Quantitative Aptitude PYQs

Shortcut Trick

Increasing a number by 100% makes it 200% of its original base value.

200% of the number = 2820

100% of the number = 2820 ÷ 2

Original Number = 1410

∴ The correct answer is 1410.

Alternate Method

Given:

A number when increased by 100% gives 2820.

Original (100%)

• Increase (100%)

Total = 200% → 2820

? = 1410

Calculations:

Let the original number be x.

⇒ x + 100% of x = 2820

⇒ x + x = 2820

⇒ 2x = 2820

⇒ x = 2820 ÷ 2

⇒ x = 1410

∴ The correct answer is 1410.

Additional Information

Percentage Increase

When a value increases by R%, the new value is (100 + R)% of the original base value.

Percentage Decrease

When a value decreases by R%, the new value is (100 − R)% of the original base value.

Successive Percentage Change

If a number is subjected to two successive changes of A% and B%, the net percentage change is given by (A + B + A × B ÷ 100)%.

Shortcut Trick

Let original pen cost = x and pencil cost = y.

8x + 10y = 132 ⇒ 4x + 5y = 66 ⇒ 24x + 30y = 396

17(x − 2) + 6(y + 7) = 136 ⇒ 17x + 6y = 128 ⇒ 85x + 30y = 640

Subtracting both: 61x = 244 ⇒ x = 4. Then 4(4) + 5y = 66 ⇒ y = 10.

Required cost of 12 pens and 9 pencils = 12(4) + 9(10) = 48 + 90 = 138.

∴ The correct answer is ₹138.

Alternate Method

Given:

Cost of 8 pens and 10 pencils = ₹132

Cost of 17 pens (rate decreased by ₹2) and 6 pencils (rate increased by ₹7) = ₹136

Case 1: Original Cost
8 Pens (x) + 10 Pencils (y) = ₹132

Case 2: Changed Cost
17 (x − 2) + 6 (y + 7) = ₹136

Calculation:

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

⇒ 8x + 10y = 132

⇒ 4x + 5y = 66     --- (Equation 1)

⇒ Cost of a pen decreases by ₹2, new cost = (x − 2)

⇒ Cost of a pencil increases by ₹7, new cost = (y + 7)

⇒ 17(x − 2) + 6(y + 7) = 136

⇒ 17x − 34 + 6y + 42 = 136

⇒ 17x + 6y + 8 = 136

⇒ 17x + 6y = 128     --- (Equation 2)

⇒ Multiply Equation 1 by 6 and Equation 2 by 5 to eliminate y:

⇒ (4x + 5y) × 6 = 66 × 6 ⇒ 24x + 30y = 396

⇒ (17x + 6y) × 5 = 128 × 5 ⇒ 85x + 30y = 640

⇒ Subtracting the two equations:

⇒ (85x + 30y) − (24x + 30y) = 640 − 396

⇒ 61x = 244

⇒ x = 244 ÷ 61 = 4

⇒ Substitute x = 4 into Equation 1:

⇒ 4(4) + 5y = 66

⇒ 16 + 5y = 66

⇒ 5y = 50 ⇒ y = 10

⇒ The original cost of a pen is ₹4 and a pencil is ₹10.

⇒ Cost of 12 pens and 9 pencils = 12x + 9y

⇒ 12(4) + 9(10) = 48 + 90 = 138

∴ The correct answer is ₹138.

Additional Information

Linear Equations in Two Variables

A system of two linear equations (ax + by = c) can be easily solved using substitution, elimination, or cross-multiplication methods depending on the coefficients.

Elimination Method

This method involves multiplying the equations by suitable non-zero constants to make the coefficients of one variable mathematically equal, and then adding or subtracting the equations to eliminate that variable.

Shortcut Trick

In right-angled ▵BDC, the sum of the acute angles is exactly 90°.

⇒ ∠C = 90° − ∠DBC = 90° − 16° = 74°

By the Exterior Angle Theorem on ▵AEC, the exterior angle equals the sum of the remote interior angles.

⇒ ∠AEB = ∠CAE + ∠C

⇒ ∠AEB = 51° + 74° = 125°

∴ The correct answer is 125°.

Alternate Method

16°

74°

51°

125°

A
B
C
D
E

Given:

⇒ BD ⊥ AC at D in ▵ABC, which means ∠BDC = 90°.

⇒ ∠DBC = 16°

⇒ ∠CAE = 51°

Step 1: Calculate ∠C using ▵BDC

⇒ The sum of angles in any triangle is 180°.

