Quantitative Aptitude PYQs

Shortcut Trick

A number is completely divisible by 3 if the sum of its digits is a multiple of 3.

Sum of digits of 253118139 = 2 + 5 + 3 + 1 + 1 + 8 + 1 + 3 + 9 = 33.

Since 33 is exactly divisible by 3, the entire number is divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

The given number is 253118139.

Method Used:

We check the divisibility of the given number by testing each option using basic divisibility rules.

Calculations:

⇒ Divisibility by 12: A number must be divisible by both 3 and 4. For 4, the last two digits must be divisible by 4. Here, the last two digits form 39, which is not divisible by 4.

⇒ Divisibility by 5: A number is divisible by 5 only if its unit digit is 0 or 5. Here, the unit digit is 9, so it is not divisible by 5.

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

⇒ Sum of digits = 2 + 5 + 3 + 1 + 1 + 8 + 1 + 3 + 9 = 33.

⇒ 33 ÷ 3 = 11. Since the sum is perfectly divisible by 3, the number is divisible by 3.

⇒ Divisibility by 8: A number is divisible by 8 if its last three digits form a number divisible by 8. Here, 139 is an odd number and cannot be divided by an even number like 8.

∴ The correct answer is 3.

Additional Information

Divisibility Rule of 4 and 8

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

Divisibility Rule of 9

Similar to the rule of 3, a number is exactly divisible by 9 if the sum of all its digits is a multiple of 9.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.

Shortcut Trick

Let the cost price for A be 100 units.

Selling price to B at 60% loss = 100 − 60 = 40 units.

Selling price to C at 25% profit = 40 + (25% of 40) = 50 units.

Given that C purchased it for ₹47,236, we get 50 units = 47,236.

Cost price for A (100 units) = 47,236 × (100 ÷ 50) = 47,236 × 2 = 94,472.

∴ The correct answer is ₹94,472.

Alternate Method

Given:

Loss percentage for A = 60%

Profit percentage for B = 25%

Cost price for C = ₹47,236

CP of A

−60%

CP of B

+25%

CP of C

?
₹47,236

Net Multiplier = 0.4 × 1.25 = 0.5

Formula Used:

CP of A × (100 − L%) ÷ 100 × (100 + P%) ÷ 100 = CP of C

Calculations:

⇒ CP of A × (100 − 60) ÷ 100 × (100 + 25) ÷ 100 = 47236

⇒ CP of A × 40 ÷ 100 × 125 ÷ 100 = 47236

⇒ CP of A × 2 ÷ 5 × 5 ÷ 4 = 47236

⇒ CP of A × 1 ÷ 2 = 47236

⇒ CP of A = 47236 × 2

⇒ CP of A = 94472

∴ The correct answer is ₹94,472.

Additional Information

Successive Profit/Loss

When an item passes through multiple hands, the final price is calculated by multiplying the successive fractional factors: Final CP = Initial CP × (1 ± P₁/100) × (1 ± P₂/100).

Effective Percentage Change

The net percentage change for two successive changes x% and y% is given by the formula: Net % Change = x + y + (x × y) ÷ 100. (Use negative values for loss).

Fraction Equivalents

Converting percentages to fractions speeds up calculations. A loss of 60% leaves 2/5 of the value, and a profit of 25% makes the new value 5/4 of the previous value.

Shortcut Trick

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

Here, the number of terms n = 170.

⇒ Average = 170 + 1 = 171

∴ The correct answer is 171.

Alternate Method

Given:

Number of terms (n) = 170

The sequence of the first 170 even numbers is: 2, 4, 6, ..., 340

Formula Used:

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

2
4
6
. . . . . . .
340
Average = (2 + 340) ÷ 2 = 171
First 170 Even Numbers

Calculations:

⇒ First term (a) = 2

⇒ Last term (l) = 170 × 2 = 340

⇒ Average = (2 + 340) ÷ 2

⇒ Average = 342 ÷ 2

⇒ Average = 171

∴ The correct answer is 171.

Additional Information

Average of first n natural numbers

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

Average of first n odd numbers

The average of the first n consecutive odd numbers starting from 1 is exactly equal to n.

Average of an Arithmetic Progression

For any given sequence of numbers that maintains a constant difference between consecutive terms, the average is strictly derived by (First Term + Last Term) ÷ 2.

