Quantitative Aptitude PYQs

Shortcut Trick

For Compound Interest, the multiplying factor over 2 years is the square of the annual factor.

Ratio of Amount to Principal (A/P) = 12321 / 10000

Annual multiplier = √(12321 / 10000) = 111 / 100

Rate % = ((111 − 100) / 100) × 100% = 11%

∴ The correct answer is 11%.

Alternate Method

P = ₹10,000

A = ₹12,321

t = 0
t = 2 years

Rate = R% p.a.

Given:

Principal (P) = ₹10,000

Amount (A) = ₹12,321

Time (n) = 2 years

Formula Used:

Amount (A) = P × (1 + R/100)ⁿ

Calculations:

⇒ 12321 = 10000 × (1 + R/100)²

⇒ 12321 / 10000 = (1 + R/100)²

⇒ √(12321 / 10000) = 1 + R/100

⇒ 111 / 100 = 1 + R/100

⇒ R/100 = 111/100 − 1

⇒ R/100 = 11 / 100

⇒ R = 11%

∴ The correct answer is 11%.

Additional Information

CI vs SI Difference Formula

For 2 years, the difference between CI and SI is given by: D = P × (R/100)². For 3 years, D = P × (R/100)² × (3 + R/100).

Half-Yearly Compounding

If the interest is compounded half-yearly, the Rate becomes R/2 and the Time period becomes 2n.

Effective Rate for 2 Years

The effective compound interest rate for 2 years at R% p.a. can be calculated directly as: (2R + R²/100)%.

Shortcut Trick

Using Percentage Interchangeability: 65% of 75 = 75% of 65

⇒ 3/4 × 65 = 195 / 4 = 48.75

Calculate fraction part directly: 2/5 of 45 = 2 × 9 = 18

Difference = 48.75 − 18 = 30.75

∴ The correct answer is 30.75.

Alternate Method

Given:

First value = 65% of 75

Second value = 2/5 of 45

Calculations:

Evaluate the first value:

⇒ First value = (65 / 100) × 75

⇒ First value = (13 / 20) × 75

⇒ First value = (13 × 15) / 4

⇒ First value = 195 / 4 = 48.75

Evaluate the second value:

⇒ Second value = (2 / 5) × 45

⇒ Second value = 2 × 9 = 18

Calculate the required difference between the two values:

⇒ Required Difference = First value − Second value

⇒ Required Difference = 48.75 − 18

⇒ Required Difference = 30.75

∴ The correct answer is 30.75.

Additional Information

Percentage Interchangeability

A very useful trick for faster mathematical calculations is x% of y = y% of x. For example, computing 75% of 65 is easier because 75% directly simplifies to the fraction 3/4.

Fraction to Percentage Conversion

Fractions can be easily converted to percentages by multiplying the fraction by 100. For instance, the fraction 2/5 is equivalent to (2/5) × 100% = 40%.

BODMAS Rule

Always follow the proper order of arithmetic operations to avoid errors: Brackets, Orders (powers and roots), Division and Multiplication (left to right), and Addition and Subtraction (left to right).

Shortcut Trick

Let pen cost = x and pencil cost = y. The first condition gives 6x + 19y = 275.

After the price changes, 15(x − 7) + 4(y + 7) = 132, which simplifies to 15x + 4y = 209.

Multiply the first equation by 5 and the second by 2 to balance coefficients of x.

We get 30x + 95y = 1375 and 30x + 8y = 418. Subtracting them yields 87y = 957 ⇒ y = 11.

Substituting y = 11 in the first equation gives x = 11. Cost of 6 pens and 2 pencils = 6(11) + 2(11) = 88.

∴ The correct answer is ₹88.

Alternate Method

Given:

Original cost of 6 pens and 19 pencils = ₹275

New cost of 15 pens (after ₹7 decrease) and 4 pencils (after ₹7 increase) = ₹132

Calculation:

Let the original cost of 1 pen be ₹x and 1 pencil be ₹y.