⇒ In right-angled ▵BDC: ∠BDC + ∠DBC + ∠C = 180°

⇒ 90° + 16° + ∠C = 180°

⇒ 106° + ∠C = 180°

⇒ ∠C = 180° − 106° = 74°

Step 2: Find ∠AEB using ▵AEC

⇒ Since E is a point on BC, the line segment extends from C through E to B.

⇒ This makes ∠AEB an exterior angle to ▵AEC at vertex E.

⇒ According to the exterior angle theorem: ∠AEB = ∠CAE + ∠C

⇒ ∠AEB = 51° + 74°

⇒ ∠AEB = 125°

∴ The correct answer is 125°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is perfectly equal to the sum of the measures of its two remote interior angles. (Formula: ∠Ext = ∠A + ∠B)

Triangle Angle Sum Property

The sum of all three interior angles in any standard triangle in Euclidean geometry is always exactly 180°.

Altitudes & Right Triangles

An altitude is a line drawn from a vertex meeting the opposite side at exactly 90°. It splits the main triangle into smaller right-angled triangles, enabling quick angle evaluations since the remaining two acute angles must sum to 90°.

Shortcut Trick

Given, 3 × A = 9 × B = 4 × C.

To find the direct ratio, take the LCM of the coefficients 3, 9, and 4, which is 36.

Ratio of A : B : C = (36 ÷ 3) : (36 ÷ 9) : (36 ÷ 4) = 12 : 4 : 9.

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

Value of 1 part = ₹75,400 ÷ 25 = ₹3,016.

Share of A = 12 parts = 12 × 3,016 = ₹36,192.

∴ The correct answer is ₹36,192.

Alternate Method

Given:

Total amount to be divided = ₹75,400

Condition for division: 3 × A = 9 × B = 4 × C

Formula Used:

If xA = yB = zC, then the direct ratio A : B : C = 1/x : 1/y : 1/z

A (12 Parts)
B (4 Parts)
C (9 Parts)

?

Total = 25 Parts = ₹75,400
Proportional Division

Calculations:

⇒ Let 3A = 9B = 4C = k

⇒ A = k/3, B = k/9, C = k/4

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

⇒ To convert fractions to integers, multiply the ratio by the LCM of denominators (3, 9, 4), which is 36.

⇒ A : B : C = (1/3 × 36) : (1/9 × 36) : (1/4 × 36)

⇒ A : B : C = 12 : 4 : 9

⇒ Let the respective shares be 12x, 4x, and 9x.

⇒ Total sum = 12x + 4x + 9x = 25x

⇒ 25x = 75400

⇒ x = 75400 ÷ 25

⇒ x = 3016

⇒ Share of A = 12x = 12 × 3016

⇒ Share of A = 36192

∴ The correct answer is ₹36,192.

Additional Information

Inverse Ratio

If two quantities are in the ratio a : b, their inverse ratio is formed by swapping the terms and is written as 1/a : 1/b or b : a.

Compound Ratio

When two or more ratios are multiplied together term by term, the resultant ratio is called a compound ratio. For a : b and c : d, it is ac : bd.

Duplicate Ratio

The compound ratio of two identical ratios is termed the duplicate ratio. For a given ratio a : b, its duplicate ratio is expressed as a² : b².

Shortcut Trick

The given number to be divided is 774359327.

Its unit digit is 7, which means the given number is an odd number.

An odd number can never be exactly divided by an even number.

Among the given options, 12, 2, and 16 are even numbers.

By elimination, 7 is the only possible odd number that can divide it.

∴ The correct answer is 7.

Alternate Method

Given:

The number to be divided is 774359327.

Calculation:

⇒ Let us check the divisibility by the given options one by one.

⇒ Option 3 (2): A number is divisible by 2 if its last digit is even. Since 7 is an odd digit, the number is not divisible by 2.

⇒ Option 4 (16): A number divisible by 16 must also be divisible by all factors of 16, including 2. Hence, 774359327 is not divisible by 16.

⇒ Option 2 (12): A number divisible by 12 must be divisible by 3 and 4. Since it is not divisible by 2, it cannot be divisible by 4. Hence, it is not divisible by 12.

⇒ Option 1 (7): Let us perform direct division to verify.

⇒ 774359327 ÷ 7 = 110622761

⇒ Since the remainder is 0, the number is completely divisible by 7.

∴ The correct answer is 7.

Additional Information

Divisibility Rule for 2, 4, 8 and 16

A number is divisible by 2ⁿ if the number formed by its last n digits is divisible by 2ⁿ. For 2, check the last digit; for 4, check the last two digits.

