Quantitative Aptitude PYQs

Shortcut Trick

We can eliminate the options rapidly using basic divisibility rules.

Unit digit is 7 (odd) → Not divisible by 6.

Unit digit is 7 → Not divisible by 5.

11's rule: (8 + 2 + 2 + 4 + 7) − (6 + 0 + 6 + 1) = 23 − 13 = 10 → Not divisible by 11.

By elimination, the given number must be exactly divisible by 7.

∴ The correct answer is 7.

Alternate Method

Given number = 862026417

Checking Divisibility by 6:

⇒ For a number to be divisible by 6, it must be even (divisible by 2) and divisible by 3.

⇒ Since the last digit is 7 (odd), the number is not divisible by 2. Thus, it is not divisible by 6.

Checking Divisibility by 5:

⇒ A number is divisible by 5 if its last digit is either 0 or 5.

⇒ The last digit is 7, so it is not divisible by 5.

Checking Divisibility by 11:

⇒ Sum of digits at odd places (from right) = 7 + 4 + 2 + 2 + 8 = 23

⇒ Sum of digits at even places (from right) = 1 + 6 + 0 + 6 = 13

⇒ Difference = 23 − 13 = 10

⇒ Since 10 is not a multiple of 11, the number is not divisible by 11.

Checking Divisibility by 7:

⇒ Split the number into groups of 3 digits from right to left: 862, 026, 417

⇒ Add the alternate groups: 862 + 417 = 1279

⇒ Subtract the remaining group: 1279 − 026 = 1253

⇒ Divide the result by 7: 1253 ÷ 7 = 179

⇒ Since 1253 is perfectly divisible by 7, the entire original number is also divisible by 7.

∴ The correct answer is 7.

Additional Information

Divisibility Rule of 7, 11 and 13

Make groups of 3 digits from right to left and find the alternating sum or difference of these groups. If the final result is divisible by 7, 11, or 13, then the original number is divisible by 7, 11, or 13 respectively.

Divisibility Rule of 3 and 9

A number is divisible by 3 or 9 if the sum of all its digits is perfectly divisible by 3 or 9 respectively.

Divisibility Rule of 4 and 8

A number is divisible by 4 if its last two digits are divisible by 4. It is divisible by 8 if its last three digits are divisible by 8.

Shortcut Trick

The given expression is: 2/5 + [1 ÷ (1 + 5/7)] − 2/5

Notice that the +2/5 and −2/5 cancel each other out perfectly.

The expression simplifies directly to the middle term: 1 ÷ (1 + 5/7)

First, calculate the denominator: 1 + 5/7 = 12/7

The reciprocal of 12/7 is 7/12.

∴ The correct answer is 7/12.

Alternate Method

Given:

The expression: 2/5 + [1 ÷ (1 + 5/7)] − 2/5

Method Used:

Applying the BODMAS rule to solve the bracket first and simplify the fractions.

Calculations:

⇒ 2/5 + [1 ÷ ((7 + 5) ÷ 7)] − 2/5

⇒ 2/5 + [1 ÷ (12/7)] − 2/5

⇒ 2/5 + 7/12 − 2/5

Rearranging the expression to group similar terms:

⇒ (2/5 − 2/5) + 7/12

⇒ 0 + 7/12

⇒ 7/12

∴ The correct answer is 7/12.

Additional Information

BODMAS Rule

BODMAS determines the sequence of mathematical operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction. Expressions inside brackets are always evaluated first.

Fraction Reciprocal

When dividing 1 by a fraction a/b, the result is its reciprocal, which is b/a.

Additive Inverse

The sum of any number and its exact negative (additive inverse) is always zero. Identifying these pairs early helps in quick cancellations during simplifications.

Shortcut Trick

Let A = 30°, then p = cot 30° + cos 30° = √3 + √3/2 = 3√3/2 and q = cot 30° − cos 30° = √3 − √3/2 = √3/2.

Value of p² − q² = (3√3/2)² − (√3/2)² = 27/4 − 3/4 = 24/4 = 6.

Now evaluate the product pq = (3√3/2) × (√3/2) = 9/4.

From the given options, evaluating 4√pq = 4 × √(9/4) = 4 × 3/2 = 6, which perfectly matches our required value.