⇒ 6x + 19y = 275 --- (Equation 1)

After the given price changes, the new cost of a pen = (x − 7) and new cost of a pencil = (y + 7).

⇒ 15(x − 7) + 4(y + 7) = 132

⇒ 15x − 105 + 4y + 28 = 132

⇒ 15x + 4y − 77 = 132

⇒ 15x + 4y = 209 --- (Equation 2)

To eliminate x, multiply Equation 1 by 5 and Equation 2 by 2:

⇒ 30x + 95y = 1375

⇒ 30x + 8y = 418

Subtracting the modified Equation 2 from the modified Equation 1:

⇒ 87y = 957

⇒ y = 957 ÷ 87 = 11

Substitute y = 11 in Equation 1:

⇒ 6x + 19(11) = 275

⇒ 6x + 209 = 275

⇒ 6x = 66

⇒ x = 11

The original cost of 1 pen is ₹11 and 1 pencil is ₹11.

Now, calculate the cost of 6 pens and 2 pencils:

⇒ 6x + 2y = 6(11) + 2(11)

⇒ 66 + 22 = 88

∴ The correct answer is ₹88.

Additional Information

Linear Equations in Two Variables

An equation of the form ax + by = c is a linear equation. A system of two linear equations can be solved using substitution, elimination, or cross-multiplication methods.

Elimination Method

Multiply both equations by suitable constants so that the coefficients of one variable become mathematically equal, then add or subtract the equations to cleanly eliminate that specific variable.

Shortcut Trick

To find the square of a number ending in 5 (say N5):

The first part of the answer is N × (N + 1) and the last part is always 25.

For 145, N = 14.

First part = 14 × 15 = 210.

Combining both parts gives 21025.

∴ The correct answer is 21025.

Alternate Method

Given:

The expression is 145².

Formula Used:

(a − b)² = a² − 2ab + b²

Calculation:

⇒ 145 = 150 − 5

⇒ 145² = (150 − 5)²

⇒ 145² = 150² − 2 × 150 × 5 + 5²

⇒ 145² = 22500 − 1500 + 25

⇒ 145² = 21000 + 25

⇒ 145² = 21025

∴ The correct answer is 21025.

Additional Information

Squares of numbers ending in 5

For a number formed by digits 'a' followed by 5, the square is given by a(a + 1) followed by 25. Example: 85² → 8 × 9 = 72, so 7225.

Algebraic Identities for Squares

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². These identities are highly useful to square large numbers mentally by breaking them into simpler nearest-base parts.

Base Method for Squaring

To square a number near a base (e.g., 150), find the deviation (d). Using the formula, Square = Base × (Number + d) + d². For 145, d = −5, Base = 150. Result = 150 × (145 − 5) + 5² = 150 × 140 + 25 = 21000 + 25 = 21025.

Shortcut Trick

Let Mother's age = M and Daughter's age = D.

Simplified given equations: M − 2D = 13 and M − 3D = 7.

Subtracting the equations directly gives: D = 6 years.

Substitute D = 6 into the first equation: M = 2(6) + 13 = 25 years.

Difference between their ages = 25 − 6 = 19 years.

∴ The correct answer is 19 years.

Alternate Method

Given:

2 times the mother's age is 26 years more than 4 times the daughter's age.

3 times the daughter's age is 7 years less than the mother's age.

Mother

25 years
Daughter

6 yrs

Difference = 19 years

Calculations:

Let the mother's age be M and the daughter's age be D.

⇒ 2M = 4D + 26

⇒ M = 2D + 13   --- (Equation 1)

⇒ 3D = M − 7

⇒ M = 3D + 7   --- (Equation 2)

Equating Equation 1 and Equation 2:

⇒ 2D + 13 = 3D + 7

⇒ 3D − 2D = 13 − 7

⇒ D = 6

Substitute D = 6 in Equation 2:

⇒ M = 3(6) + 7

⇒ M = 18 + 7 = 25

Difference in their ages = M − D

⇒ Difference = 25 − 6 = 19

∴ The correct answer is 19 years.