Divisibility Rule for Composite Numbers

To check if a number is divisible by a composite number (like 12), split it into co-prime factors (such as 3 and 4). The number must be divisible by both co-prime factors.

Divisibility Rule for 7

Double the last digit of the number and subtract it from the remaining truncated number. If the obtained result is divisible by 7, the original number is also divisible by 7.

Shortcut Trick

Apply the divisibility rule of 3 directly by checking the sum of the digits.

Sum of digits = 5 + 4 + 5 + 7 + 1 + 4 + 1 + 5 + 1 = 33

Since 33 is exactly divisible by 3, the entire number 545714151 is also divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 545714151

Concept Used:

Test the number against the standard divisibility rules for the given options.

Calculations:

⇒ Checking Option 1 (10): A number is divisible by 10 only if its unit digit is 0. The given unit digit is 1 → Not divisible by 10.

⇒ Checking Option 2 (6): A number is divisible by 6 if it is divisible by both 2 and 3. Since the unit digit is 1 (an odd number), it is not divisible by 2 → Not divisible by 6.

⇒ Checking Option 3 (7): Dividing 545714151 directly by 7 yields 77959164 with a remainder of 3 → Not divisible by 7.

⇒ Checking Option 4 (3): A number is divisible by 3 if the sum of its digits is a multiple of 3.

⇒ Sum of digits = 5 + 4 + 5 + 7 + 1 + 4 + 1 + 5 + 1 = 33

⇒ Since 33 ÷ 3 = 11 (remainder 0), the number is perfectly divisible by 3.

∴ The correct answer is 3.

Additional Information

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

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 11

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

Shortcut Trick

Perimeter of a sector = Arc length + 2 × Radius = (θ ÷ 360°) × 2πr + 2r

⇒ Perimeter = (18 ÷ 360) × 2 × 3.14 × 9 + 2 × 9

⇒ Perimeter = (1 ÷ 20) × 56.52 + 18

⇒ Perimeter = 2.826 + 18 = 20.826 cm

Rounding to two decimal places, we get 20.83 cm.

∴ The correct answer is 20.83 cm.

Alternate Method

Given:

Radius (r) = 9 cm

Angle subtended at the centre (θ) = 18°

Value of π = 3.14

O9 cm9 cm18°Arc (l)Perimeter of Sector= 2r + l= 2 × Radius + Arc length

Formula Used:

Arc length (l) = (θ ÷ 360°) × 2πr

Perimeter of Sector = 2r + l

Calculations:

⇒ Arc length (l) = (18 ÷ 360) × 2 × 3.14 × 9

⇒ l = (1 ÷ 20) × 56.52

⇒ l = 2.826 cm

⇒ Perimeter of Sector = (2 × 9) + 2.826

⇒ Perimeter of Sector = 18 + 2.826 = 20.826 cm

⇒ Rounded up to two decimal places = 20.83 cm

∴ The correct answer is 20.83 cm.

Additional Information

Area of a Sector

Formula: Area = (θ ÷ 360°) × πr² or (1 ÷ 2) × l × r. It represents the region enclosed by two radii and the arc.

Area of a Segment

Formula: Area of Sector − Area of Triangle formed by the radii and the corresponding chord. This gives the region bounded by a chord and its arc.

Perimeter of a Semicircle

Formula: πr + 2r. A semicircle is a special sector where the central angle is exactly 180°.

Shortcut Trick

Let total registered voters be N.

Valid votes = N × (85/100) × (96/100) = N × (17/20) × (24/25)

Winning margin = (65% − 35%) of valid votes = 30% = 3/10 of valid votes.

Equation: N × (17/20) × (24/25) × (3/10) = 2142

⇒ N × 1224 ÷ 5000 = 2142 ⇒ N = (2142 × 5000) ÷ 1224 = 8750

∴ The correct answer is 8750 voters.

Alternate Method

Given:

Registered voters who cast their vote = 85%

Invalid votes = 4% of polled votes

Winner gets 65% of valid votes

Winning margin = 2142 votes

Formula Used:

Margin = Winner's Votes − Loser's Votes

Valid Votes = Polled Votes − Invalid Votes

Registered Voters (100x)Did Not Vote (15x)Polled Votes (85x)Invalid (3.4x)Valid (81.6x)Loser: 35%Winner: 65%

Calculations:

Let the total registered voters be 100x.