Shortcut Trick

Using the direct cover-up method for 4A = 9B = 3C:

Ratio A : B : C = (9 × 3) : (4 × 3) : (4 × 9) = 27 : 12 : 36 ⇒ 9 : 4 : 12

Total ratio parts = 9 + 4 + 12 = 25 parts.

A's Share = (9 ÷ 25) × 58300 = 9 × 2332 = 20988.

∴ The correct answer is ₹20,988.

Alternate Method

Given:

Total amount = ₹58,300

Relation between shares: 4A = 9B = 3C

Total = 25 parts = ₹58,300

A (9 parts)

B (4)

C (12 parts)

A's Share = ?

Calculations:

⇒ Let 4A = 9B = 3C = k

⇒ A = k/4, B = k/9, C = k/3

⇒ Ratio of shares A : B : C = 1/4 : 1/9 : 1/3

⇒ LCM of denominators (4, 9, 3) is 36.

⇒ Multiply the ratio by 36 to remove fractions:

⇒ A : B : C = (36/4) : (36/9) : (36/3) = 9 : 4 : 12

⇒ Total ratio parts = 9 + 4 + 12 = 25 parts

⇒ 25 parts = 58,300

⇒ 1 part = 58,300 ÷ 25 = 2,332

⇒ Share of A = 9 parts = 9 × 2,332 = 20,988

∴ The correct answer is ₹20,988.

Additional Information

Equality of Multiple Terms (Cover-up Method)

If aA = bB = cC, the ratio A : B : C can be found directly by multiplying the coefficients of the other terms: A = b × c, B = a × c, C = a × b.

Dividing in a Given Ratio

To divide a total amount T in the ratio a : b : c, the individual share is calculated as: Share = (Ratio part ÷ Sum of all ratio parts) × T.

Inverse Ratio

The inverse ratio of a : b is b : a. For three quantities a : b : c, the inverse ratio is (1/a) : (1/b) : (1/c), which simplifies to bc : ac : ab.

Shortcut Trick

If the discount is 10%, the selling price becomes 90% of the Marked Price.

Selling Price = 90% of 900

Selling Price = (90 ÷ 100) × 900

Selling Price = 810

∴ The correct answer is ₹810.

Alternate Method

Given:

Marked Price (MP) = ₹900

Discount = 10%

Formula Used:

Selling Price (SP) = MP − Discount

Discount = (Discount % ÷ 100) × MP

Marked Price
₹900

Selling Price
₹810

− 10% Discount

Calculations:

⇒ Discount amount = 10% of 900

⇒ Discount amount = (10 ÷ 100) × 900

⇒ Discount amount = ₹90

⇒ Selling Price = 900 − 90

⇒ Selling Price = ₹810

∴ The correct answer is ₹810.

Additional Information

Successive Discounts

When two discounts of a% and b% are given, the single equivalent discount is given by the formula: (a + b − ab ÷ 100)%.

Direct Relation Between MP and CP

Ratio of Marked Price to Cost Price can be directly found using: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Base Values for Profit and Discount

Profit or Loss is always calculated over the Cost Price (CP), whereas the Discount is always calculated over the Marked Price (MP).

Shortcut Trick

Using the basic rules of exponents, we can solve this simplification in a single step.

When the base is the same, subtract exponents for division and add for multiplication.

⇒ 50⁻¹³ ÷ 50¹⁵ × 50⁻¹¹

⇒ 50^{−13 − 15 + (−11)} = 50⁻³⁹

∴ The correct answer is 50⁻³⁹.

Alternate Method

Given:

50⁻¹³ ÷ 50¹⁵ × 50⁻¹¹

Formula Used:

a^m ÷ aⁿ = a^{m − n}

a^m × aⁿ = a^{m + n}

Calculation:

According to the BODMAS rule, division and multiplication have the same priority. We evaluate them from left to right.

⇒ (50⁻¹³ ÷ 50¹⁵) × 50⁻¹¹

⇒ 50^{−13 − 15} × 50⁻¹¹

⇒ 50⁻²⁸ × 50⁻¹¹

⇒ 50^{−28 + (−11)}

⇒ 50⁻³⁹

∴ The correct answer is 50⁻³⁹.

Additional Information

Product Rule of Exponents

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

Quotient Rule of Exponents

When dividing two terms with the same base, you subtract the exponent of the divisor from the exponent of the dividend: a^m ÷ aⁿ = a^{m − n}.