Additional Information

Ages & Age Difference

The numerical difference between the ages of two individuals always remains constant over time. If a person is 5 years older than someone today, they will remain 5 years older indefinitely.

Linear Equations with Two Variables

Whenever a word problem provides two unknown variables (like ages of two different people), forming two distinct algebraic equations is necessary to evaluate their exact individual values.

Ratio Multiplier Method

For questions involving specific present and future age ratios, cross-multiplying the initial and final ratio expressions helps in quickly finding the fundamental ratio unit directly without extensive steps.

Shortcut Trick

Let the profit shares be 5 units, 8 units, and 17 units.

Difference between largest and smallest share = 17 − 5 = 12 units.

Given, 12 units = ₹78,106.

Total profit = 5 + 8 + 17 = 30 units.

Total profit = ₹78,106 × (30 ÷ 12) = ₹78,106 × 2.5 = ₹1,95,265.

∴ The correct answer is ₹1,95,265.

Alternate Method

Given:

Ratio of profit distributed among three partners = 5 : 8 : 17

Difference between the smallest and largest shares = ₹78,106

5x
8x
17x

Difference (17x − 5x) = 12x = ₹78,106

Total Profit = 30x = ?

Calculations:

⇒ Let the shares of the three partners be 5x, 8x, and 17x respectively.

⇒ The smallest share = 5x

⇒ The largest share = 17x

⇒ According to the given condition, 17x − 5x = 78,106

⇒ 12x = 78,106

⇒ x = 78,106 ÷ 12

⇒ Total profit = 5x + 8x + 17x = 30x

⇒ Total profit = 30 × (78,106 ÷ 12)

⇒ Total profit = 78,106 × 2.5

⇒ Total profit = 1,95,265

∴ The correct answer is ₹1,95,265.

Additional Information

Partnership Profit Ratio

In any business partnership, the ratio in which profit is shared among the partners is directly proportional to the product of their invested capital and the time period of investment (Profit ∝ Capital × Time).

Dividing a Sum in a Ratio

If a total amount is divided in the ratio a : b : c, the individual parts are calculated as (Part Ratio ÷ Sum of all Ratios) × Total Amount.

Shortcut Trick

Convert time directly to hours: 1.7 minutes = 1.7 ÷ 60 hours

Distance = Speed × Time

Length of the bridge = 54 × (1.7 ÷ 60)

Length = 0.9 × 1.7 = 1.53 km

∴ The correct answer is 1.53 km.

Alternate Method

Given:

Speed of the car = 54 km/hr

Time taken to cross the bridge = 1.7 minutes

Formula Used:

Distance = Speed × Time

Speed = 54 km/hr
Time = 1.7 minutes

Bridge Length = ?

Calculations:

⇒ First, let us convert the speed of the car into meters per second (m/s).

⇒ Speed in m/s = 54 × (5 ÷ 18)

⇒ Speed = 3 × 5 = 15 m/s

⇒ Next, convert the given time from minutes into seconds.

⇒ Time in seconds = 1.7 × 60

⇒ Time = 102 seconds

⇒ We know that Length of the bridge = Distance covered by the car

⇒ Distance = Speed × Time

⇒ Distance = 15 × 102

⇒ Distance = 1530 meters

⇒ Finally, convert the distance back into kilometers (since options are in km).

⇒ Distance in km = 1530 ÷ 1000 = 1.53 km

∴ The correct answer is 1.53 km.

Additional Information

Unit Conversion

To quickly convert speed from km/hr to m/s, multiply the given value by 5/18. Conversely, to convert from m/s to km/hr, multiply by 18/5.

Relative Speed

When two objects move in the same direction, Relative Speed = |S₁ − S₂|. When moving in opposite directions, Relative Speed = S₁ + S₂.

Average Speed

Average Speed = Total Distance ÷ Total Time. If two equal distances are covered at different speeds x and y, the Average Speed is given by (2xy) ÷ (x + y).