⇒ Votes polled = 85% of 100x = 85x

⇒ Invalid votes = 4% of 85x = 3.4x

⇒ Valid votes = 85x − 3.4x = 81.6x

⇒ Winner's votes = 65% of 81.6x = 53.04x

⇒ Loser's votes = (100% − 65%) of 81.6x = 35% of 81.6x = 28.56x

⇒ Margin of victory = Winner's votes − Loser's votes = 53.04x − 28.56x = 24.48x

According to the question, winning margin is 2142 votes.

⇒ 24.48x = 2142

⇒ x = 2142 ÷ 24.48 = 87.5

⇒ Total registered voters = 100x = 100 × 87.5 = 8750

∴ The correct answer is 8750 voters.

Additional Information

Election Percentage Breakdown

In election questions, always track the shifting base: Total Voters → Polled Votes → Valid Votes. The percentages at each step apply to the remaining base from the previous step.

Winning Margin Rule

When there are only two candidates, the winning margin percentage is always the difference between the winner's share and the loser's share. Margin % = Winner % − Loser %.

Successive Multiplying Factor

Use successive multiplication to find the final value quickly. Valid Votes = Total Voters × (Polled %) × (Valid %). Avoid calculating intermediate discrete values.

Shortcut Trick

For any two numbers 'a' and 'b', their third proportional 'c' is directly given by the formula c = b² ÷ a.

Here, the given numbers are a = 16 and b = 56.

Third proportional (x) = 56² ÷ 16

x = (56 × 56) ÷ 16 = 3136 ÷ 16 = 196

∴ The correct answer is 196.

Alternate Method

Given:

First number = 16

Second number = 56

Formula Used:

When three numbers a, b, and c are in continued proportion, then a : b :: b : c

16

:

56

::

56

:

x

16 ÷ 56
56 ÷ x

Concept of Continued Proportion
16 × x = 56 × 56

Calculations:

Let the third proportional be x.

⇒ 16 / 56 = 56 / x

⇒ 16 × x = 56 × 56

⇒ x = (56 × 56) ÷ 16

⇒ x = 3136 ÷ 16

⇒ x = 196

∴ The correct answer is 196.

Additional Information

Mean Proportional

For any two numbers 'a' and 'b', the mean proportional 'x' is defined such that a : x :: x : b, which gives the formula x = √(ab).

Fourth Proportional

For three numbers 'a', 'b', and 'c', the fourth proportional 'x' satisfies the ratio a : b :: c : x, resulting in the calculation x = (b × c) ÷ a.

First Proportional

If 'x' is the first proportional to 'a' and 'b', it is written as x : a :: a : b, yielding the direct relation x = a² ÷ b.

Compound Ratio

The compound ratio of two individual ratios (a : b) and (c : d) is calculated by multiplying the respective antecedents and consequents to get (ac : bd).

Shortcut Trick

Let total work = LCM(26, 13, 10) = 130 units.

Efficiencies: Priya = 5, Priya + Varsha = 10, Priya + Varsha + Nisha = 13 units/day.

Nisha's efficiency = Combined (Priya + Varsha + Nisha) − (Priya + Varsha) = 13 − 10 = 3 units/day.

Combined efficiency of Priya + Nisha = 5 + 3 = 8 units/day.

Time for Priya + Nisha = Total Work ÷ combined efficiency = 130 ÷ 8 = 65/4 days.

∴ The correct answer is 65/4 days.

Alternate Method

Given:

Time taken by Priya = 26 days

Time taken by Priya and Varsha = 13 days

Time taken by Priya, Varsha and Nisha = 10 days

Formula Used:

Total Work = Time × Efficiency

Calculation:

⇒ Let Total Work be the LCM of 26, 13 and 10.

⇒ Efficiency of Nisha = Efficiency of (Priya + Varsha + Nisha) − Efficiency of (Priya + Varsha)

⇒ Nisha's efficiency = 13 − 10 = 3 units/day

⇒ Combined efficiency of Priya and Nisha = 5 + 3 = 8 units/day

⇒ Time taken by Priya and Nisha = Total Work ÷ Combined efficiency

⇒ Time taken = 130 ÷ 8 = 65/4 days

∴ The correct answer is 65/4 days.

Additional Information

Pipes & Cisterns

Works exactly like Time & Work. Inlet pipes have positive efficiency, and outlet pipes (leaks) have negative efficiency.

Work & Wages

Wages are distributed in the ratio of the work done. If all individuals work for the same number of days, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

When individuals work on alternate days, calculate the work done in a complete cycle (e.g., 2 days for 2 people) and determine how many cycles are needed to reach the total work.

