Quantitative Aptitude PYQs

Shortcut Trick

Value 1 = 65% of 65 = (65 × 65) ÷ 100 = 4225 ÷ 100 = 42.25

Value 2 = (1 ÷ 5) of 15 = 3

Required difference = 42.25 − 3 = 39.25

∴ The correct answer is 39.25.

Alternate Method

Given:

First expression: 65% of 65

Second expression: (1 ÷ 5) of 15

Calculations:

⇒ Value of first expression = (65 ÷ 100) × 65

⇒ Value of first expression = 4225 ÷ 100 = 42.25

⇒ Value of second expression = (1 ÷ 5) × 15 = 15 ÷ 5 = 3

⇒ Required difference = Value of first expression − Value of second expression

⇒ Required difference = 42.25 − 3 = 39.25

∴ The correct answer is 39.25.

Additional Information

BODMAS Rule

BODMAS specifies the order of operations: Brackets, Order (powers/roots and 'of'), Division, Multiplication, Addition, and Subtraction. The word 'of' is treated as multiplication but solved before standard division and multiplication.

Squaring Numbers Ending in 5

To square any two-digit number ending in 5 (e.g., 65), multiply the tens digit by the next consecutive integer (6 × 7 = 42) and append 25 at the end to get 4225.

Percentage to Fraction

To evaluate a percentage expression, convert the percentage into a fraction by dividing by 100. Thus, 65% is equivalent to 65 ÷ 100.

Shortcut Trick

Total work = LCM(30, 21, 14) = 210 units.

Efficiency: Jaya = 7, Jaya + Sushma = 10, Jaya + Sushma + Hina = 15.

Efficiency of Hina = (Jaya + Sushma + Hina) − (Jaya + Sushma) = 15 − 10 = 5 units/day.

Combined efficiency of Jaya and Hina = 7 + 5 = 12 units/day.

Time taken by Jaya and Hina = 210 ÷ 12 = 35/2 days.

∴ The correct answer is 35/2 days.

Alternate Method

Given:

Time taken by Jaya = 30 days

Time taken by Jaya and Sushma = 21 days

Time taken by Jaya, Sushma, and Hina = 14 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Efficiency of Jaya = 7 units/day

⇒ Efficiency of Jaya + Sushma = 10 units/day

⇒ Efficiency of Jaya + Sushma + Hina = 15 units/day

⇒ Efficiency of Hina = Efficiency of (Jaya + Sushma + Hina) − Efficiency of (Jaya + Sushma)

⇒ Efficiency of Hina = 15 − 10 = 5 units/day

⇒ Combined efficiency of Jaya and Hina = Efficiency of Jaya + Efficiency of Hina

⇒ Combined efficiency of Jaya and Hina = 7 + 5 = 12 units/day

⇒ Time taken by Jaya and Hina = Total Work ÷ Combined Efficiency

⇒ Time taken by Jaya and Hina = 210 ÷ 12 = 35/2 days

∴ The correct answer is 35/2 days.

Additional Information

Pipes & Cisterns

Similar to Time & Work, filling pipes have positive efficiency while emptying pipes have negative efficiency. The total capacity of the tank acts as the Total Work.

Work & Wages

Wages are always distributed in the direct ratio of the total work done by individuals. If they work together for the same number of days, wages are directly proportional to their efficiencies.

Alternate Day Work

When individuals work on alternate days, find the work done in one complete cycle (e.g., 2 days for 2 people) and divide the total work by this cycle's work to determine the number of complete cycles.

Shortcut Trick

Successive discounts can be applied directly using multiplying factors.

Final SP = MP × (100 − D₁) / 100 × (100 − D₂) / 100

Final SP = 5000 × (90 / 100) × (95 / 100)

Final SP = 50 × 9 × 9.5 = 4275

∴ The correct answer is ₹4,275.

Alternate Method

Given:

Marked Price (MP) = ₹5,000

Successive Discounts: a = 10%, b = 5%

Formula Used:

Effective Single Discount % = a + b − (a × b) / 100

Selling Price (SP) = MP − Discount Amount

MP = 5000

−10%

4500

−5%

SP = 4275

Effective Discount = 14.5%

Calculations:

⇒ Effective Single Discount = 10 + 5 − (10 × 5) / 100

⇒ Effective Discount = 15 − 0.5 = 14.5%

⇒ Discount Value = 14.5% of 5000

⇒ Discount Value = (14.5 / 100) × 5000 = 725

⇒ Selling Price = 5000 − 725 = 4275

∴ The correct answer is ₹4,275.