∴ The correct answer is 4√pq.

Alternate Method

Given:

p = cot A + cos A

q = cot A − cos A

Formula Used:

x² − y² = (x + y)(x − y)

Calculations:

⇒ p + q = (cot A + cos A) + (cot A − cos A) = 2 cot A

⇒ p − q = (cot A + cos A) − (cot A − cos A) = 2 cos A

⇒ p² − q² = (2 cot A) × (2 cos A) = 4 cot A cos A

⇒ pq = (cot A + cos A) × (cot A − cos A)

⇒ pq = cot² A − cos² A

⇒ pq = (cos² A / sin² A) − cos² A

⇒ pq = cos² A × (1 / sin² A − 1)

⇒ pq = cos² A × (cosec² A − 1)

⇒ pq = cos² A × cot² A

⇒ √pq = cos A cot A

⇒ p² − q² = 4(cos A cot A) = 4√pq

∴ The correct answer is 4√pq.

Additional Information

Trigonometric Identities

The primary Pythagorean identity is sin²θ + cos²θ = 1, which leads to 1 + cot²θ = cosec²θ and 1 + tan²θ = sec²θ.

Quotient Relations

The tangent and cotangent functions can be effectively converted using terms of sine and cosine as tanθ = sinθ / cosθ and cotθ = cosθ / sinθ.

Algebraic Difference of Squares

The foundational rule a² − b² = (a − b)(a + b) dramatically simplifies polynomial operations involving conjugate pairings.

Shortcut Trick

Use the successive discount multiplier method to find the final price directly.

Selling Price = MP × (100 − D₁)% × (100 − D₂)%

⇒ Final Price = 2000 × (90 ÷ 100) × (95 ÷ 100)

⇒ Final Price = 2000 × 0.9 × 0.95 = 1710

∴ The correct answer is ₹1,710.

Alternate Method

MP
₹2000

−10%

Price 1
₹1800

−5%

SP
₹1710

Given:

Marked Price (MP) = ₹2,000

First discount (D₁) = 10%

Second discount (D₂) = 5%

Formula Used:

Effective Discount % = D₁ + D₂ − (D₁ × D₂) ÷ 100

Selling Price (SP) = MP − Discount Amount

Calculations:

⇒ Effective Discount % = 10 + 5 − (10 × 5) ÷ 100

⇒ Effective Discount % = 15 − 0.5 = 14.5%

⇒ Total Discount Amount = 14.5% of 2000 = (14.5 ÷ 100) × 2000 = ₹290

⇒ Final Selling Price = 2000 − 290 = 1710

∴ The correct answer is ₹1,710.

Additional Information

Successive Discount Formula

For two successive discounts of x% and y%, the single equivalent discount is x + y − (x × y) ÷ 100.

Three Successive Discounts

Find the equivalent of the first two discounts (say d%), then apply the formula again between d% and the third discount z%.

Relationship between MP and CP

MP × (100 − Discount%) = CP × (100 + Profit%). This is a highly useful direct relation formula for shopkeeper profit/loss questions.

Shortcut Trick

Directly expand the brackets and mentally group the like terms to save time:

Total x² terms → 5x² + 7x² = 12x²

Total x terms → −9x

Constant terms → −14 + 17 = 3

Combine these terms directly to get the simplified expression.

∴ The correct answer is 12x² − 9x + 3.

Alternate Method

Given:

x(5x − 9) + 7(x² − 2) + 17

Calculations:

⇒ Apply the distributive property to open the brackets:

⇒ (x × 5x) − (x × 9) + (7 × x²) − (7 × 2) + 17

⇒ 5x² − 9x + 7x² − 14 + 17

⇒ Group the like terms together:

⇒ (5x² + 7x²) − 9x + (17 − 14)

⇒ 12x² − 9x + 3

∴ The correct answer is 12x² − 9x + 3.

Additional Information

Distributive Property

The rule a(b + c) = ab + ac is fundamental in algebra for removing brackets by distributing the multiplication over addition or subtraction.

Like Terms

Terms having the same variables raised to identical exponents.

Only like terms can be added or subtracted to simplify an algebraic expression.