Power of a Power Rule

To raise a power to another power, multiply the exponents: (a^m)ⁿ = a^{m × n}.

Negative Exponent Rule

A negative exponent indicates the reciprocal of the base raised to the positive exponent: a^−m = 1 ÷ a^m.

Shortcut Trick

Let the original number be 100%.

When increased by 70%, the new value becomes 100% + 70% = 170%.

We are given that 170% corresponds to 3910.

1% = 3910 ÷ 170 = 23

Original number (100%) = 23 × 100 = 2300

∴ The correct answer is 2300.

Alternate Method

Given:

A number when increased by 70% results in 3910.

Formula Used:

Final Value = Initial Value × (1 + Percentage Increase ÷ 100)

Original = 100%

• 70% Increase

x = ?

Total = 170% = 3910

100% = 2300

Calculations:

⇒ Let the number be x.

⇒ x + (70% of x) = 3910

⇒ x × (1 + 70 ÷ 100) = 3910

⇒ x × 1.70 = 3910

⇒ x = 3910 ÷ 1.70

⇒ x = 39100 ÷ 17

⇒ x = 2300

∴ The correct answer is 2300.

Additional Information

Percentage Increase or Decrease

Percentage change is calculated as: (Difference ÷ Original Value) × 100.

Successive Percentage Change

When a number undergoes two successive changes of A% and B%, the net change is calculated using the formula: A + B + (A × B) ÷ 100.

Base Change Rule

If quantity A is R% more than quantity B, then B is [R ÷ (100 + R)] × 100% less than A.

Shortcut Trick

Simplify both given statements into linear equations for Mother (M) and Daughter (D):

First

Statement:

2M = 4D + 26 ⇒ M − 2D = 13

Second

Statement:

M − 3D = 2

Subtracting the second equation from the first directly gives D = 11.

Substituting D = 11 into the second equation gives M = 33 + 2 = 35.

The difference in their ages is 35 − 11 = 24.

∴ The correct answer is 24 years.

Alternate Method

Given:

Let the Mother's age be M and the Daughter's age be D.

Condition 1: 2 × M = 4 × D + 26

Condition 2: 3 × D = M − 2

Mother (M)Daughter (D)Eq 1: M - 2D = 13Eq 2: M - 3D = 2

Calculations:

⇒ From Condition 1, divide the entire equation by 2 to simplify:

⇒ M = 2D + 13

⇒ Substitute the value of M into Condition 2:

⇒ 3D = (2D + 13) − 2

⇒ 3D − 2D = 11

⇒ D = 11 years

⇒ Now, substitute D back into M = 2D + 13:

⇒ M = 2 × 11 + 13

⇒ M = 22 + 13 = 35 years

⇒ Required difference in their ages = M − D

⇒ 35 − 11 = 24 years

∴ The correct answer is 24 years.

Additional Information

Ages and Linear Equations

Age problems are translated into linear equations by replacing phrases like 'times', 'more than', or 'less than' with multiplication, addition, and subtraction respectively.

Constant Age Difference

The absolute difference in age between any two individuals always remains constant across time. If person A is 24 years older than person B today, the age gap will always be 24 years regardless of future or past shifts.

Solving Simultaneous Equations

Linear equations with two unknown variables can be effectively resolved using the substitution method (expressing one variable in terms of the other) or the elimination method (adding/subtracting aligned equations to cancel out a variable).

Shortcut Trick

In ΔABC, ∠BCA = 180° − (93° + 18°) = 69°.

Due to BC ∥ AD, alternate interior angle ∠CAD = ∠BCA = 69°.

In isosceles ΔACD (AC = CD), base angles are equal: ∠ADC = ∠CAD = 69°.

Thus, ∠ACD = 180° − (69° + 69°) = 42°.

∴ The correct answer is 42°.

Alternate Method

Given:

Trapezium ABCD with BC ∥ AD

AC = CD

∠ABC = 18°, ∠BAC = 93°

Formula Used:

Sum of angles in a triangle = 180°

Alternate interior angles for parallel lines are equal.

Calculations:

⇒ In ΔABC, the sum of angles is 180°.

⇒ ∠BCA = 180° − (∠ABC + ∠BAC)

⇒ ∠BCA = 180° − (18° + 93°) = 180° − 111° = 69°

⇒ Since BC ∥ AD and AC is a transversal line, the alternate interior angles are equal.