Shortcut Trick

When two identical cubes of side 'a' are joined, the newly formed cuboid has exactly 10 visible faces.

This is because 2 out of the 12 total faces (6 faces per cube × 2) are completely hidden inside the joint.

⇒ Total Surface Area = 10a²

⇒ Total Surface Area = 10 × (5)²

⇒ Total Surface Area = 10 × 25 = 250 cm²

∴ The correct answer is 250 cm².

Alternate Method

l = 10 cm

h = 5 cm

b = 5 cm

Given:

Edge of the individual cube (a) = 5 cm

When two identical cubes are joined end to end, the dimensions of the newly formed cuboid are:

⇒ Length (l) = 5 cm + 5 cm = 10 cm

⇒ Breadth (b) = 5 cm

⇒ Height (h) = 5 cm

Formula Used:

Total Surface Area of Cuboid = 2(lb + bh + hl)

Calculations:

⇒ TSA = 2 × (10 × 5 + 5 × 5 + 5 × 10)

⇒ TSA = 2 × (50 + 25 + 50)

⇒ TSA = 2 × 125

⇒ TSA = 250 cm²

∴ The correct answer is 250 cm².

Additional Information

Volume of a Cuboid

Volume is the total 3D space contained within the cuboid, calculated using the formula V = l × b × h.

Diagonal of a Cuboid

The length of the longest possible line segment that can fit inside the cuboid is calculated as Diagonal = √(l² + b² + h²).

Lateral Surface Area (LSA)

The Lateral Surface Area refers exclusively to the area of the four upright side walls of the cuboid, omitting the top and bottom faces. It is given by LSA = 2h(l + b).

Total Surface Area of a Cube

Since all 6 faces of a standard cube are identical squares of side length 'a', the total area formula simplifies to TSA = 6a².

Shortcut Trick

Time for P to fill tank = 24 × (3/2) = 36 hr. Q = 16 × (3/2) = 24 hr. Total capacity = LCM(36, 24) = 72 units.

Efficiency of P = +2, Q = +3. Water filled in 4 hr = 4 × (2 + 3) = 20 units.

R empties 20 units in 5 hr, so efficiency of R = −20 ÷ 5 = −4.

Combined efficiency of (P + Q + R) = 2 + 3 − 4 = +1.

Time taken to fill the tank together = 72 ÷ 1 = 72 hours.

∴ The correct answer is 72 hours.

Alternate Method

Given:

Pipe P fills 2/3 of the tank in 24 hours.

Pipe Q fills 2/3 of the tank in 16 hours.

Both P and Q were kept open for 4 hours, and then pipe R emptied the filled water in 5 hours.

Formula Used:

Total Work = LCM of time taken

Efficiency = Total Work ÷ Time taken

Calculation:

⇒ Time taken by P to fill the full tank = 24 × (3/2) = 36 hours

⇒ Time taken by Q to fill the full tank = 16 × (3/2) = 24 hours

⇒ Water filled by P and Q in 4 hours = 4 × (2 + 3) = 20 units

⇒ Pipe R empties these 20 units in 5 hours.

⇒ Efficiency of pipe R = −20 ÷ 5 = −4 units/hr (Negative sign indicates emptying)

⇒ Combined efficiency of pipes P, Q, and R together = 2 + 3 − 4 = 1 unit/hr

⇒ Time required for P, Q, and R to fill the tank = Total Work ÷ Combined Efficiency

⇒ Time = 72 ÷ 1 = 72 hours

∴ The correct answer is 72 hours.

Additional Information

Pipes & Cisterns Concept

Inlet pipes add water to the tank and are assigned a positive efficiency. Outlet pipes remove water and are assigned a negative efficiency.

LCM Method for Total Capacity

The total capacity of a tank is assumed to be the Lowest Common Multiple (LCM) of the times taken by individual pipes to fill or empty it.

Combined Work Rule

When multiple pipes operate together, their effective efficiency is the algebraic sum of their individual efficiencies. Total time = Total Capacity ÷ Combined Efficiency.