BODMAS Rule

It defines the correct sequence to solve mathematical expressions: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

The volume of a cylinder is directly proportional to r²h.

Net multiplying factor for volume = (Multiplying factor of r)² × (Multiplying factor of h)

Net multiplying factor = 0.73 × 0.73 × 2.65 = 1.412185

Percentage increase = (1.412185 − 1) × 100 = 41.2185%

The closest integer to 41.2185% is 41%.

∴ The correct answer is 41%.

Alternate Method

Given:

Decrease in radius = 27%

Increase in height = 165%

Original Cylinder
r = 100, h = 100
V = 1,000,000π

→

New Cylinder
r = 73, h = 265
V = 1,412,185π

Formula Used:

Volume of a right circular cylinder = πr²h

Calculations:

Let the original radius be 100x and the original height be 100y.

⇒ Original Volume (V₁) = π × (100x)² × (100y) = 1000000πx²y

⇒ New radius (r') = 100x − 27% of 100x = 73x

⇒ New height (h') = 100y + 165% of 100y = 265y

⇒ New Volume (V₂) = π × (73x)² × (265y)

⇒ V₂ = π × (5329x²) × (265y) = 1412185πx²y

⇒ Increase in Volume = V₂ − V₁ = 1412185πx²y − 1000000πx²y = 412185πx²y

⇒ Percentage Increase = (Increase in Volume ÷ Original Volume) × 100

⇒ Percentage Increase = (412185πx²y ÷ 1000000πx²y) × 100 = 41.2185%

⇒ Closest integer = 41%

∴ The correct answer is 41%.

Additional Information

Percentage Change Formula

Percentage Change = [(Final Value − Initial Value) ÷ Initial Value] × 100. Multiplying factors help to quickly calculate successive changes.

Successive Percentage Change

If a value is changed successively by x% and y%, the net percentage change is given directly by the formula: x + y + (x × y) ÷ 100.

Volume of a Right Circular Cylinder

The volume (V = πr²h) means the overall volume is subjected to a two-dimensional change with respect to the radius and a one-dimensional change with respect to its height.

Shortcut Trick

Volume of a cylinder includes the terms π (contains factor 11) and r² (contains factor 25).

Since the height (21 cm) is completely divisible by 7, the final volume must be a direct multiple of both 11 and 25.

Checking the options for divisibility by 25 (last two digits) and 11 (difference of alternating sums):

Option 1: 1200 (not divisible by 11) | Option 3: 1500 (not divisible by 11) | Option 4: 1320 (not divisible by 25)

Only 1650 is exactly divisible by both 11 and 25.

∴ The correct answer is 1650 cm³.

Alternate Method

r = 5 cm

h = 21 cm

Volume = ?

Given:

Radius of cylinder (r) = 5 cm

Height of cylinder (h) = 21 cm

Formula Used:

Volume of a cylinder = π × r² × h

Calculation:

⇒ Volume = (22 ÷ 7) × (5)² × 21

⇒ Volume = (22 ÷ 7) × 25 × 21

⇒ Volume = 22 × 25 × (21 ÷ 7)

⇒ Volume = 22 × 25 × 3

⇒ Volume = 550 × 3

⇒ Volume = 1650

∴ The correct answer is 1650 cm³.

Additional Information

Curved Surface Area (CSA)

The area of the curved lateral surface of a cylinder. Formula: CSA = 2 × π × r × h.

Total Surface Area (TSA)

The total area including the curved surface and the two circular bases. Formula: TSA = 2 × π × r × (r + h).

Volume of a Cone

A cone with the same base radius and height has exactly one-third the volume of a cylinder. Formula: V = (1 ÷ 3) × π × r² × h.

Shortcut Trick

Total sum of 5 numbers = 655

Sum of the first two numbers = 80 × 2 = 160

Sum of remaining two numbers = 655 − (160 + 107) = 388

Average of remaining two numbers = 388 ÷ 2 = 194

∴ The correct answer is 194.