Shortcut Trick

Ratio of speeds of Arjun and Malaika = 138 : 92 = 3 : 2

Since the distance is constant, the time taken is inversely proportional to the speed.

Ratio of time taken by Arjun and Malaika = 2 : 3

Time taken by Arjun (2 units) = 8 hours ⇒ 1 unit = 4 hours.

Time taken by Malaika (3 units) = 3 × 4 = 12 hours.

∴ The correct answer is 12 hours.

Alternate Method

Delhi

Kanpur

Total Distance (D) is Constant

Arjun
138 km/hr | 8 hrs

Malaika
92 km/hr | T hrs

Given:

Speed of Arjun (S₁) = 138 km/hr

Time taken by Arjun (T₁) = 8 hours

Speed of Malaika (S₂) = 92 km/hr

Formula Used:

Distance = Speed × Time

When Distance is constant, S₁ × T₁ = S₂ × T₂

Calculations:

⇒ Total Distance = 138 × 8 = 1104 km

Let the time taken by Malaika be T₂.

⇒ T₂ = Distance ÷ Speed of Malaika

⇒ T₂ = 1104 ÷ 92

⇒ T₂ = 12 hours

∴ The correct answer is 12 hours.

Additional Information

Relative Speed

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

Average Speed

If equal distances are covered at speeds x and y, Average Speed = 2xy ÷ (x + y). For different distances covered in different times, Average Speed = Total Distance ÷ Total Time.

Proportionality in Speed, Time and Distance

If Distance is constant, Speed is inversely proportional to Time (S₁ ÷ S₂ = T₂ ÷ T₁). If Time is constant, Distance is directly proportional to Speed.

Shortcut Trick

Let the standard cost price of 100 units of weight be 100.

Actual Cost Price (due to 8% less weight) = 100 − 8 = 92.

Selling Price (at 47% claimed profit) = 100 + 47 = 147.

Profit = 147 − 92 = 55.

Profit % = (55 ÷ 92) × 100 = 59.78% ≈ 60%.

∴ The correct answer is 60%.

Alternate Method

Given:

Claimed profit percentage = 47%

Underweight used by the trader = 8% less than the actual weight

CP = 92Actual WeightBase = 100SP = 147Claimed GainTotal Profit = 147 − 92 = 55Profit % = (55 ÷ 92) × 100 ≈ 60%

Calculations:

⇒ Let the true cost price (CP) of 100 g of flour be 100 units.

⇒ The trader claims a profit of 47%, so the Selling Price (SP) for 100 g = 100 + (47 × 100 ÷ 100) = 147 units.

⇒ However, he dishonestly gives a weight which is 8% less, so the actual weight given = 100 − 8 = 92 g.

⇒ The actual Cost Price (CP) for the 92 g of flour given to the customer = 92 units.

⇒ Total Profit earned = Selling Price − Actual Cost Price

⇒ Total Profit = 147 − 92 = 55 units.

⇒ Overall Profit Percentage = (Total Profit ÷ Actual Cost Price) × 100

⇒ Profit Percentage = (55 ÷ 92) × 100

⇒ Profit Percentage = 5500 ÷ 92 = 59.7826%

⇒ Rounding off to the nearest whole number, the profit percentage is 60%.

∴ The correct answer is 60%.

Additional Information

Dishonest Dealer Concept

When a dealer claims to sell goods at x% profit but uses y% less weight, the overall Actual Profit % is given directly by the formula: [(x + y) ÷ (100 − y)] × 100.

Marked Price and Profit Relation

Selling Price (SP) is always calculated based on Markup or Profit over Cost Price (CP) using: SP = CP × [(100 + Profit %) ÷ 100].

Successive Changes Rule

When two successive factors impact the total value by a% and b%, the effective percentage change is calculated using: a + b + (a × b ÷ 100)%.

Shortcut Trick

Let Mary's age = M and Diya's age = D.

Given: 3M − 5D = 20 and M − 2D = 2

Solving simultaneously: (3M − 5D) − 3(M − 2D) = 20 − 3(2)

This directly gives D = 14, and substituting yields M = 2 × 14 + 2 = 30.

Age difference = 30 − 14 = 16 years.

∴ The correct answer is 16 years.

Alternate Method

Given:

⇒ 3 × Mary's age = 5 × Diya's age + 20

⇒ 2 × Diya's age = Mary's age − 2

Calculations:

Let Mary's present age be M and Diya's present age be D.