Additional Information

Marked Price & Discount

Discount is always given on the Marked Price (MP) of an article. The selling price is calculated as SP = MP − Discount.

Successive Discount Rule

When two consecutive discounts d₁% and d₂% are offered, the overall effective discount is calculated using the formula: d₁ + d₂ − (d₁ × d₂) / 100.

Direct CP to MP Relation

A common and highly useful relation to connect Cost Price (CP) and Marked Price (MP) is: MP × (100 − Discount%) = CP × (100 + Profit%).

Shortcut Trick

M's Profit = (1359 + 453) ÷ 2 = ₹906 and N's Profit = 1359 − 906 = ₹453.

Profit Ratio (M : N) = 906 : 453 = 2 :

1. Capital Ratio = (Profit_M ÷ Time_M) : (Profit_N ÷ Time_N) = (2 ÷ 3) : (1 ÷ 4) = 8 : 3.

Difference in parts = 8 − 3 = 5 parts = ₹46,000 ⇒ 1 part = ₹9,200.

Capital of M = 8 parts = 8 × 9200 = ₹73,600.

∴ The correct answer is ₹73,600.

Alternate Method

Given:

Total profit = ₹1,359

M's profit − N's profit = ₹453

M's time = 3 months, N's time = 4 months

M's capital − N's capital = ₹46,000

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Partner M
Capital: x + 46000
Time: 3 months
Profit: ₹906

Partner N
Capital: x
Time: 4 months
Profit: ₹453

Capital & Time Distribution

Calculations:

⇒ M's profit + N's profit = 1359

⇒ M's profit − N's profit = 453

⇒ Adding the above two equations: 2 × M's profit = 1812

⇒ M's profit = 906

⇒ N's profit = 1359 − 906 = 453

⇒ Profit ratio of M to N = 906 : 453 = 2 : 1

⇒ Let N's capital be x. Then M's capital is x + 46000.

⇒ [(x + 46000) × 3] ÷ [x × 4] = 2 ÷ 1

⇒ 3x + 138000 = 8x

⇒ 5x = 138000

⇒ x = 27600

⇒ M's capital = x + 46000 = 27600 + 46000 = 73600

∴ The correct answer is ₹73,600.

Additional Information

Profit Sharing Ratio in Partnership

When partners invest varying capital amounts for different time periods, the profit is shared in the ratio of the product of their capital and time period: Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An active partner actively manages the business and frequently gets a salary or commission before the profit division, whereas a sleeping partner solely contributes capital and earns a share of the remaining profits based on their investment.

Equivalent Capital

To simplify complex partnerships, the overall investment is converted into equivalent capital for a standard unit of time (generally 1 month) by multiplying the invested principal amount with the corresponding number of months.

Shortcut Trick

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

Mean proportional = √(24 × 150)

⇒ Mean proportional = √3600

⇒ 60

∴ The correct answer is 60.

Alternate Method

Given:

The two given numbers are 24 and 150.

Formula Used:

If x is the mean proportional between two numbers a and b, then they are in continued proportion such that a : x = x : b, which implies x² = a × b.

Mean Proportional Concept

a = 24

b = 150

24 : x = x : 150

x = √(24 × 150)

x = 60

Calculations:

Let the mean proportional be x.

⇒ 24 : x = x : 150

⇒ 24 ÷ x = x ÷ 150

⇒ x² = 24 × 150

⇒ x² = 3600

⇒ x = √3600

⇒ x = 60

∴ The correct answer is 60.

Additional Information

Mean Proportional

For any two positive numbers a and b, the mean proportional x satisfies a : x = x : b, resulting in the rule x = √(a × b).

Third Proportional

If three numbers a, b, and c are in continued proportion (a : b = b : c), then c is called the third proportional. The formula used is c = b² ÷ a.

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b = c : d), then d is known as the fourth proportional. The formula used is d = (b × c) ÷ a.