Alternate Method

Given:

⇒ Total sum of five numbers = 655

⇒ Average of the first two numbers = 80

⇒ Third number = 107

Formula Used:

⇒ Average = Sum of observations ÷ Number of observations

⇒ Sum = Average × Number of observations

Total Sum of 5 Numbers = 655

First 2
Sum = 160

3^rd No.
107

Last 2
Sum = 388

Average = 388 ÷ 2 = 194

Calculation:

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

⇒ Sum of the first three numbers = 160 + 107 = 267

⇒ Sum of the remaining two numbers = Total sum − Sum of first three numbers

⇒ Sum of the remaining two numbers = 655 − 267 = 388

⇒ Average of the remaining two numbers = 388 ÷ 2 = 194

∴ The correct answer is 194.

Additional Information

Combined Average

If group 1 has n₁ items with average A₁ and group 2 has n₂ items with average A₂, then the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Concept of Deviation

The sum of deviations of all observations from their arithmetic mean is always zero. This is a highly useful technique for calculating averages or missing values without heavy multiplication.

Weighted Average

When specific values hold greater importance or frequency, multiply each value by its weight, sum the results, and divide by the total weight: Σ(xw) ÷ Σw.

Shortcut Trick

Total Profit % = [(Claimed Profit % + Less Weight %) ÷ (100 − Less Weight %)] × 100

Total Profit % = [(29 + 30) ÷ (100 − 30)] × 100

Total Profit % = (59 ÷ 70) × 100 = 590 ÷ 7 ≈ 84.28%

Rounding off to the nearest whole number, we get 84%.

∴ The correct answer is 84%.

Alternate Method

Given:

Claimed profit = 29%

Weight used is 30% less than the actual weight.

Actual CP = ₹700

SP = ₹1290

• ₹590 Profit Total Profit % = (590 ÷ 700) × 100 = 84%

Calculations:

⇒ Let the true cost price of 1000 g of wheat be ₹1000.

⇒ Selling price of 1000 g (as claimed) = ₹1000 + 29% of 1000 = ₹1290.

⇒ Actual weight given by the trader = 1000 g − 30% of 1000 g = 700 g.

⇒ Actual cost price for the trader (for 700 g) = ₹700.

⇒ Profit earned = Selling Price − Actual Cost Price

⇒ Profit = 1290 − 700 = ₹590.

⇒ Total Profit % = (Profit ÷ Actual CP) × 100

⇒ Total Profit % = (590 ÷ 700) × 100 = 590 ÷ 7

⇒ Total Profit % ≈ 84.28%

⇒ Rounding off to the nearest whole number, we get 84%.

∴ The correct answer is 84%.

Additional Information

Dishonest Dealer (Selling at CP)

If a trader sells goods at cost price but uses a false weight, the profit is: Profit % = [Error ÷ (True Weight − Error)] × 100.

Successive Profit

When a dealer makes consecutive profits, the effective net percentage change is calculated as: Net % = a + b + (a × b) ÷ 100.

Marked Price & Discount Relation

The exact relationship between Cost Price, Marked Price, Profit, and Discount is structured as: MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %).

Shortcut Trick

When the distance is constant, speed is inversely proportional to time.

Time ratio = 7 : 14 = 1 : 2

Speed ratio = Inverse of time ratio = 2 : 1

Given speed (2 ratio parts) = 86 km/hr

Required speed (1 ratio part) = 86 ÷ 2 = 43 km/hr

∴ The correct answer is 43 km/hr.

Alternate Method

A
B
Total Distance (D) = Constant

Case 1: S = 86 km/hr
T = 7 hours

Case 2: S = ? km/hr
T = 14 hours

Given:

Initial Speed (S₁) = 86 km/hr

Initial Time (T₁) = 7 hours

New Time (T₂) = 14 hours

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Total Distance = S₁ × T₁

⇒ Distance = 86 × 7

⇒ Let the new speed be S₂ to cover the same distance in 14 hours.

⇒ Distance = S₂ × T₂

⇒ 86 × 7 = S₂ × 14

⇒ S₂ = (86 × 7) ÷ 14

⇒ S₂ = 86 ÷ 2

⇒ S₂ = 43 km/hr

∴ The correct answer is 43 km/hr.

Additional Information

Unit Conversions

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

Average Speed

Average Speed = Total Distance ÷ Total Time. If equal distances are covered at speeds a and b, Average Speed = 2ab ÷ (a + b).