Shortcut Trick

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

⇒ Mean proportional = √(19 × 76)

⇒ Mean proportional = √(19 × 19 × 4)

⇒ Mean proportional = 19 × 2 = 38

∴ The correct answer is 38.

Alternate Method

Given:

First number (a) = 19

Second number (b) = 76

Formula Used:

In a continuous proportion a : x :: x : b, the middle term 'x' is the mean proportional.

19

:

x

::

x

:

76

Continuous Proportion
Mean Proportional (x)

Calculations:

⇒ Let the mean proportional be x.

⇒ 19 : x :: x : 76

⇒ 19 ÷ x = x ÷ 76

⇒ x² = 19 × 76

⇒ x = √(19 × 76)

⇒ x = √(19 × 19 × 4)

⇒ x = 19 × 2

⇒ x = 38

∴ The correct answer is 38.

Additional Information

Mean Proportional

For any two numbers 'a' and 'b', their mean proportional 'x' is calculated as x = √(a × b).

Third Proportional

For two numbers 'a' and 'b', the third proportional 'c' is found using the relation a : b :: b : c, which simplifies to c = b² ÷ a.

Fourth Proportional

For three numbers 'a', 'b', and 'c', the fourth proportional 'd' is determined by a : b :: c : d, resulting in d = (b × c) ÷ a.

Compound Ratio

The compound ratio of two individual ratios a : b and c : d is found by multiplying their corresponding terms, yielding (a × c) : (b × d).

Shortcut Trick

To find the smallest number to be added, use: Number to be added = Divisor − Remainder.

Divide the given number 16809 by 41.

16809 ÷ 41 gives a quotient of 409 and a remainder of 40.

Smallest number to be added = 41 − 40 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

The dividend is 16809.

The divisor is 41.

Formula Used:

Dividend = (Divisor × Quotient) + Remainder

Smallest number to be added = Divisor − Remainder

Calculations:

⇒ Perform the division of 16809 by 41.

⇒ 16809 = 41 × 400 + 409

⇒ 409 = 41 × 9 + 40

⇒ Therefore, 16809 = 41 × 409 + 40

⇒ The remainder obtained is 40.

⇒ For the number to be completely divisible by 41, the remainder must be exactly 0. Currently, it is short of the next multiple.

⇒ The required number to reach the next complete multiple = Divisor − Remainder.

⇒ Number to be added = 41 − 40 = 1.

⇒ Verification: 16809 + 1 = 16810. And 16810 ÷ 41 = 410.

∴ The correct answer is 1.

Additional Information

Number to be Subtracted

To find the smallest number to be subtracted from a given number so that the result is perfectly divisible by a divisor, simply subtract the remainder obtained from the division.

Division Algorithm

The division algorithm states that: Dividend = (Divisor × Quotient) + Remainder. This is a fundamental mathematical rule used to verify division calculations.

Divisibility and Remainders

When a given number is not perfectly divisible by a divisor, adding the difference between the divisor and the remainder bridges the exact gap required to reach the next perfect multiple.

Shortcut Trick
Given, 16 × SP = 8 × CP
⇒ SP / CP = 8 / 16 = 1 / 2
Here, CP = 2 units and SP = 1 unit. Since CP > SP, there is a loss.
Loss % = [(CP − SP) / CP] × 100
⇒ Loss % = (1 / 2) × 100 = 50%
∴ The correct answer is 50% loss.
Alternate Method
Given:
Selling price of 16 books = Cost price of 8 books
Formula Used:
Loss = Cost Price (CP) − Selling Price (SP)
Loss % = (Loss ÷ CP) × 100

Cost Price (2 units)

SP (1 unit)

Loss (1 unit)