⇒ 3M = 5D + 20 → (Equation 1)

⇒ 2D = M − 2

⇒ M = 2D + 2 → (Equation 2)

Substitute the value of M from Equation 2 into Equation 1:

⇒ 3(2D + 2) = 5D + 20

⇒ 6D + 6 = 5D + 20

⇒ 6D − 5D = 20 − 6

⇒ D = 14 years

Now, substitute D = 14 back into Equation 2:

⇒ M = 2 × 14 + 2

⇒ M = 28 + 2 = 30 years

Mary's Age (M = 30)
Diya's Age (D = 14)

Difference = 16 years

The difference between their ages:

⇒ M − D = 30 − 14 = 16 years

∴ The correct answer is 16 years.

Additional Information

Problems on Ages

If the present age is x, then the age 'n' years ago was (x − n) and the age 'n' years from now will be (x + n).

Ratio of Ages

If the ratio of the present ages of A and B is a : b, their ages can be assumed as ax and bx. After n years, the new age ratio becomes (ax + n) ÷ (bx + n).

Linear Equations in Two Variables

A system of two linear equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ can be efficiently solved using algebraic techniques such as substitution or elimination.

Shortcut Trick

Convert the given discount percentages into fraction multipliers: 20% = 4/5, 25% = 3/4, and 15% = 17/20.

Find the combined multiplier: (4/5) × (3/4) × (17/20) = 51/100.

This means the Final Selling Price is 51% of the Marked Price.

Equivalent Discount = 100% − 51% = 49%.

∴ The correct answer is 49%.

Alternate Method

Given:

Three successive discounts of 20%, 25%, and 15%.

Formula Used:

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

Discount % = [(MP − SP) / MP] × 100

100
MP

80

60

51
SP

− 20%
− 25%
− 15%

Total Discount = 49%

Calculations:

⇒ Let the Marked Price (MP) = 100.

⇒ Price after 1^st discount of 20% = 100 − 20 = 80

⇒ Price after 2^nd discount of 25% = 80 − (25% of 80) = 80 − 20 = 60

⇒ Price after 3^rd discount of 15% = 60 − (15% of 60) = 60 − 9 = 51

⇒ Final Selling Price (SP) = 51

⇒ Total Equivalent Discount = MP − SP = 100 − 51 = 49

⇒ Since the base MP is 100, the equivalent discount % = 49%.

∴ The correct answer is 49%.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the equivalent single discount can be directly calculated using the formula: [a + b − (a × b) / 100]%.

Marked Price (MP) vs Selling Price (SP)

Discount is always calculated on the Marked Price (or list price) of the item. Profit or loss, on the other hand, is always calculated on the Cost Price (CP).

Buy X Get Y Free Offers

In promotional offers, the effective discount percentage is calculated as: [Free Items ÷ Total Items] × 100.

Shortcut Trick

List possible prime candidates by avoiding even numbers and multiples of 5: 331, 333, 337, 339, 341, 343.

Eliminate multiples of 3 (sum of digits is divisible by 3): 333 and 339 are removed.

Check the remaining against primes up to √345 ≈ 18 (7, 11, 13, 17).

341 = 11 × 31 and 343 = 7 × 49, so both are not prime.

The remaining numbers 331 and 337 are prime, giving a total of 2 numbers.

∴ The correct answer is 2 prime numbers.

Alternate Method

Given:

The range of numbers is from 331 to 345 (both included).

Method to find prime numbers:

A number is prime if it is greater than 1 and has no divisors other than 1 and itself.

⇒ Total numbers in the range: 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345.

⇒ Eliminate all even numbers: 331, 333, 335, 337, 339, 341, 343, 345.

⇒ Eliminate multiples of 5 (ending in 5): 331, 333, 337, 339, 341, 343.

⇒ Eliminate multiples of 3 (sum of digits ÷ 3): 333 (sum = 9) and 339 (sum = 15).

⇒ Remaining candidates: 331, 337, 341, 343.

⇒ Check divisibility by other primes (7, 11, 13, 17) up to √345 ≈ 18:

⇒ 341 ÷ 11 = 31 (Not prime)

⇒ 343 ÷ 7 = 49 (Not prime)

⇒ 331 and 337 are not divisible by any prime ≤ 18, so they are prime numbers.

⇒ Total number of prime numbers = 2.

∴ The correct answer is 2 prime numbers.

Additional Information

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two distinct divisors: 1 and itself.