Relative Speed

When moving in the same direction, relative speed = S₁ − S₂. When moving in opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Effective simple interest rate for 6 months = 3% × 6 = 18%.

Simple Interest = 18% of ₹1200.

⇒ SI = (18 ÷ 100) × 1200 = 216

∴ The correct answer is ₹216.

Alternate Method

Given:

Principal (P) = ₹1200

Rate of Interest (R) = 3% per month

Time (T) = 6 months

Formula Used:

Simple Interest = (P × R × T) ÷ 100

*(

Note:

Since the rate is given per month and time is in months, both units are matching and no conversion to years is needed.)*

₹1200
Principal

3% / month
6 months

₹216
Simple Interest

Total Interest = 18% of ₹1200

Calculation:

⇒ SI = (1200 × 3 × 6) ÷ 100

⇒ SI = 12 × 18

⇒ SI = 216

∴ The correct answer is ₹216.

Additional Information

Unit Consistency in Simple Interest

Always ensure that the Time (T) unit strictly matches the Rate (R) period. If Rate is monthly, Time must be in months. If Rate is per annum, Time must be in years.

Total Amount Formula

Amount (A) = Principal (P) + Simple Interest (SI). This indicates the total sum that must be repaid after the time duration.

Simple vs Compound Interest

Simple Interest is charged only on the original principal sum for the entire tenure, whereas Compound Interest adds previously earned interest to the principal before calculating the next period's interest.

Shortcut Trick

A 60% increase means the new value is 160% of the original number.

160% = 3590

1% = 3590 ÷ 160

100% (Original Number) = (3590 ÷ 160) × 100 = 2243.75

∴ The correct answer is 2243.75.

Alternate Method

Given:

Final number after increase = 3590

Percentage increase = 60%

Formula Used:

Original Number = (Final Value × 100) ÷ (100 + Increase %)

100% (Original Number)

+60% Increase

Total 160% = 3590
100% = (3590 ÷ 160) × 100 = 2243.75

Calculations:

⇒ Let the original number be x.

⇒ x + (60% of x) = 3590

⇒ x × (1 + 0.60) = 3590

⇒ 1.6x = 3590

⇒ x = 3590 ÷ 1.6

⇒ x = 35900 ÷ 16

⇒ x = 2243.75

∴ The correct answer is 2243.75.

Additional Information

Percentage Decrease

If a number is decreased by R%, the new value becomes Original × (100 − R) ÷ 100.

Successive Percentage Change

When a number is changed by A% and then by B%, the net percentage change is given by A + B + (A × B) ÷ 100.

Base Change Rule

If value X is R% more than Y, then Y is less than X by (R ÷ (100 + R)) × 100%.

Shortcut Trick

Sum of parallel sides = (Area × 2) ÷ Height = (648 × 2) ÷ 24 = 54 cm.

Given difference of parallel sides = 6 cm.

Larger side = (Sum + Difference) ÷ 2 = (54 + 6) ÷ 2 = 30 cm.

Smaller side = (Sum − Difference) ÷ 2 = (54 − 6) ÷ 2 = 24 cm.

∴ The correct answer is 30 cm, 24 cm.

Alternate Method

Given:

Area of trapezium = 648 cm²

Perpendicular distance (h) = 24 cm

Difference between parallel sides (a − b) = 6 cm

Side b
Side a
24 cm
Area = 648 cm²
a - b = 6 cm

Formula Used:

Area of Trapezium = ½ × (Sum of parallel sides) × Height

Calculation:

⇒ 648 = ½ × (a + b) × 24

⇒ 648 = 12 × (a + b)

⇒ a + b = 648 ÷ 12

⇒ a + b = 54 ...... (Equation 1)

⇒ a − b = 6 ...... (Equation 2)

Adding Equation 1 and Equation 2:

⇒ 2a = 60

⇒ a = 30 cm

Substituting the value of a in Equation 1:

⇒ 30 + b = 54

⇒ b = 54 − 30

⇒ b = 24 cm

∴ The correct answer is 30 cm, 24 cm.

Additional Information

Area of a Trapezium

The area is calculated as half the product of the sum of parallel sides and the perpendicular distance between them: A = ½ × (a + b) × h.