Loss = 50%

Calculations:
⇒ Let the cost price of 1 book be ₹1.
⇒ Then, Cost Price (CP) of 16 books = ₹16.
⇒ According to the question, Selling Price (SP) of 16 books = Cost Price of 8 books = ₹8.
⇒ Since CP > SP, there is a loss in the transaction.
⇒ Loss = ₹16 − ₹8 = ₹8
⇒ Loss % = (8 ÷ 16) × 100
⇒ Loss % = (1 ÷ 2) × 100 = 50%
∴ The correct answer is 50% loss.
Additional Information
Profit and Loss Percentage
Profit % = (Profit ÷ CP) × 100 | Loss % = (Loss ÷ CP) × 100. Both are always calculated on the Cost Price (CP) unless specified otherwise.
Marked Price and Discount
Discount is always calculated on the Marked Price (MP). Final Selling Price is calculated as SP = MP − Discount.
Dishonest Dealer Concept
If a dealer claims to sell at CP but uses a false weight, Profit % = [Error ÷ (True Value − Error)] × 100.

Shortcut Trick

Effective interest rate = Rate × Time

⇒ Effective rate = 4% × 5.5 = 22%

⇒ Simple Interest = 22% of Principal

⇒ SI = 22% of 400 = 88

∴ The correct answer is ₹88.

Alternate Method

Given:

Principal (P) = ₹400

Rate (R) = 4% per annum

Time (T) = 5.5 years

Year 0
Year 5.5

P = ₹400

SI = ?

Rate = 4% p.a.
Time = 5.5 Years

• Total Simple Interest

Simple Interest Timeline

Formula Used:

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

Calculation:

⇒ SI = (400 × 4 × 5.5) ÷ 100

⇒ SI = 4 × 4 × 5.5

⇒ SI = 16 × 5.5

⇒ SI = 88

∴ The correct answer is ₹88.

Additional Information

Amount in Simple Interest

The total amount paid or received is the sum of the principal and the simple interest. Formula: A = P + SI = P[1 + (RT ÷ 100)]

Compound Interest (CI)

Interest calculated on the initial principal and also on the accumulated interest of previous periods. Formula: A = P(1 + R ÷ 100)^T

Difference Between SI and CI (for 2 years)

The difference between compound interest and simple interest for exactly 2 years on the same principal is calculated as: D = P(R ÷ 100)²

Shortcut Trick

Using the successive discount formula: D = x + y − (x × y) / 100

Here, the successive discounts are x = 20% and y = 10%.

Equivalent single discount = 20 + 10 − (20 × 10) / 100

Equivalent single discount = 30 − 2 = 28%

∴ The correct answer is 28%.

Alternate Method

Given:

First discount = 20%

Second discount = 10%

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) / 100

Equivalent Discount % = [(MP − Final SP) / MP] × 100

MP = 100

SP₁ = 80

SP₂ = 72

−20%

−10%

Equivalent Discount = 28%

Calculations:

⇒ Let the Marked Price (MP) of the article be 100.

⇒ Price after 1^st discount of 20% = 100 − (20 / 100 × 100) = 80

⇒ Price after 2^nd discount of 10% = 80 − (10 / 100 × 80) = 80 − 8 = 72

⇒ Final Selling Price (SP) = 72

⇒ Total Discount offered = MP − Final SP = 100 − 72 = 28

⇒ Single Equivalent Discount % = (28 / 100) × 100 = 28%

∴ The correct answer is 28%.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the equivalent single discount is directly given by the formula: (a + b − ab/100)%.

Marked Price vs Selling Price

A discount is always calculated on the Marked Price (or Tag Price). The final transaction value after applying discounts is the Selling Price.

Ratio Method for Discounts

Alternatively, discounts can be solved by converting percentages to fractions. Multiply the initial Marked Price ratios and the final Selling Price ratios to find the overall net change.

Three Successive Discounts

For three discounts a%, b%, and c%, first find the equivalent discount of a% and b% (let it be x%). Then apply the same formula between x% and c%.

Shortcut Trick

Total work = LCM(17, 16, 4) = 272 units.

Efficiency of A = 272 ÷ 17 = 16 units/day.

Efficiency of B = 272 ÷ 16 = 17 units/day.

Efficiency of A, B, and C together = 272 ÷ 4 = 68 units/day.

