Quantitative Aptitude PYQs

Shortcut Trick

If cosecθ − cotθ = 1/11, then cosecθ + cotθ = 11.

Adding both: 2 cosecθ = 11 + 1/11 = 122/11 ⇒ cosecθ = 61/11.

Subtracting both: 2 cotθ = 11 − 1/11 = 120/11 ⇒ cotθ = 60/11.

cosθ = cotθ / cosecθ = (60/11) / (61/11) = 60/61.

122 cosθ = 122 × (60/61) = 2 × 60 = 120.

∴ The correct answer is 120.

Alternate Method

Given: cosecθ − cotθ = 1/11

Formula Used: cosec²θ − cot²θ = 1 ⇒ (cosecθ − cotθ)(cosecθ + cotθ) = 1

⇒ (1/11) × (cosecθ + cotθ) = 1

⇒ cosecθ + cotθ = 11

⇒ (cosecθ + cotθ) + (cosecθ − cotθ) = 11 + 1/11

⇒ 2 cosecθ = 122/11 ⇒ cosecθ = 61/11

⇒ sinθ = 1/cosecθ = 11/61

⇒ cosθ = √(1 − sin²θ) = √(1 − (11/61)²) = √((3721 − 121)/3721) = √(3600/3721) = 60/61

⇒ Value = 122 × (60/61) = 120

∴ The correct answer is 120.

Additional Information

Trigonometric Identity

The identity 1 + cot²θ = cosec²θ implies that cosecθ + cotθ and cosecθ − cotθ are reciprocals of each other.

Pythagorean Triplets

Common triplets like (11, 60, 61) satisfy a² + b² = c² and are frequently used in right-angled triangle problems.

Secant and Tangent Relation

Similarly, sec²θ − tan²θ = 1, which means secθ + tanθ = 1 / (secθ − tanθ).

Shortcut Trick

Age ratio (Teacher : Student) Now = 4 : 1 (diff = 3)

Age ratio 5 years ago = 7 : 1 (diff = 6)

Equalize difference: Now (4:1)×2 = 8 : 2. Ratio change = 8−7 = 1 unit.

1 unit = 5 years. Teacher current age = 8 × 5 = 40 years.

∴ The correct answer is 40 years.

Alternate Method

Given: Current age ratio is 4:1 and 5 years ago it was 7:1.

Let the current age of the student be x.

⇒ Teacher current age = 4x

⇒ 5 years ago: (4x − 5) = 7 × (x − 5)

⇒ 4x − 5 = 7x − 35

⇒ 3x = 30

⇒ x = 10

⇒ Teacher current age = 4 × 10 = 40 years

∴ The correct answer is 40 years.

Additional Information

Ratio Difference Concept

The age difference between two people remains constant over time. Always equate the difference in ratio parts to solve faster.

Linear Equations in Ages

For "n" years ago, subtract n from current ages. For "n" years hence, add n to current ages.

Shortcut Trick

For two numbers in the ratio a : b with HCF = H, the LCM is given by H × a × b.

Given: Ratio = 2 : 3 and HCF = 6.

⇒ LCM = 6 × 2 × 3

⇒ LCM = 36

∴ The correct answer is 36.

Alternate Method

Given:

Ratio of numbers = 2 : 3

HCF = 6

Calculations:

⇒ Let the numbers be 2x and 3x.

⇒ Since HCF is the common factor x, we have x = 6.

⇒ First number = 2 × 6 = 12

⇒ Second number = 3 × 6 = 18

⇒ LCM of 12 and 18:

⇒ 12 = 2² × 3

⇒ 18 = 2 × 3²

⇒ LCM = 2² × 3² = 4 × 9 = 36

∴ The correct answer is 36.

Additional Information

Product Rule of HCF and LCM

For any two numbers A and B, Product of Numbers = HCF × LCM. (A × B = H × L)

Co-prime Numbers Property

If two numbers are co-prime (HCF = 1), then their LCM is simply the product of the two numbers.

LCM of Fractions

LCM of Fractions = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

Total Work (LCM of 18, 12, 48) = 144 units.

Efficiency of (A+B) = 8, (A+C) = 12, (B+C) = 3.