Perimeter of a Trapezium

The perimeter is the total length of the boundary, obtained by adding the lengths of all four sides: P = a + b + c + d.

Isosceles Trapezium

A special type of trapezium in which the non-parallel sides are equal in length, making its base angles equal and its diagonals equal.

Median of a Trapezium

The line segment joining the midpoints of the non-parallel sides. Its length is exactly the average of the two parallel sides: M = (a + b) ÷ 2.

Shortcut Trick

The average of the first 'n' even natural numbers is directly given by the formula (n + 1).

Given the number of terms n = 172.

⇒ Average = 172 + 1 = 173

∴ The correct answer is 173.

Alternate Method

Given:

Sequence of the first 172 even numbers: 2, 4, 6, ..., 344

First term (a) = 2

Last term (l) = 172 × 2 = 344

Formula Used:

For an Arithmetic Progression (A.P.), Average = (First term + Last term) ÷ 2

2
4
6
344

1^st term
2^nd term
3^rd term
172^nd term

n = 172 terms

Average = (First + Last) ÷ 2

Calculations:

⇒ Average = (2 + 344) ÷ 2

⇒ Average = 346 ÷ 2

⇒ Average = 173

∴ The correct answer is 173.

Additional Information

Average of First 'n' Natural Numbers

The average of the first n natural numbers is evaluated using the formula (n + 1) ÷ 2.

Average of First 'n' Odd Numbers

The average of the first n odd natural numbers is exactly equal to n.

Average of Numbers in A.P.

If a series is in an Arithmetic Progression, its average is precisely equal to (First term + Last term) ÷ 2, which matches the middle term of the series.

Sum of First 'n' Even Numbers

The sum of the first n even natural numbers can be directly calculated using the formula n × (n + 1).

Shortcut Trick

CP of 1 ruler = 180 ÷ 12 = ₹15

Ratio of CP : SP = 15 : 24 = 5 : 8

Profit is 3 parts on a base CP of 5 parts.

Profit % = (3 ÷ 5) × 100 = 60%

∴ The correct answer is 60%.

Alternate Method

Given:

Cost price of 1 dozen (12) rulers = ₹180

Selling price of 1 ruler = ₹24

Formula Used:

Profit = SP − CP

Profit % = (Profit ÷ CP) × 100

Profit Calculation

CP = ₹15

Profit = ₹9

Selling Price (SP) = ₹24

Profit % = 60%
Result: Profit % = (9 ÷ 15) × 100 = 60%

Calculations:

⇒ Cost price of 1 ruler (CP) = 180 ÷ 12 = ₹15

⇒ Profit earned on 1 ruler = 24 − 15 = ₹9

⇒ Profit % = (9 ÷ 15) × 100

⇒ Profit % = (3 ÷ 5) × 100 = 60%

∴ The correct answer is 60%.

Additional Information

Loss Percentage

When Selling Price is less than Cost Price, Loss occurs. Loss % = (Loss ÷ CP) × 100.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

If two successive discounts of a% and b% are given, equivalent discount = a + b − (a × b ÷ 100)%.

Shortcut Trick

Total Work = LCM(24, 20, 12) = 120 units.

Efficiency of Isha = 120 ÷ 24 = 5, and Isha + Smriti = 120 ÷ 20 = 6.

Efficiency of Isha + Smriti + Ashlesha = 120 ÷ 12 = 10.

Ashlesha's efficiency = 10 − 6 = 4 units/day.

Time taken by Isha + Ashlesha = 120 ÷ (5 + 4) = 40/3 days.

∴ The correct answer is 40/3 days.

Alternate Method

Given:

Time taken by Isha = 24 days

Time taken by Isha and Smriti = 20 days

Time taken by Isha, Smriti and Ashlesha = 12 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Efficiency of Isha = 5 units/day

⇒ Efficiency of Isha and Smriti = 6 units/day

⇒ Efficiency of Isha, Smriti and Ashlesha = 10 units/day

⇒ Efficiency of Ashlesha = 10 − 6 = 4 units/day

⇒ Combined efficiency of Isha and Ashlesha = 5 + 4 = 9 units/day

⇒ Time taken by Isha and Ashlesha = 120 ÷ 9 = 40/3 days

∴ The correct answer is 40/3 days.