Primality Test Shortcut

To check if a number N is prime, we only need to test its divisibility by all prime numbers less than or equal to √N.

Co-prime Numbers

Two numbers are said to be co-prime or mutually prime if their Highest Common Factor (HCF) is 1. They do not necessarily have to be prime numbers.

Twin Primes

Twin primes are a pair of prime numbers that have a difference of 2 (e.g., 11 and 13, or 41 and 43).

Shortcut Trick

CP of 1 eraser = ₹150 ÷ 12 = ₹12.5

SP of 1 eraser = ₹18

Ratio of CP to SP = 12.5 : 18 = 25 : 36

Profit in ratio = 36 − 25 = 11 parts

Profit % = (11 ÷ 25) × 100 = 44%

∴ The correct answer is 44%.

Alternate Method

Given:

Cost price of 1 dozen (12) erasers = ₹150

Selling price of 1 eraser = ₹18

Formula Used:

Profit = SP − CP

Profit % = (Profit ÷ CP) × 100

CP = ₹12.5
(150 ÷ 12)

SP = ₹18

• ₹5.5 Profit
  Profit % = (5.5 ÷ 12.5) × 100 = 44%

Calculations:

⇒ CP of 1 eraser = 150 ÷ 12 = ₹12.5

⇒ Profit = 18 − 12.5 = ₹5.5

⇒ Profit % = (5.5 ÷ 12.5) × 100

⇒ Profit % = (55 ÷ 125) × 100

⇒ Profit % = (11 ÷ 25) × 100 = 11 × 4 = 44%

∴ The correct answer is 44%.

Additional Information

Marked Price & Discount

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

Successive Discount

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

Dishonest Dealer

If a dealer professes to sell goods at CP but uses false weight, Profit % = (True Weight − False Weight) ÷ False Weight × 100.

Shortcut Trick

Convert the decimals to fractions in their lowest terms: 15.4 = 154/10 = 77/5 and 0.006 = 6/1000 = 3/500.

Apply the fraction formula: LCM = (LCM of Numerators) ÷ (HCF of Denominators).

LCM of (77, 3) = 231.

HCF of (5, 500) = 5.

Required LCM = 231 ÷ 5 = 46.2.

∴ The correct answer is 46.2.

Alternate Method

Given:

The numbers are 15.4 and 0.006.

Method Used:

To find the LCM of decimals, make the number of decimal places equal, remove the decimals to form integers, find their LCM, and then place the decimal point back at the same position.

Calculations:

⇒ The maximum number of decimal places in the given values is 3 (in 0.006).

⇒ Convert 15.4 into 3 decimal places: 15.400

⇒ Treat both numbers as integers by removing the decimal points: 15400 and 6.

⇒ Find the LCM of 15400 and 6.

⇒ 15400 = 2 × 7700

⇒ 6 = 2 × 3

⇒ LCM(15400, 6) = 2 × 7700 × 3 = 46200

⇒ Since we shifted the decimal by 3 places, place the decimal point 3 places from the right in the result.

⇒ Required LCM = 46200 ÷ 1000 = 46.2

∴ The correct answer is 46.2.

Additional Information

HCF of Fractions

HCF = (HCF of Numerators) ÷ (LCM of Denominators). It is critical to always convert the given fractions to their simplest forms before applying this rule.

Relation between HCF and LCM

For any two numbers 'a' and 'b', the product of the numbers is equal to the product of their HCF and LCM. That is: a × b = HCF(a, b) × LCM(a, b).

Co-prime Numbers

If two numbers share no common factors other than 1, they are called co-prime numbers. For co-prime numbers, their LCM is directly equal to their product.

Shortcut Trick

Divide the given equation (sin θ + cos θ) / (sin θ − 4 cos θ) = 11 by cos θ.

⇒ (tan θ + 1) / (tan θ − 4) = 11 ⇒ 10 tan θ = 45 ⇒ tan θ = 9/2

Given expression: (13 sin² θ + 3 cos² θ − 3) / cos² θ

⇒ 13 tan² θ + 3 − 3 sec² θ = 13 tan² θ + 3 − 3(1 + tan² θ) = 10 tan² θ

⇒ 10 × (9/2)² = 10 × (81/4) = 405/2

∴ The correct answer is 405/2.