⇒ ∠CAD = ∠BCA = 69°

⇒ In ΔACD, it is given that AC = CD. Thus, the angles opposite to these sides are equal.

⇒ ∠ADC = ∠CAD = 69°

⇒ Taking ΔACD, the sum of angles is 180°.

⇒ ∠ACD = 180° − (∠CAD + ∠ADC)

⇒ ∠ACD = 180° − (69° + 69°) = 180° − 138° = 42°

∴ The correct answer is 42°.

Additional Information

Isosceles Triangle Properties

In an isosceles triangle, the angles opposite to the equal sides are always equal. This is known as the base angles theorem.

Parallel Lines & Transversals

When two parallel lines are intersected by a transversal, the alternate interior angles are equal, and the consecutive interior angles sum to 180°.

Angle Sum Property

The sum of all interior angles of a triangle is exactly 180°, regardless of the side lengths or triangle type.

Shortcut Trick

Substitute given values directly: (4^0.72)^z = (4^0.86)⁸.

Equating powers of 4: 0.72 × z = 0.86 × 8.

⇒ 0.72z = 6.88.

⇒ z = 6.88 ÷ 0.72 = 688 ÷ 72 ≈ 9.555...

∴ The correct answer is 9.56.

Alternate Method

Given:

x = 4^0.72

y = 4^0.86

x^z = y⁸

Formula Used:

(a^m)ⁿ = a^mn

If a^m = aⁿ, then m = n.

Calculations:

⇒ Substitute the values of x and y in the equation x^z = y⁸.

⇒ (4^0.72)^z = (4^0.86)⁸

⇒ 4^{0.72 × z} = 4^{0.86 × 8}

⇒ 4^0.72z = 4^6.88

Since the bases are equal, we can equate the exponents:

⇒ 0.72z = 6.88

⇒ z = 6.88 ÷ 0.72

⇒ z = 688 ÷ 72

⇒ z = 86 ÷ 9

⇒ z = 9.555...

Rounding off to two decimal places, we get z ≈ 9.56.

∴ The correct answer is 9.56.

Additional Information

Laws of Exponents (Power of a Power)

When raising a power to another power, multiply the exponents: (a^m)ⁿ = a^mn.

Equality of Exponents

If the bases on both sides of an equation are equal and strictly greater than 1, their exponents must also be equal: a^x = a^y ⇒ x = y.

Laws of Exponents (Multiplication & Division)

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

Shortcut Trick

Let the assumed average be 50 kg.

Net deviation = (40 − 50) + (49 − 50) + (56 − 50) + (75 − 50) + (39 − 50)

Net deviation = −10 − 1 + 6 + 25 − 11 = +9

Average = Assumed average + (Net deviation ÷ Count)

Average = 50 + (9 ÷ 5) = 50 + 1.8 = 51.8 kg

∴ The correct answer is 51.8 kg.

Alternate Method

Given:

Weights of the family members = 40 kg, 49 kg, 56 kg, 75 kg and 39 kg

Number of members (n) = 5

Formula Used:

Average = Sum of all observations ÷ Total number of observations

40 kg

49 kg

56 kg

75 kg

39 kg

Sum = 259 kg

Average = 51.8 kg

Calculations:

⇒ Total sum of weights = 40 + 49 + 56 + 75 + 39

⇒ Total sum of weights = 259 kg

⇒ Average weight = 259 ÷ 5

⇒ Average weight = 51.8 kg

∴ The correct answer is 51.8 kg.

Additional Information

Combined Average

If two groups have sizes n₁ and n₂ with averages A₁ and A₂ respectively, the combined average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Deviation Method

A highly efficient exam technique where an assumed average is chosen arbitrarily. The true average is then calculated as: True Average = Assumed Average + (Sum of Deviations ÷ Total Count). This method minimizes large calculations and saves time.

Impact of Addition or Subtraction

If every observation in a dataset is uniformly increased or decreased by a constant value 'k', the new average of the dataset also strictly increases or decreases by the exact same constant 'k'.

Shortcut Trick

Convert the percentage discounts into multiplying fractions: 10% = 1/10 and 5% = 1/20.

Multiplying factors after discount become: (1 − 1/10) = 9/10 and (1 − 1/20) = 19/20.

Selling Price (SP) = MP × 9/10 × 19/20

Selling Price = 1200 × 171/200 = 6 × 171 = 1026

∴ The correct answer is ₹1,026.