Efficiency of C = 68 − (16 + 17) = 35 units/day.

∴ The correct answer is 7 ²⁷⁄₃₅ days.

Alternate Method

Given:

Time taken by A = 17 days.

Time taken by B = 16 days.

Time taken by A, B, and C together = 4 days.

Method Used:

We use the LCM method to find the total work and the efficiency of each person.

Total Work = LCM of days taken by given persons or groups.

Efficiency = Total Work ÷ Number of Days.

Calculations:

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

⇒ Efficiency of C = 68 − (16 + 17)

⇒ Efficiency of C = 68 − 33 = 35 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time taken by C alone = 272 ÷ 35

⇒ Time taken by C alone = 7 ²⁷⁄₃₅ days

∴ The correct answer is 7 ²⁷⁄₃₅ days.

Additional Information

Pipes and Cisterns

Concepts are similar to Time and Work. Fill pipes have positive efficiency (adding work), and empty pipes have negative efficiency (removing work).

Work and Wages

Total wages are distributed in direct proportion to the actual amount of work done by each individual. If they work for an equal number of days, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

If persons work on alternate days, you group the work into cycles (e.g., A works on Day 1, B works on Day 2 → 1 cycle = 2 days). Find the combined work done in one cycle and scale it up to reach close to the total work.

Shortcut Trick

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

Side opposite to 60° (Height) = 75 m ⇒ √3 units = 75 m.

Side opposite to 90° (String length) = 2 units ⇒ (75/√3) × 2.

⇒ 25√3 × 2 = 50√3 m.

∴ The correct answer is 50√3 m.

Alternate Method

Given: Height of the kite (AB) = 75 m, Angle of elevation (θ) = 60°.

Formula Used: sin θ = Perpendicular / Hypotenuse

⇒ Let the length of the string be L.

⇒ sin 60° = 75 / L

⇒ √3 / 2 = 75 / L

⇒ L = (75 × 2) / √3

⇒ L = 150 / √3

⇒ L = (150 × √3) / (√3 × √3) = 150√3 / 3

⇒ L = 50√3 m.

∴ The correct answer is 50√3 m.

Additional Information

Trigonometric Ratios

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

Standard Values

Common values for 60°: sin 60° = √3/2, cos 60° = 1/2, and tan 60° = √3.

Angle of Elevation

The angle of elevation is the angle between the horizontal line of sight and the object when looking upwards.

Shortcut Trick

A number divisible by 45 must be divisible by both 5 and 9.

Divisibility by 9 depends only on the sum of digits.

Rearranging the digits does not change the sum of digits, so the new number remains divisible by 9.

Since the new number is divisible by 9, it must also be divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given: An 8-digit number is divisible by 45.

Formula Used: Divisibility rule of 45 (divisible by 5 and 9). Divisibility rule of 9 (sum of digits is divisible by 9).

⇒ Divisibility by 45 → Divisible by 5 and 9 (since 5 × 9 = 45 and gcd(5,9)=1).

⇒ Divisibility rule of 9: The sum of the digits of the 8-digit number must be a multiple of 9.

⇒ When digits are rearranged, the sum of digits remains the same.

⇒ Therefore, the rearranged number is definitely divisible by 9.

⇒ Any number divisible by 9 is also divisible by its factor, 3.

⇒

Note:

Divisibility by 5 depends on the last digit; rearranging might move the 0 or 5 from the unit place, so divisibility by 5, 15, or 45 is not guaranteed.

∴ The correct answer is 3.

Additional Information

Divisibility by 9 and 3

A number is divisible by 9 (or 3) if and only if the sum of its digits is divisible by 9 (or 3). This property is independent of the position of the digits.

Divisibility by 5

A number is divisible by 5 if its last digit is either 0 or 5. Rearranging digits can change the last digit, thus losing divisibility by 5.

Composite Divisibility

If a number is divisible by n, it is also divisible by every factor of n. Since 3 is a factor of 45, the number is always divisible by 3.