Compound Ratio

The compound ratio of two different ratios a : b and c : d is the product of their respective antecedents and consequents, calculated directly as (a × c) : (b × d).

Shortcut Trick

Time taken by Train A = 7 hours, and by Train B = 5 hours.

Let Total Distance = LCM of 7 and 5 = 35 units.

Speed of Train A = 5 units/hr, Speed of Train B = 7 units/hr.

Distance covered by Train A in 2 hours (8:25 AM to 10:25 AM) = 2 × 5 = 10 units.

Remaining distance = 25 units. Relative speed = 5 + 7 = 12 units/hr.

Time to meet = 25 ÷ 12 = 2 hours 5 minutes. Meeting Time = 10:25 AM + 2 hours 5 minutes = 12:30 PM.

∴ The correct answer is 12:30 PM.

Alternate Method

Given:

Train A start time = 8:25 AM, arrival time = 3:25 PM

Train B start time = 10:25 AM, arrival time = 3:25 PM

Formula Used:

Speed = Distance ÷ Time

Relative Speed (Opposite Directions) = S₁ + S₂

Station M
Train A
8:25 AM

Station N
Train B
10:25 AM

A at 10:25 AM

10 km

Remaining: 25 km

Meeting Point

Calculations:

⇒ Total journey time for Train A = 3:25 PM − 8:25 AM = 7 hours

⇒ Total journey time for Train B = 3:25 PM − 10:25 AM = 5 hours

⇒ Let the total distance between M and N be the LCM of 7 and 5 = 35 km

⇒ Speed of Train A = 35 ÷ 7 = 5 km/h

⇒ Speed of Train B = 35 ÷ 5 = 7 km/h

⇒ Train A starts 2 hours earlier than Train B (from 8:25 AM to 10:25 AM).

⇒ Distance covered by Train A in these 2 hours = 5 × 2 = 10 km

⇒ Remaining distance to be covered together = 35 − 10 = 25 km

⇒ Relative speed of both trains = 5 + 7 = 12 km/h

⇒ Time taken to meet after 10:25 AM = Distance ÷ Relative speed = 25 ÷ 12 hours

⇒ 25 ÷ 12 hours = 2 hours and (1/12) × 60 minutes = 2 hours 5 minutes

⇒ Meeting time = 10:25 AM + 2 hours 5 minutes = 12:30 PM

∴ The correct answer is 12:30 PM.

Additional Information

Relative Speed (Opposite Direction)

When two objects move towards each other, their relative speed is the sum of their individual speeds: S_R = S₁ + S₂.

Relative Speed (Same Direction)

When two objects move in the same direction, their relative speed is the difference between their individual speeds: S_R = S₁ − S₂.

Average Speed

If an object covers equal distances at different speeds S₁ and S₂, the overall average speed is calculated as (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

Divisibility rule of 3: A number is divisible by 3 if the sum of its digits is a multiple of 3.

Sum of digits of 327087 = 3 + 2 + 7 + 0 + 8 + 7 = 27

Since 27 is divisible by 3, the given number 327087 is perfectly divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 327087

Checking the given options:

⇒ Option 1 (5): A number is divisible by 5 if its last digit is 0 or 5. Here, the last digit is 7, so it is not divisible by 5.

⇒ Option 2 (8): A number is divisible by 8 if its last three digits are divisible by 8. The last three digits are 087, which is not perfectly divisible by 8.

⇒ Option 4 (12): A number must be divisible by both 4 and 3 to be divisible by 12. Since the last digit (7) is odd, the number is not divisible by 4, and hence not divisible by 12.

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

⇒ Sum of digits = 3 + 2 + 7 + 0 + 8 + 7 = 27

⇒ 27 is perfectly divisible by 3 (27 ÷ 3 = 9).

∴ The correct answer is 3.

Additional Information

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 9

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

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
The greatest number leaving the same remainder is the HCF of the differences between the given numbers.
Given numbers are 49, 97, and 189.
Differences = |97 − 49|, |189 − 97|, and |189 − 49|.
The differences are 48, 92, and 140.
HCF(48, 92, 140) = 4.
∴ The correct answer is 4.

