%) = 20% of the valid votes.

Total Voters × 80% × (100% − 15%) × 20% = 1394

⇒ Total × 0.8 × 0.85 × 0.2 = 1394

⇒ Total × 0.136 = 1394

⇒ Total = 1394 ÷ 0.136 = 10250

∴ The correct answer is 10250 voters.

Alternate Method

Given:

Voted = 80% of registered, Invalid = 15% of polled votes.

Winner = 60% of valid votes, Margin = 1394 votes.

80%
20%
85%
15%
60%
40%

Total Voters = 100u

Voted = 80u

Not Voted = 20u

Valid = 68u

Invalid = 12u

Winner = 40.8u

Loser = 27.2u

Margin = 13.6u
13.6 units = 1394
1 unit = 102.5
100 units = 10250

⇒ Let the total number of registered voters be 100u.

⇒ Votes polled = 80% of 100u = 80u

⇒ Invalid votes = 15% of 80u = 12u

⇒ Valid votes = 80u − 12u = 68u

⇒ Winner's votes = 60% of 68u = 40.8u

⇒ Loser's votes = (100% − 60%) of 68u = 40% of 68u = 27.2u

⇒ Winning Margin = 40.8u − 27.2u = 13.6u

⇒ Given, 13.6u = 1394 ⇒ u = 102.5

⇒ Total registered voters = 100u = 100 × 102.5 = 10250

∴ The correct answer is 10250 voters.

Additional Information

Valid Votes Base

In election problems, percentages secured by the candidates are typically calculated on the total valid votes unless specifically stated as a percentage of total registered voters.

Successive Percentage Change

When events occur sequentially (e.g., total → voted → valid), the ultimate value is found by multiplying all the respective fractional factors to the original base.