Combined efficiency 2(A+B+C) = 8 + 12 + 3 = 23 ⇒ (A+B+C) = 11.5.

Time taken = 144 ÷ 11.5 ≈ 12.52 days.

∴ The correct answer is 12.52 days.

Alternate Method

Given:

(A + B) can do work in 18 days.

(A + C) can do work in 12 days.

(B + C) can do work in 48 days.

Calculations:

⇒ Total efficiency of (A+B) + (A+C) + (B+C) = 8 + 12 + 3 = 23

⇒ 2(A + B + C) = 23

⇒ Efficiency of (A + B + C) = 23 ÷ 2 = 11.5

⇒ Time taken by A, B, and C together = Total Work ÷ Efficiency

⇒ Time = 144 ÷ 11.5

⇒ Time = 12.5217... ≈ 12.52 days

∴ The correct answer is 12.52 days.

Additional Information

Combined Efficiency Rule

When pairs of workers are given, the sum of their efficiencies equals twice the efficiency of the individuals working together: 2(A+B+C) = Eff(A+B) + Eff(B+C) + Eff(C+A).

Work and Days Relationship

Time taken is inversely proportional to efficiency. Time = Total Work ÷ Efficiency.

LCM Method for Work

Taking the Least Common Multiple of given time periods as the Total Work simplifies calculations by providing integer efficiency values.

Shortcut Trick

Convert all percentages and fractions into a single multiplying factor.

Total product factor = 1/10 × 3/10 × 2/3 = 1/50.

Number × 1/50 = 537.

Number = 537 × 50 = 26850.

∴ The correct answer is 26850.

Alternate Method

Given:

10% of 30% of 2/3 of x = 537.

Calculations:

⇒ (10/100) × (30/100) × (2/3) × x = 537

⇒ (1/10) × (3/10) × (2/3) × x = 537

⇒ (1 × 3 × 2) / (10 × 10 × 3) × x = 537

⇒ (6/300) × x = 537

⇒ (1/50) × x = 537

⇒ x = 537 × 50

⇒ x = 26850

∴ The correct answer is 26850.

Additional Information

Percentage to Fraction Conversion

To convert x% to a fraction, divide by 100 (e.g., 20% = 20/100 = 1/5).

Successive Percentage Calculation

When multiple percentages are applied to a number, they are multiplied together as fractions: Result = N × (P₁/100) × (P₂/100).

Reverse Calculation

To find the original number when the final value is known, multiply the result by the reciprocal of the combined fraction.

Shortcut Trick

Marked Price (MP) is 100%.

After a 20% discount, the Selling Price (SP) is 80% of MP.

SP = 8,460 × 0.80 = 6,768.

∴ The correct answer is ₹6,768.

Alternate Method

Given: Marked Price (MP) = ₹8,460, Discount% = 20%

Formulas: Discount = (Discount% ÷ 100) × MP | SP = MP − Discount

⇒ Discount = (20 ÷ 100) × 8,460 = 0.2 × 8,460 = ₹1,692

⇒ Selling Price (SP) = 8,460 − 1,692 = ₹6,768

∴ The correct answer is ₹6,768.

Additional Information

Marked Price & Discount

The Marked Price is the price printed on the label. Discount is always calculated on the Marked Price unless stated otherwise. Formula: SP = MP × (100 − D%) ÷ 100.

Relation between CP and MP

The ratio of MP to CP is given by: MP ÷ CP = (100 + P%) ÷ (100 − D%), where P% is profit percentage and D% is discount percentage.

Shortcut Trick

Total Cost Price (CP) of 9 articles = ₹16.

Total Selling Price (SP) of 9 articles = 9 × 2.4 = ₹21.6.

Profit = 21.6 − 16 = ₹5.6.

Profit% = (Profit ÷ CP) × 100 = (5.6 ÷ 16) × 100.

Profit% = 0.35 × 100 = 35%.

∴ The correct answer is 35%.

Alternate Method

Given: Total CP = ₹16, Number of articles = 9, SP per article = ₹2.4.
Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100.