Alternate Method
Given:
The numbers are 49, 97, and 189.
Formula Used:
Required Number = HCF of (y − x), (z − y), and (z − x) where x < y < z.
Calculation:
⇒ Let x = 49, y = 97, and z = 189
⇒ y − x = 97 − 49 = 48
⇒ z − y = 189 − 97 = 92
⇒ z − x = 189 − 49 = 140
⇒ Now, we need to find the highest common factor (HCF) of 48, 92, and 140.
⇒ Prime factorization of 48 = 24 × 3
⇒ Prime factorization of 92 = 22 × 23
⇒ Prime factorization of 140 = 22 × 5 × 7
⇒ The common prime factor with the lowest power is 22.
⇒ HCF = 22 = 4
∴ The correct answer is 4.

Additional Information
HCF for Same Remainder
When a number divides x, y, and z leaving the same remainder, the greatest such number is the HCF of the absolute differences: |x − y|, |y − z|, and |z − x|.
HCF for Specific Remainders
When a number divides x, y, and z leaving specific remainders a, b, and c respectively, the greatest such number is the HCF of (x − a), (y − b), and (z − c).
LCM with Specific Remainder
The smallest number which when divided by x, y, and z leaves the same remainder r in each case is obtained by finding LCM(x, y, z) + r.

Shortcut Trick

Multiply out the first and second bracket terms directly.

⇒ (7x² − 75x + 36) − (7x² + 63x − 91)

⇒ The x² terms cancel out: 7x² − 7x² = 0

⇒ Collect x terms and constants: −75x − 63x + 36 + 91 = −138x + 127

∴ The correct answer is −138x + 127.

Alternate Method

Given:

Expression: 3((7/3)x² − 25x + 12) − 7(x² + 9x − 13)

Calculation:

⇒ Expand the first term by multiplying each element by 3:

⇒ 3 × (7/3)x² − 3 × 25x + 3 × 12 = 7x² − 75x + 36

⇒ Expand the second term by multiplying each element by 7:

⇒ 7 × x² + 7 × 9x − 7 × 13 = 7x² + 63x − 91

⇒ Subtract the second expanded expression from the first:

⇒ 7x² − 75x + 36 − (7x² + 63x − 91)

⇒ 7x² − 75x + 36 − 7x² − 63x + 91

⇒ Group the like terms together:

⇒ (7 − 7)x² + (−75 − 63)x + (36 + 91)

⇒ 0x² − 138x + 127

⇒ −138x + 127

∴ The correct answer is −138x + 127.

Additional Information

BODMAS Rule

Simplification of algebraic expressions follows the BODMAS rule: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction, resolved from left to right.

Distributive Property

The rule states that a(b + c) = ab + ac. It allows you to multiply a sum or difference by multiplying each addend separately and then bringing them together.

Like Terms

Only terms that have identical variable parts (like x, x², xy) can be added or subtracted to simplify the expression.

Shortcut Trick

Given, SP of 20 books = CP of 5 books

⇒ 20 × SP = 5 × CP

⇒ SP ÷ CP = 1 ÷ 4

Loss = CP − SP = 4 − 1 = 3 units

Loss % = (3 ÷ 4) × 100 = 75%

∴ The correct answer is 75% loss.

Alternate Method

Given:

Selling price of 20 books = Cost price of 5 books

Formula Used:

Loss = CP − SP

Loss % = (Loss ÷ CP) × 100

Cost Price (CP) = 4 units

SP = 1 unit

Loss = 3 units

Loss % = (3 ÷ 4) × 100 = 75%

Calculations:

Let SP be the selling price of 1 book and CP be the cost price of 1 book.

⇒ 20 × SP = 5 × CP

⇒ SP ÷ CP = 5 ÷ 20 = 1 ÷ 4

⇒ Let SP = 1 unit and CP = 4 units.

⇒ Loss = 4 − 1 = 3 units

⇒ Loss % = (3 ÷ 4) × 100

⇒ Loss % = 75%

∴ The correct answer is 75% loss.

Additional Information

Cost Price (CP) & Selling Price (SP)

CP is the price at which goods are bought. SP is the price at which goods are sold. Profit or loss is always calculated on the CP unless specifically stated otherwise.

Profit & Loss Formulas

Profit = SP − CP and Profit % = (Profit ÷ CP) × 100. Loss = CP − SP and Loss % = (Loss ÷ CP) × 100.