Alternate Method

Given:

(sin θ + cos θ) / (sin θ − 4 cos θ) = 11

Formula Used:

sin² θ + cos² θ = 1

tan θ = sin θ / cos θ

Calculations:

⇒ sin θ + cos θ = 11(sin θ − 4 cos θ)

⇒ sin θ + cos θ = 11 sin θ − 44 cos θ

⇒ 45 cos θ = 10 sin θ

⇒ sin θ / cos θ = 45 / 10

⇒ tan θ = 9/2

Now, solve the required expression:

⇒ (13 sin² θ + 3 cos² θ − 3) / cos² θ

Substitute 3 = 3(sin² θ + cos² θ):

⇒ [13 sin² θ + 3 cos² θ − 3(sin² θ + cos² θ)] / cos² θ

⇒ (13 sin² θ + 3 cos² θ − 3 sin² θ − 3 cos² θ) / cos² θ

⇒ 10 sin² θ / cos² θ

⇒ 10 tan² θ

⇒ 10 × (9/2)²

⇒ 10 × 81 / 4

⇒ 810 / 4 = 405 / 2

∴ The correct answer is 405/2.

Additional Information

Pythagorean Identities

The fundamental trigonometric identity relating sine and cosine is: sin² θ + cos² θ = 1.

Tangent and Secant Relation

By dividing the basic Pythagorean identity by cos² θ, we obtain another standard identity: 1 + tan² θ = sec² θ.

Cotangent and Cosecant Relation

By dividing the basic Pythagorean identity by sin² θ, we obtain the third standard identity: 1 + cot² θ = cosec² θ.

Shortcut Trick

The diameter of the largest circle inscribed in a rectangle equals its shorter side (breadth).

Diameter (d) = 14 m

Circumference = π × d

Circumference = (22 ÷ 7) × 14 = 44 m

∴ The correct answer is 44 m.

Alternate Method

Length = 116 mBreadth = 14 md = 14 m

Given:

Length of the rectangle (l) = 116 m

Breadth of the rectangle (b) = 14 m

Concept Used:

The largest circle that can be drawn completely inside a rectangle is limited by the shorter side of the rectangle. Hence, the diameter of the circle will be exactly equal to the breadth of the rectangle.

Diameter (d) = 14 m

Radius (r) = d ÷ 2 = 14 ÷ 2 = 7 m

Formula Used:

Circumference of a circle = 2 × π × r

Calculation:

⇒ Circumference = 2 × (22 ÷ 7) × 7

⇒ Circumference = 2 × 22

⇒ Circumference = 44 m

∴ The correct answer is 44 m.

Additional Information

Area of Inscribed Circle

The area of the largest circle inscribed in a rectangle of dimensions l × b (where b is the shorter side) is given by the formula π × (b ÷ 2)².

Circumcircle of a Rectangle

A circle drawn passing through all four vertices of a rectangle is called its circumcircle. The diameter of this circumcircle is exactly equal to the diagonal of the rectangle: d = √(l² + b²).

Largest Square in a Circle

The diagonal of the largest square that can be inscribed in a given circle is equal to the diameter of the circle. The side of this square is d ÷ √2.

Shortcut Trick

Effective interest rate for 5 years = 4% × 5 = 20%.

Total amount in percentage = 100% (Principal) + 20% (Interest) = 120%.

Given that 120% of the principal corresponds to ₹2,400.

Principal (100%) = (2400 ÷ 120) × 100 = 20 × 100 = 2000.

∴ The correct answer is ₹2,000.

Alternate Method

P = ?
Year 0

A = ₹2400
Year 5

• SI (4% × 5 = 20%)

Amount = P + 20% of P = 120% of P

Given:

Amount (A) = ₹2,400

Rate of Interest (R) = 4% p.a.

Time (T) = 5 years

Formula Used:

Amount = Principal × [1 + (R × T) ÷ 100]

Calculations:

⇒ 2400 = P × [1 + (4 × 5) ÷ 100]

⇒ 2400 = P × [1 + 20 ÷ 100]

⇒ 2400 = P × [1 + 1/5]

⇒ 2400 = P × (6/5)

⇒ P = (2400 × 5) ÷ 6

⇒ P = 400 × 5

⇒ P = 2000

∴ The correct answer is ₹2,000.

Additional Information

Simple Interest (SI) Formula

The general formula to calculate simple interest is SI = (P × R × T) ÷ 100, where P is the principal, R is the rate per annum, and T is the time in years.

Effective Rate of Interest

In simple interest, the effective interest rate over a given period is directly proportional to time and is calculated as R × T. The total amount returned becomes (100 + R × T)% of the principal.

Relation between Amount and Principal

Amount (A) is the total sum returned at the end of the time period. The formula is A = P + SI = P × [1 + (R × T) ÷ 100].