Additional Information

Pipes & Cisterns

Similar to Time and Work, but an emptying pipe has negative efficiency. Net efficiency = Sum of Filling rates − Sum of Emptying rates.

Work & Wages

Wages are distributed in the ratio of the total work done by each person. If everyone works for the same number of days, wages are directly proportional to their efficiencies.

Alternate Day Work

When people work on alternate days, find the total work done in one complete cycle and multiply it to approach the Total Work, then calculate the remaining work for the next person in sequence.

Shortcut Trick

Total Registered Voters = Margin ÷ (Polled% × Valid% × Margin% of Valid)

Margin% = 51% − (100% − 51%) = 2%

Valid% = 100% − 21% = 79%

Total Voters = 1817 ÷ (80% × 79% × 2%)

Total Voters = 1817 ÷ 0.01264 = 143750

∴ The correct answer is 143750 voters.

Alternate Method

Given:

Registered voters who cast votes = 80%

Invalid votes = 21%

Winning candidate's share = 51% of valid votes

Winning margin = 1817 votes

Registered Voters

80% Polled

Polled Votes

79% Valid

Valid Votes

2% Margin

Margin = 1817 Votes

Calculations:

⇒ Let the total registered voters be 100 units.

⇒ Votes polled = 80% of 100 = 80 units

⇒ Valid votes = (100 − 21)% of 80 units = 79% of 80 = 63.2 units

⇒ Winner's votes = 51% of 63.2 units

⇒ Loser's votes = (100 − 51)% = 49% of 63.2 units

⇒ Winning margin = (51 − 49)% of 63.2 units = 2% of 63.2 units

⇒ Margin = 0.02 × 63.2 = 1.264 units

⇒ 1.264 units = 1817

⇒ 1 unit = 1817 ÷ 1.264 = 1437.5

⇒ Total registered voters (100 units) = 1437.5 × 100 = 143750

∴ The correct answer is 143750 voters.

Additional Information

Election Percentage Concept

In election-based problems, always clearly distinguish between "Total Registered Voters", "Polled Votes", and "Valid Votes". Calculations are typically a chain of percentages applied consecutively on the reduced base.

Winning Margin Rule

The margin of victory is the difference in percentage between the winner and the loser. If a winner gets x% of valid votes, the loser gets (100 − x)%, and the direct margin equals (2x − 100)% of the valid votes.

Successive Percentage Trick

When multiple fractional drops are applied one after another, the final value can be directly found using continuous multiplying factors: Final = Initial × (a ÷ 100) × (b ÷ 100) × (c ÷ 100).

Shortcut Trick

Given 8A = 3B = 6C, the ratio A : B : C is derived from the inverses: (1/8) : (1/3) : (1/6).

Multiply each term by the LCM of (8, 3, 6) = 24 → A : B : C = 3 : 8 : 4.

Total ratio parts = 3 + 8 + 4 = 15 parts, which corresponds to the total sum of ₹42,100.

A's share is 3 parts out of 15, which means exactly 1/5^th of the total amount.

Share of A = 42,100 ÷ 5 = ₹8,420.

∴ The correct answer is ₹8,420.

Alternate Method

Given:

Total amount to be divided = ₹42,100

8 × (Share of A) = 3 × (Share of B) = 6 × (Share of C)

A (3)
B (8)
C (4)

Total = 15 parts = ₹42,100

A's Share = ?

Calculations:

⇒ Let 8A = 3B = 6C = k (where k is a constant)

⇒ A = k/8, B = k/3, C = k/6

⇒ Ratio of their shares A : B : C = k/8 : k/3 : k/6 = 1/8 : 1/3 : 1/6

⇒ To simplify the ratio, multiply by the LCM of the denominators (8, 3, 6) which is 24.

⇒ A : B : C = (1/8 × 24) : (1/3 × 24) : (1/6 × 24)

⇒ A : B : C = 3 : 8 : 4

⇒ Total sum of ratio parts = 3 + 8 + 4 = 15 parts

⇒ Value of 15 parts = ₹42,100

⇒ Share of A = (Ratio parts of A ÷ Total parts) × Total Amount

⇒ Share of A = (3 ÷ 15) × 42,100

⇒ Share of A = (1 ÷ 5) × 42,100

⇒ Share of A = 8,420

∴ The correct answer is ₹8,420.