Marked Price (MP) & Discount

Discount is an allowance given on the Marked Price (MP) of an article. Discount = MP − SP. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

Total Work = LCM(12, 18) = 36 units.

Efficiency of A = 3 units/day, Efficiency of B = 2 units/day.

Combined work in 4 days = (3 + 2) × 4 = 20 units.

Remaining work = 36 − 20 = 16 units.

Fraction of work left = 16/36 = 4/9.

∴ The correct answer is 4/9.

Alternate Method

Given:

⇒ Time taken by A to complete the work = 12 days

⇒ Time taken by B to complete the work = 18 days

Formula Used:

⇒ Total Work = LCM of given days

⇒ Efficiency = Total Work ÷ Time

Calculations:

⇒ Total efficiency of (A + B) = 3 + 2 = 5 units/day.

⇒ Work completed by A and B working together for 4 days = 5 × 4 = 20 units.

⇒ Remaining work = Total Work − Completed Work = 36 − 20 = 16 units.

⇒ Fraction of work left = Remaining work ÷ Total Work = 16 ÷ 36 = 4/9.

∴ The correct answer is 4/9.

Additional Information

Total Work Formula

Total Work = Time Taken × Efficiency. Efficiency is defined as the work done per unit time.

Combined Work Concept

If person A can do a work in X days and person B can do it in Y days, their combined 1 day work is equal to 1/X + 1/Y.

Work and Wages

If workers complete a job together, the wages are divided in the ratio of the total work done by each individual.

Pipes and Cisterns

This topic follows the same principles as Time and Work, but note that outlet (emptying) pipes have a negative efficiency.

Shortcut Trick

In ΔABC, ∠BCA = 180º − (137º + 19º) = 24º.

Since BC || AD, the alternate interior angles are equal ⇒ ∠CAD = ∠BCA = 24º.

In ΔACD, given AC = CD, angles opposite to equal sides are equal ⇒ ∠CDA = ∠CAD = 24º.

Finally, ∠ACD = 180º − (24º + 24º) = 132º.

∴ The correct answer is 132º.

Alternate Method

Given:

ABCD is a trapezium with BC || AD.

AC = CD

∠ABC = 19º and ∠BAC = 137º

A
B
C
D
19º
137º
24º
24º
24º
132º

Calculations:

In ΔABC, the sum of all internal angles is 180º.

⇒ ∠ABC + ∠BAC + ∠BCA = 180º

⇒ 19º + 137º + ∠BCA = 180º

⇒ 156º + ∠BCA = 180º

⇒ ∠BCA = 180º − 156º = 24º

Since BC || AD, the alternate interior angles are equal.

⇒ ∠CAD = ∠BCA = 24º

Now, consider ΔACD. It is given that AC = CD.

Angles opposite to equal sides of a triangle are equal.

⇒ ∠CDA = ∠CAD = 24º

The sum of all internal angles in ΔACD is 180º.

⇒ ∠ACD + ∠CAD + ∠CDA = 180º

⇒ ∠ACD + 24º + 24º = 180º

⇒ ∠ACD + 48º = 180º

⇒ ∠ACD = 180º − 48º = 132º

∴ The correct answer is 132º.

Additional Information

Properties of Trapezium

A trapezium is a quadrilateral with at least one pair of parallel sides. The adjacent angles along the non-parallel sides sum up to 180º.

Alternate Interior Angles

When a transversal line intersects two parallel lines, the alternate interior angles formed on opposite sides of the transversal are equal.

Isosceles Triangle Properties

A triangle with two equal side lengths is called an isosceles triangle. The interior angles opposite to these equal sides are also strictly equal to each other.

Shortcut Trick

For divisibility by 11, difference of sum of alternate digits must be 0 or a multiple of 11.

Sum of odd placed digits = N + 3 + 9 = N + 12

Sum of even placed digits = 8 + M + 2 = M + 10

Equating their difference to 0: (M + 10) − (N + 12) = 0

⇒ M − N − 2 = 0 ⇒ M − N = 2

∴ The correct answer is M − N = 2.

Alternate Method

Given:

6-digit number: N83M92

The number is divisible by 11.

Formula Used:

A number is divisible by 11 if the difference of 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.