Alternate Method

Given:

Marked Price (MP) of the bottle = ₹1,200

First discount (D₁) = 10%

Second successive discount (D₂) = 5%

Formula Used:

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

−10%
−5%

Marked Price
₹1200

1^st SP
₹1080

Final SP
₹1026

Successive Discount Flow

Calculations:

⇒ SP = 1200 × [(100 − 10) / 100] × [(100 − 5) / 100]

⇒ SP = 1200 × (90 / 100) × (95 / 100)

⇒ SP = 1200 × 0.90 × 0.95

⇒ SP = 1080 × 0.95

⇒ SP = 1026

∴ The correct answer is ₹1,026.

Additional Information

Single Equivalent Discount

When two successive discounts of a% and b% are offered, their single equivalent discount can be calculated directly using the formula: Equivalent Discount = (a + b − ab/100)%.

Successive Discount Mechanism

In a successive discount scheme, every subsequent discount is applied on the freshly reduced price after deducting the previous discount, rather than on the original Marked Price (MP).

Base Reference Rules

Discounts are universally calculated over the Marked Price (MP), whereas terms like Profit or Loss are always computed over the base Cost Price (CP).

Shortcut Trick

In an equilateral triangle, all three sides are equal in length.

Perimeter = 3 × side

Perimeter = 3 × 13 = 39 cm

∴ The correct answer is 39 cm.

Alternate Method

Given:

Side of the equilateral triangle (a) = 13 cm

A
B
C
13 cm
13 cm
13 cm

Formula Used:

Perimeter of a triangle = Sum of all its sides

Calculation:

⇒ Perimeter = a + a + a

⇒ Perimeter = 13 + 13 + 13

⇒ Perimeter = 39 cm

∴ The correct answer is 39 cm.

Additional Information

Area of Equilateral Triangle

The area is calculated using the formula: Area = (√3 ÷ 4) × a², where 'a' is the side of the triangle.

Height (Altitude) of Equilateral Triangle

The perpendicular distance from any vertex to the opposite side is given by: Height = (√3 ÷ 2) × a.

Inradius & Circumradius

The radius of the inscribed circle (Inradius) = a ÷ (2√3) and the radius of the circumscribed circle (Circumradius) = a ÷ √3.

Shortcut Trick

Total work = LCM(30, 18, 9) = 90 units.

Efficiency of Anjali = 90 ÷ 30 = 3 units/day.

Efficiency of (Anjali + Ayushi + Ankita) = 90 ÷ 9 = 10 units/day.

Efficiency of Ankita = 10 − (90 ÷ 18) = 10 − 5 = 5 units/day.

Time taken by Anjali and Ankita = 90 ÷ (3 + 5) = 90 ÷ 8 = 45/4 days.

∴ The correct answer is 45/4 days.

Alternate Method

Given:

Time taken by Anjali = 30 days

Time taken by Anjali and Ayushi = 18 days

Time taken by Anjali, Ayushi and Ankita = 9 days

Formula Used:

Total Work = LCM of given days

Efficiency = Total Work ÷ Time

Calculation:

⇒ Efficiency of Ankita = Efficiency of (Anjali + Ayushi + Ankita) − Efficiency of (Anjali + Ayushi)

⇒ Efficiency of Ankita = 10 − 5 = 5 units/day

⇒ Combined efficiency of Anjali and Ankita = 3 + 5 = 8 units/day

⇒ Time taken by Anjali and Ankita together = Total Work ÷ Combined Efficiency

⇒ Time taken = 90 ÷ 8 = 45/4 days

∴ The correct answer is 45/4 days.

Additional Information

Pipes & Cisterns

Works identically to Time & Work. The inlet pipe performs positive work (+ efficiency), while the outlet pipe performs negative work (− efficiency).

Work & Wages

Wages are distributed in direct proportion to the actual amount of work done by each person. When all persons work for the same total time, wages are divided based on their ratio of efficiency.

Alternate Days Work

Always find the total work done in one complete cycle (e.g., 2 days if two people work alternately). Multiply cycles to reach closest to the total work, then calculate the time required for remaining work.