⇒ Calculate total Selling Price: 9 articles × ₹2.4/article = ₹21.6.
⇒ Calculate Profit amount: ₹21.6 − ₹16 = ₹5.6.
⇒ Apply Profit percentage formula: (5.6 ÷ 16) × 100.
⇒ Profit% = 35%.

∴ The correct answer is 35%.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price (SP) is greater than Cost Price (CP). Loss occurs when CP is greater than SP.

Percentage Calculation

Profit or Loss percentage is always calculated on the Cost Price unless otherwise specified in the problem.

Shortcut Trick

Age ratio (Present) Father : Son = 3 : 1 (Difference = 2 units)

Age ratio (After 12 years) Father : Son = 2 : 1 (Difference = 1 unit)

Equalize differences by multiplying the second ratio by 2 → 4 : 2.

Increase in units = 4 − 3 = 1 unit → 12 years.

Present ages: Father = 3 × 12 = 36 years; Son = 1 × 12 = 12 years.

∴ The correct answer is 36 and 12 years.

Alternate Method

Given: Present age ratio is 3:1 and ratio after 12 years is 2:1.

Formula: Age after n years = Present Age + n.

⇒ Let Son's present age = x years and Father's present age = 3x years.

⇒ According to the question: (3x + 12) = 2 × (x + 12)

⇒ 3x + 12 = 2x + 24

⇒ 3x − 2x = 24 − 12

⇒ x = 12 years (Son's age)

⇒ Father's age = 3 × 12 = 36 years.

∴ The correct answer is 36 and 12 years.

Additional Information

Age Difference Constant

The difference between the ages of two persons remains constant throughout their lives, regardless of the number of years passed.

Ratio Balancing

To solve age problems quickly, multiply the ratios so that the numerical difference between the ratio parts is equal for both individuals over time.

Shortcut Trick

Assume an Assumed Mean (A) = 60.

Find deviations from 60: +28, +39, −13, +19, −48, +19, +16, −43, +1.

Sum of deviations = (28 + 39 + 19 + 19 + 16 + 1) − (13 + 48 + 43) = 122 − 104 = +18.

Actual Mean = Assumed Mean + (Sum of Deviations ÷ Total Count) = 60 + (18 ÷ 9) = 62.

∴ The correct answer is 62.

Alternate Method

Given Observations: 88, 99, 47, 79, 12, 79, 76, 17, 61

Number of observations (n) = 9

Formula: Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum = 88 + 99 + 47 + 79 + 12 + 79 + 76 + 17 + 61

⇒ Sum = 558

⇒ Mean = 558 ÷ 9

⇒ Mean = 62

∴ The correct answer is 62.

Additional Information

Sum of Deviations

The sum of the deviations of a set of observations from their arithmetic mean is always zero: ∑(x_i − Mean) = 0.

Change of Origin and Scale

If every observation is increased/decreased by a constant k, the mean also increases/decreases by k. If multiplied by k, the mean is also multiplied by k.

Mean of First n Natural Numbers

The arithmetic mean of the first n natural numbers is given by the formula: Mean = (n + 1) ÷ 2.

Shortcut Trick

Convert all times to hours: 12 min = 12/60 hr, 22 min = 22/60 hr, remaining 16 min = 16/60 hr.

Total distance = ∑ (Speed × Time) ⇒ 30 = x×(12/60) + 2x×(22/60) + x×(16/60).

30 = (12x + 44x + 16x) ÷ 60 ⇒ 30 = 72x ÷ 60 ⇒ 30 = 1.2x.

x = 30 ÷ 1.2 = 25 km/hr.

∴ The correct answer is 25 km/hr.

Alternate Method

Given: Total distance = 30 km, Total time = 50 min. Journey segments: 12 min at x, 22 min at 2x, and (50 − 12 − 22) = 16 min at x.

Formula Used: Distance = Speed × Time

Calculations:

⇒ Distance₁ = x × (12/60) km

⇒ Distance₂ = 2x × (22/60) km

⇒ Distance₃ = x × (16/60) km

⇒ Total Distance = (12x/60) + (44x/60) + (16x/60) = 30

⇒ (12x + 44x + 16x) / 60 = 30

⇒ 72x / 60 = 30

⇒ 1.2x = 30

⇒ x = 30 / 1.2 = 25

∴ The correct answer is 25 km/hr.