Calculation:

⇒ Sum of digits at odd places (from left) = N + 3 + 9 = N + 12

⇒ Sum of digits at even places (from left) = 8 + M + 2 = M + 10

⇒ Difference = (N + 12) − (M + 10)

⇒ Difference = N − M + 2

⇒ For the number to be divisible by 11, the difference must be equal to 0, 11, −11, etc.

⇒ Case 1: N − M + 2 = 0

⇒ M − N = 2 (This option is available)

⇒ Case 2: N − M + 2 = 11

⇒ N − M = 9 (This option is not available)

∴ The correct answer is M − N = 2.

Additional Information

Divisibility Rule for 3 and 9

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

Divisibility Rule for 4 and 8

A number is divisible by 4 if its last 2 digits form a number divisible by 4. It is completely divisible by 8 if its last 3 digits form a number divisible by 8.

Combined Divisibility Rule for 7, 11 and 13

For any large number, if the difference between the number formed by its last 3 digits and the remaining digits is a multiple of 7, 11, or 13, then the entire number is divisible by 7, 11, or 13 respectively.

Shortcut Trick

No profit or loss means the Selling Price (SP) is equal to the Cost Price (CP) = ₹3,500.

A discount of 20% means the SP is exactly 80% of the Marked Price (MP).

80% of MP = ₹3,500 ⇒ MP = 3500 ÷ 0.8 = ₹4,375.

∴ The correct answer is ₹4,375.

Alternate Method

Given:

Cost Price (CP) = ₹3,500

Discount = 20%

Profit/Loss = Nil

Formula Used:

Selling Price (SP) = CP + Profit (or − Loss)

MP = (SP × 100) ÷ (100 − Discount%)

Marked Price
(MP = 100%)

Selling Price
(SP = ₹3500)

−20% Discount

80% of MP = ₹3500

Calculations:

⇒ Since there is no profit or loss, the shopkeeper sells the bicycle exactly at the Cost Price.

⇒ Selling Price (SP) = CP = ₹3,500

⇒ We know that SP is calculated after allowing a discount on the Marked Price (MP).

⇒ SP = MP − (20% of MP) = 80% of MP

⇒ 80% of MP = 3500

⇒ (80 ÷ 100) × MP = 3500

⇒ MP = (3500 × 100) ÷ 80

⇒ MP = 350000 ÷ 80

⇒ MP = ₹4,375

∴ The correct answer is ₹4,375.

Additional Information

Marked Price & Discount

Discount is always offered strictly on the Marked Price (MP). The formula to find the Selling Price is SP = MP − Discount, which can also be written as SP = MP × (100 − Discount%) ÷ 100.

No Profit No Loss Concept

This situation typically occurs when the total selling price of goods exactly matches the total cost price (i.e., SP = CP). As a result, both Net Profit and Net Loss equal zero.

Direct Relation Between CP and MP

The ratio of Marked Price to Cost Price can be directly derived using the formula: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%). In the case of no profit, simply use Profit% = 0.

Shortcut Trick

Given expression: 10² + √(2²) − 4√25 − 3

Directly evaluate powers and square roots first:

= 100 + 2 − (4 × 5) − 3

= 102 − 20 − 3 = 79

∴ The correct answer is 79.

Alternate Method

Given:

Expression: 10² + √(2²) − 4√25 − 3

Formula Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

Solving the orders (powers and roots) first:

⇒ 100 + 2 − 4 × 5 − 3

Solving multiplication next:

⇒ 100 + 2 − 20 − 3

Solving addition from left to right:

⇒ 102 − 20 − 3

Solving subtraction successively:

⇒ 82 − 3

⇒ 79

∴ The correct answer is 79.

Additional Information

BODMAS Rule

BODMAS determines the sequence of operations for mathematical expressions: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Square Roots

The square root of a perfect square is an integer. For example, √25 = 5. Moreover, taking the square root of a squared positive number results in the number itself: √(x²) = x.

Exponents

An exponent indicates how many times a base number is multiplied by itself. For example, 10² means 10 × 10 = 100.

Shortcut Trick

For any triangle with a given base and height, the area is directly calculated as half of their product.

Area = (Base × Height) ÷ 2

Area = (12 × 8) ÷ 2 = 96 ÷ 2 = 48 cm².

∴ The correct answer is 48 cm².