Shortcut Trick

Total Work = LCM(19, 16, 5) = 19 × 16 × 5 = 1520 units

Efficiency of A = 1520 ÷ 19 = 80, Efficiency of B = 1520 ÷ 16 = 95

Combined Efficiency (A + B + C) = 1520 ÷ 5 = 304

Efficiency of C = 304 − (80 + 95) = 304 − 175 = 129 units/day

Time required for C = 1520 ÷ 129 = 11¹⁰¹/₁₂₉ days

∴ The correct answer is 11¹⁰¹/₁₂₉ days.

Alternate Method

Given:

Time taken by A = 19 days

Time taken by B = 16 days

Time taken by A, B, and C together = 5 days

Formula Used:

Total Work = LCM of all individual days

Efficiency = Total Work ÷ Days

Calculations:

⇒ Efficiency of A + B = 80 + 95 = 175 units/day

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

⇒ Efficiency of C = 304 − 175 = 129 units/day

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

⇒ Time taken by C alone = 1520 ÷ 129

⇒ Time taken by C alone = 11¹⁰¹/₁₂₉ days

∴ The correct answer is 11¹⁰¹/₁₂₉ days.

Additional Information

Pipes & Cisterns

Inlet pipes have positive efficiency while outlet pipes (leaks) have negative efficiency. Total capacity = LCM of individual times.

Work & Wages

Wages are always distributed in the ratio of the total work done by each person. If they work for the exact same duration, wages are distributed directly in the ratio of their daily efficiencies.

Alternate Day Work

Find the total work done in a complete cycle (e.g., 2 days for 2 people) and multiply by the maximum possible integer before the remaining work is done individually.

Shortcut Trick

Simplified investment ratio of A : B : C = 63 : 35 : 42 = 9 : 5 : 6

Total profit ratio parts = 9 + 5 + 6 = 20 parts

20 parts = ₹5,120 ⇒ 1 part = 5120 ÷ 20 = ₹256

Difference between the profit share of C and B = 6 parts − 5 parts = 1 part

∴ The correct answer is ₹256.

Alternate Method

Given:

Investment ratio of A, B, and C = 63 : 35 : 42

Total profit earned at the end of the year = ₹5,120

Formula Used:

Since the time period of investment is the same for all partners, Profit Ratio = Investment Ratio

Profit Distribution (Total = 20 parts = ₹5,120)

A = 9 parts
B = 5 parts
C = 6 parts

Difference (C − B) = 6 − 5 = 1 part
1 part = 5120 ÷ 20 = ₹256

Calculation:

⇒ Simplifying the given investment ratio by dividing by 7:

⇒ A : B : C = 63 : 35 : 42 = 9 : 5 : 6

⇒ Let the profit share of A, B, and C be 9x, 5x, and 6x respectively.

⇒ Total profit = 9x + 5x + 6x = 20x

⇒ 20x = 5120

⇒ x = 5120 ÷ 20 = 256

⇒ Difference between the share of B and C = 6x − 5x = x

⇒ Difference = ₹256

∴ The correct answer is ₹256.

Additional Information

Simple Partnership

If all partners invest their capital for the exact same time period, the final profit is distributed strictly in the ratio of their invested amounts.

Compound Partnership

If partners invest for different time periods, the profit ratio is calculated as the product of capital and time: (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A working partner manages the business and receives a fixed salary or percentage of the profit beforehand, while a sleeping partner only receives a share based on their investment.

Shortcut Trick

The divisibility rule of 5 states that any number ending with a 0 or 5 is exactly divisible by 5.

The given number is 245015, and its unit digit is 5.

Hence, the number is exactly divisible by 5 without the need to check other complex options.

∴ The correct answer is 5.

Alternate Method

Given:

Number = 245015

Options to check: 2, 15, 5, 11

Method Used:

We will verify the standard divisibility rules for each given option to find the exact divisor.

Calculations:

⇒ Checking divisibility by 2: The unit digit is 5 (which is an odd number). Thus, 245015 is not divisible by 2.

⇒ Checking divisibility by 15: For a number to be divisible by 15, it must be divisible by both 3 and 5. The sum of the digits = 2 + 4 + 5 + 0 + 1 + 5 = 17, which is not divisible by 3. Thus, 245015 is not divisible by 15.

⇒ Checking divisibility by 5: The unit digit is 5. Thus, 245015 is exactly divisible by 5.

⇒ Checking divisibility by 11: The difference between the sum of digits at odd positions (from right) and even positions = (5 + 0 + 4) − (1 + 5 + 2) = 9 − 8 = 1. Since 1 is neither 0 nor a multiple of 11, it is not divisible by 11.