Shortcut Trick

Ratio of efficiencies of pipes Z : C : B = (3×2) : 2 : 1 = 6 : 2 :

1. Total efficiency = 6 + 2 + 1 = 9 units/hr.

Total Capacity of Tank = Total efficiency × Total time = 9 × 12 = 108 units.

Time taken by Pipe B alone = Total Capacity ÷ Efficiency of B = 108 ÷ 1 = 108 hours.

∴ The correct answer is 108 hr.

Alternate Method

Given:

Time taken by (Z + C + B) together = 12 hours

Efficiency of Pipe C = 2 × Efficiency of Pipe B

Efficiency of Pipe Z = 3 × Efficiency of Pipe C = 3 × (2 × Efficiency of B) = 6 × Efficiency of B

Calculations:

⇒ Total work (Tank capacity) = Combined efficiency × Time

⇒ Total work = 9 × 12 = 108 units

⇒ Time for Pipe B = Total work ÷ Efficiency of B

⇒ Time for Pipe B = 108 ÷ 1 = 108 hours

∴ The correct answer is 108 hr.

Additional Information

Efficiency and Time Relation

The efficiency of a pipe is inversely proportional to the time taken to fill the tank. If efficiency ratio is x:y, the time ratio is y:x.

Total Work Formula

Total Capacity = LCM of individual times or Efficiency × Time. In Pipes and Cisterns, filling pipes have positive efficiency and leakage/outlet pipes have negative efficiency.

Shortcut Trick

Total percentage = 100%

Remaining percentage for C and D = 100% − (40% + 50%) = 10%

Since votes are split equally, Candidate C percentage = 10% ÷ 2 = 5%

Votes for C = 5% of 20,000 = 1000

∴ The correct answer is 1000 votes.

Alternate Method

Given:

Total number of voters = 20,000

Votes for Candidate A = 40%

Votes for Candidate B = 50%

Calculations:

⇒ Total percentage of votes for A and B = 40% + 50% = 90%

⇒ Total percentage of votes for C and D = 100% − 90% = 10%

⇒ Since C and D split remaining votes equally, C gets = 10% ÷ 2 = 5%

⇒ Number of votes for Candidate C = (5 ÷ 100) × 20,000

⇒ Number of votes for Candidate C = 5 × 200 = 1000

∴ The correct answer is 1000 votes.

Additional Information

Percentage Base Change

When calculating percentages of a fixed total, the values are additive. Percentage = (Part ÷ Whole) × 100.

Ratio in Elections

If votes are split equally, the ratio between the candidates is 1:1, meaning each gets half of the combined remainder.

Valid vs Total Votes

In complex election problems, always distinguish between Total Voters, Votes Polled, and Valid Votes.

Shortcut Trick

Total Distance = 60 + 80 + 100 = 240 km

Total Time = (60/50) + (80/40) + (100/80) = 1.2 + 2 + 1.25 = 4.45 hours

Average Speed = Total Distance ÷ Total Time = 240 ÷ 4.45 ≈ 53.93 km/hr

∴ The correct answer is 53.93 km/hr.

Alternate Method

Given:

Distance₁ = 60 km, Speed₁ = 50 km/hr

Distance₂ = 80 km, Speed₂ = 40 km/hr

Distance₃ = 100 km, Speed₃ = 80 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Time₁ = 60 ÷ 50 = 1.2 hours

⇒ Time₂ = 80 ÷ 40 = 2 hours

⇒ Time₃ = 100 ÷ 80 = 1.25 hours

⇒ Total Time = 1.2 + 2 + 1.25 = 4.45 hours

⇒ Total Distance = 60 + 80 + 100 = 240 km

⇒ Average Speed = 240 ÷ 4.45 = 53.9325...

∴ The correct answer is 53.93 km/hr.

Additional Information

Average Speed (Equal Distances)

If a person covers two equal distances at speeds x and y, the Average Speed = 2xy ÷ (x + y).

Relative Speed

When two objects move in the same direction, relative speed is (u − v). In opposite directions, it is (u + v).

Conversion of Units

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