Alternate Method

A
B
C
Base = 12 cm
Height = 8 cm

Given:

Base of the triangle (b) = 12 cm

Height of the triangle (h) = 8 cm

Formula Used:

Area of a triangle = 1/2 × Base × Height

Calculation:

⇒ Area = 1/2 × 12 × 8

⇒ Area = 6 × 8

⇒ Area = 48 cm²

∴ The correct answer is 48 cm².

Additional Information

Area of an Equilateral Triangle

For a triangle with all three sides equal (side = a), the area is calculated using the formula: Area = (√3 ÷ 4) × a².

Heron's Formula for Scalene Triangles

When all three sides (a, b, c) of a triangle are known, Area = √[s(s−a)(s−b)(s−c)], where the semi-perimeter s = (a + b + c) ÷ 2.

Right-Angled Triangle Area

If the triangle contains a 90° angle, the area is simply half the product of the legs forming the right angle: Area = 1/2 × Base × Perpendicular.

Shortcut Trick

Let the present ages of the father and son be F and S. We need to find 'x' years from now when F + x = 2(S + x), which simplifies to x = F − 2S.

From the condition 3 years ago: F − 3 = 2(S − 3) + 24.

Simplifying this directly gives: F − 2S = 21.

Since x = F − 2S, the required number of years 'x' is exactly 21.

∴ The correct answer is 21 years.

Alternate Method

3 Years Ago
Present
x Years Hence

F − 3 = 2(S − 3) + 24
F − 2S = 21

Ages: F, S

F + x = 2(S + x)
x = F − 2S

Substitute to get x = 21

Given:

Three years ago, the father was 24 years older than twice his son's age.

Let the present age of the Father be F and the Son be S.

Formula Used:

Age x years ago = Present Age − x

Age x years hence = Present Age + x

Calculations:

Father's age 3 years ago = F − 3

Son's age 3 years ago = S − 3

According to the given past condition:

⇒ F − 3 = 2 × (S − 3) + 24

⇒ F − 3 = 2S − 6 + 24

⇒ F − 3 = 2S + 18

⇒ F − 2S = 21    ... (Equation 1)

Let after 'x' years, the Father's age be twice the Son's age.

⇒ Father's future age = F + x

⇒ Son's future age = S + x

According to the required condition:

⇒ F + x = 2 × (S + x)

⇒ F + x = 2S + 2x

⇒ x = F − 2S

Substitute the value of (F − 2S) from Equation 1:

⇒ x = 21

∴ The correct answer is 21 years.

Additional Information

Age Timeline Principle

If the present age is P, then age 'n' years ago is (P − n) and age 'n' years hence is (P + n). Always base equations on the present ages to seamlessly connect different timelines.

Constant Difference Rule

The age difference between two individuals remains constant throughout their lifetime. If a father is 30 years older than his son today, he will remain 30 years older even 50 years later.

Linear Equations in Ages

Even when variables like individual ages cannot be uniquely determined, composite values (like F − 2S) can evaluate exactly to solve for a specific timeline shift, making it a system of dependent equations.

Shortcut Trick

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

Given equations: 12x + 7y = 285 and 11(x − 6) + 3(y + 5) = 118 ⇒ 11x + 3y = 169

Using cross-multiplication: x = (285 × 3 − 169 × 7) ÷ (12 × 3 − 11 × 7) = −328 ÷ −41 = 8

Substituting x in the second equation: 11 × 8 + 3y = 169 ⇒ 3y = 81 ⇒ y = 27

Original cost of 2 pens + 2 pencils = 2 × (8 + 27) = 2 × 35 = 70

∴ The correct answer is ₹70.