∴ The correct answer is 5.

Additional Information

Divisibility Rule of 3 and 9

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

Divisibility Rule of 4 and 8

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if the last three digits form a number divisible by 8.

Divisibility Rule of 6

A number is divisible by 6 if it satisfies the divisibility rules of both 2 (must be an even number) and 3 (sum of digits divisible by 3).

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.

Shortcut Trick

Diameter of the largest circle in a rectangle = Breadth of the rectangle = 126 m.

Circumference = π × d

⇒ 22/7 × 126

⇒ 396 m

∴ The correct answer is 396 m.

Alternate Method

Given:

Length of the rectangle = 226 m

Breadth of the rectangle = 126 m

226 m
126 m
d = 126 m

Formula Used:

Circumference of a circle = 2 × π × r

Calculations:

⇒ For the largest circle to be completely inside the rectangle, its diameter must be equal to the smaller side (breadth) of the rectangle.

⇒ Diameter of the circle (d) = 126 m

⇒ Radius (r) = d ÷ 2 = 126 ÷ 2 = 63 m

⇒ Circumference = 2 × (22/7) × 63

⇒ Circumference = 2 × 22 × 9

⇒ Circumference = 44 × 9 = 396 m

∴ The correct answer is 396 m.

Additional Information

Largest Circle inside a Rectangle

The diameter is always equal to the shorter side (breadth) of the rectangle.

Largest Circle inside a Square

The diameter is equal to the side of the square. Area = π × (a/2)².

Largest Square inside a Circle

The diagonal of the square is equal to the diameter of the circle. Side = d ÷ √2.

Shortcut Trick

When the distance covered is equal for both parts of the journey, the average speed formula is:

Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

⇒ Average Speed = (2 × 72 × 90) ÷ (72 + 90)

⇒ Average Speed = 12960 ÷ 162 = 80

∴ The correct answer is 80 km/hr.

Alternate Method

Kanpur

Lucknow

Distance = D

Speed = 72 km/hr

Speed = 90 km/hr

Given:

Speed from Kanpur to Lucknow (S₁) = 72 km/hr

Speed from Lucknow to Kanpur (S₂) = 90 km/hr

Method: LCM Approach

⇒ Let the distance between Kanpur and Lucknow be the LCM of the speeds 72 and 90.

⇒ LCM of 72 and 90 = 360

⇒ Let total distance one way = 360 km

⇒ Time taken for onward journey = 360 ÷ 72 = 5 hours

⇒ Time taken for return journey = 360 ÷ 90 = 4 hours

⇒ Total Distance for the entire journey = 360 + 360 = 720 km

⇒ Total Time taken = 5 + 4 = 9 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = 720 ÷ 9 = 80 km/hr

∴ The correct answer is 80 km/hr.

Additional Information

Average Speed for Three Equal Distances

If a person covers three equal distances at speeds x, y, and z, the average speed is (3 × x × y × z) ÷ (xy + yz + zx).

Average Speed for Unequal Distances

When different distances are covered, the average speed is always calculated as Total Distance ÷ Total Time.

Relative Speed

When two bodies move in opposite directions, their relative speed is S₁ + S₂. When moving in the same direction, it is |S₁ − S₂|.

Shortcut Trick

Put β = 45° in the given expression:

⇒ sec²(45°) − 1 / (cosec²(45°) − 1)

⇒ (√2)² − 1 / ((√2)² − 1)

⇒ 2 − 1 / (2 − 1)

⇒ 2 − 1 / 1 = 1

∴ The correct answer is 1.

Alternate Method

Given:

sec²β − 1 / (cosec²β − 1)

Formula Used:

cosec²θ − cot²θ = 1 ⇒ cosec²θ − 1 = cot²θ

1 / cotθ = tanθ

sec²θ − tan²θ = 1

Calculations:

⇒ sec²β − 1 / (cosec²β − 1)

⇒ sec²β − 1 / (cot²β)

⇒ sec²β − tan²β

⇒ 1

∴ The correct answer is 1.

Additional Information

Basic Trigonometric Identities

sin²θ + cos²θ = 1

Reciprocal Relations

sinθ = 1 / cosecθ | cosθ = 1 / secθ | tanθ = 1 / cotθ

Quotient Relations

tanθ = sinθ / cosθ | cotθ = cosθ / sinθ