Additional Information

Inverse Ratio Concept

If pA = qB = rC, then the direct ratio A : B : C is (1/p) : (1/q) : (1/r). It can also be found quickly by hiding the coefficient of one variable and multiplying the remaining coefficients.

Proportional Division

When a given total amount is divided in a specific ratio a : b : c, the individual share of the first entity is always equal to Total Amount × [a ÷ (a + b + c)].

Shortcut Trick

For two given numbers a and b, the third proportional x is calculated using the direct formula: x = b² ÷ a.

Here, the given numbers are a = 4 and b = 16.

x = (16 × 16) ÷ 4

x = 256 ÷ 4 = 64

∴ The correct answer is 64.

Alternate Method

Given:

The first number (a) is 4.

The second number (b) is 16.

The third proportional is x.

4
1^st Term

:

16
2^nd Term

::

16
2^nd Term

:

x
3^rd Term

Proportional Relationship: a : b :: b : x

Formula Used:

If three terms a, b, and x are in continued proportion, then a : b :: b :

x. ⇒ a ÷ b = b ÷ x

Calculations:

⇒ 4 ÷ 16 = 16 ÷ x

⇒ 1 ÷ 4 = 16 ÷ x

⇒ x = 16 × 4

⇒ x = 64

∴ The correct answer is 64.

Additional Information

Mean Proportional

The mean proportional (or geometric mean) between two numbers a and b is x, such that a : x :: x : b. The formula is x = √(a × b).

Fourth Proportional

If x is the fourth proportional to three distinct numbers a, b, and c, then a : b :: c :

x. The formula is x = (b × c) ÷ a.

First Proportional

If x is the first proportional to a and b, then x : a :: a : b. The required value is computed using the formula x = a² ÷ b.

Duplicate Ratio

The duplicate ratio of a : b is given by a² : b², while its sub-duplicate ratio is given by √a : √b.

Shortcut Trick

Total Distance = 774 + 567 + 135 = 1476 km

Total Time = (774 ÷ 86) + (567 ÷ 63) + (135 ÷ 27) = 9 + 9 + 5 = 23 hours

Average Speed = Total Distance ÷ Total Time = 1476 ÷ 23 ≈ 64.1739 km/hr

∴ The correct answer is 64.17 km/hr.

Alternate Method

Given:

Distance 1 (D₁) = 774 km, Speed 1 (S₁) = 86 km/hr

Distance 2 (D₂) = 567 km, Speed 2 (S₂) = 63 km/hr

Distance 3 (D₃) = 135 km, Speed 3 (S₃) = 27 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = Total Distance ÷ Total Time

774 km
86 km/hr
9 hr

567 km
63 km/hr
9 hr

135 km
27 km/hr
5 hr

Total Distance = 1476 km
Total Time = 23 hr

Calculations:

⇒ Total Distance = 774 + 567 + 135 = 1476 km

⇒ Total Time = (774 ÷ 86) + (567 ÷ 63) + (135 ÷ 27)

⇒ Total Time = 9 + 9 + 5 = 23 hours

⇒ Average Speed = 1476 ÷ 23

⇒ Average Speed ≈ 64.1739 km/hr

⇒ Rounding off to two decimal places, we get 64.17 km/hr.

∴ The correct answer is 64.17 km/hr.

Additional Information

Average Speed (Different Distances)

When a journey is broken into multiple parts with varying distances and speeds, the average speed is not the arithmetic mean of the speeds. It is always determined using: Average Speed = Total Distance ÷ Total Time.

Average Speed (Equal Distances)

If two strictly equal distances are covered at speeds x and y respectively, the average speed for the whole journey simplifies directly to the formula: 2xy ÷ (x + y).

Relative Speed Concept

When two moving bodies travel in the same direction, their effective relative speed is the difference between their speeds |x − y|. When they travel towards each other in opposite directions, their relative speed is the sum of their speeds (x + y).