Alternate Method

Given:

Cost of 12 pens and 7 pencils = ₹285

Cost of 11 pens (decreased by ₹6) and 3 pencils (increased by ₹5) = ₹118

Calculation:

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

⇒ 12x + 7y = 285     ... (Equation 1)

After price change, the new cost of a pen = (x − 6) and a pencil = (y + 5)

⇒ 11(x − 6) + 3(y + 5) = 118

⇒ 11x − 66 + 3y + 15 = 118

⇒ 11x + 3y − 51 = 118

⇒ 11x + 3y = 169     ... (Equation 2)

Multiply Equation 1 by 3 and Equation 2 by 7 to eliminate y:

⇒ 36x + 21y = 855     ... (Equation 3)

⇒ 77x + 21y = 1183     ... (Equation 4)

Subtracting Equation 3 from Equation 4:

⇒ 77x − 36x = 1183 − 855

⇒ 41x = 328

⇒ x = 328 ÷ 41 = 8

Substitute the value of x in Equation 2:

⇒ 11(8) + 3y = 169

⇒ 88 + 3y = 169

⇒ 3y = 169 − 88 = 81

⇒ y = 81 ÷ 3 = 27

The original cost of a pen is ₹8 and a pencil is ₹27.

Now, calculate the required original cost of 2 pens and 2 pencils:

⇒ Cost = 2x + 2y = 2(8) + 2(27)

⇒ Cost = 16 + 54 = 70

∴ The correct answer is ₹70.

Additional Information

Linear Equations in Two Variables

The standard form is a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. The exact solution represents the point of intersection of two straight lines on a graph.

Cross-Multiplication Trick

For basic equations of form a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the direct value of x is solved as (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁).

Consistency of Equations

If a₁/a₂ ≠ b₁/b₂, the system has a unique solution. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and have no solution.

Shortcut Trick

Since the rate is given per month and time is in months, Total Effective Rate = 4% × 9 = 36%.

Simple Interest = 36% of ₹1200

⇒ SI = (36 ÷ 100) × 1200

⇒ SI = 432

∴ The correct answer is ₹432.

Alternate Method

Given:

Principal (P) = ₹1200

Rate of Interest (R) = 4% per month

Time (T) = 9 months

Principal
₹1200

Total Rate
4% × 9 = 36%

Simple Interest
₹432

Formula Used:

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

Calculation:

⇒ SI = (1200 × 4 × 9) ÷ 100

⇒ SI = (1200 × 36) ÷ 100

⇒ SI = 12 × 36

⇒ SI = 432

∴ The correct answer is ₹432.

Additional Information

Total Amount Formula

Total Amount = Principal + Simple Interest, which can be written as A = P[1 + (RT ÷ 100)].

Unit Consistency Rule

Always ensure that the unit of Time (T) matches the unit of the Rate of Interest (R). If the rate is given per month, the time must strictly be evaluated in months.

Compound Interest

Unlike Simple Interest, Compound Interest is calculated on the principal plus previously accumulated interest. Formula: CI = P(1 + R ÷ 100)^T − P.

Shortcut Trick

Cost price of 1 pencil = ₹180 ÷ 12 = ₹15

Selling price of 1 pencil = ₹17

Profit on 1 pencil = ₹17 − ₹15 = ₹2

Profit % = (Profit ÷ CP) × 100 = (2 ÷ 15) × 100 = 13.33%

∴ The correct answer is 13.33%.

Alternate Method

Given:

Cost price of a dozen (12 pencils) = ₹180

Selling price of 1 pencil = ₹17

Formula Used:

Cost Price (CP) of 1 unit = Total CP ÷ Quantity

Profit = Selling Price (SP) − Cost Price (CP)

Profit Percentage = (Profit ÷ CP) × 100

CP = ₹15

SP = ₹17

• ₹2 (Profit)

Profit % = (2 ÷ 15) × 100 = 13.33%

Calculations:

⇒ CP of 1 pencil = 180 ÷ 12 = ₹15

⇒ Profit earned on 1 pencil = 17 − 15 = ₹2

⇒ Profit % = (2 ÷ 15) × 100

⇒ Profit % = 200 ÷ 15 = 40 ÷ 3 = 13.333... %

⇒ Rounding off to two decimal places, Profit % ≈ 13.33%

∴ The correct answer is 13.33%.

Additional Information

Loss Percentage

When the cost price (CP) is greater than the selling price (SP), a loss is incurred. Loss = CP − SP. Loss % = (Loss ÷ CP) × 100.

Marked Price & Discount

Discount is an allowance given by the seller to the buyer on the marked price of the goods. SP = MP − Discount. Discount % = (Discount ÷ MP) × 100.

Dishonest Dealer

When a dealer claims to sell goods at cost price but uses false weights, he makes a profit. Profit % = (Error ÷ True Weight − Error) × 100.