Additional Information

Speed-Distance Relation

Distance is directly proportional to speed when time is constant and directly proportional to time when speed is constant.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert minutes to hours, divide by 60.

Average Speed

Average Speed = Total Distance ÷ Total Time. In this case, Average Speed = 30 km ÷ (50/60) hr = 36 km/hr.

Shortcut Trick

The multiplying factors for successive discounts of 10% and 5% are 9/10 and 19/20 respectively.

Selling Price (SP) = Marked Price (MP) × Discount Multipliers

SP = 18000 × (9/10) × (19/20)

SP = 18000 × (171/200) = 90 × 171 = 15390

∴ The correct answer is Rs. 15,390.

Alternate Method

Given:

Marked Price (MP) = Rs. 18,000

First successive discount (D₁) = 10%

Second successive discount (D₂) = 5%

Formula Used:

Selling Price (SP) = MP × (1 − D₁/100) × (1 − D₂/100)

₹18,000
Marked Price

−10%

₹16,200
After 1^st Discount

−5%

₹15,390
Selling Price

Calculations:

⇒ SP = 18000 × (1 − 10/100) × (1 − 5/100)

⇒ SP = 18000 × (90/100) × (95/100)

⇒ SP = 18000 × (9/10) × (19/20)

⇒ SP = 18000 × 171/200

⇒ SP = 90 × 171

⇒ SP = 15390

∴ The correct answer is Rs. 15,390.

Additional Information

Successive Discount Formula

When two successive discounts of a% and b% are given, the equivalent single discount percentage is calculated as (a + b − ab/100)%.

Marked Price vs Cost Price

Marked price (MP) is derived by adding a markup percentage to the cost price: MP = CP × (1 + Markup/100). Discounts are always applied on the Marked Price, never on the Cost Price.

Selling Price Relation

Selling Price connects the manufacturer to the buyer. It can be found from MP via discounts: SP = MP × (1 − Discount/100), or from CP via profits: SP = CP × (1 ± Profit/Loss/100).

Shortcut Trick

The square root of a decimal has exactly half the number of decimal places as the original number.

√256 = 16

√0.0256 = 0.16 (4 decimal places → 2 decimal places)

√4.41 = 2.1 (2 decimal places → 1 decimal place)

Expression = 16 + 0.16 − 2.1 = 16.16 − 2.10 = 14.06

∴ The correct answer is 14.06.

Alternate Method

Given:

Expression = √256 + √0.0256 − √4.41

Calculations:

⇒ Expression = √256 + √(256 / 10000) − √(441 / 100)

⇒ Expression = 16 + (16 / 100) − (21 / 10)

⇒ Expression = 16 + 0.16 − 2.1

⇒ Expression = 16.16 − 2.10

⇒ Expression = 14.06

∴ The correct answer is 14.06.

Additional Information

Square Roots of Decimals

When extracting the square root of a perfect square decimal number, count the total decimal places. The square root will contain exactly half as many decimal places. For example, √0.0064 has 4 decimal places, so its root is 0.08 with 2 decimal places.

Fractional Conversion Method

Decimals can be easily converted to fractions with denominators of 10, 100, 1000 etc. to avoid errors during calculation. Example: √4.41 = √(441 / 100) = 21 / 10 = 2.1.

BODMAS Rule

In any simplification, operations must be strictly performed in the sequential order of Brackets, Orders (Roots/Indices), Division, Multiplication, Addition, and finally Subtraction.

Shortcut Trick

For a number to be divisible by 14, it must be perfectly divisible by both 2 and 7.

Eliminate Option 2 (6436639) immediately as it is an odd number and not divisible by 2.

Check the remaining even options for divisibility by 7 directly.

For Option 3: 7048048 ÷ 7 = 1006864 (Exact division, remainder is 0).

Since 7048048 is divisible by both 2 and 7, it is exactly divisible by 14.

∴ The correct answer is 7048048.

Alternate Method

Given:

Numbers to be checked: 6794266, 6436639, 7048048, 7132968

Concept Used:

A number is divisible by a composite number if it is divisible by all its co-prime factors.

The co-prime factors of 14 are 2 and 7 (Since 2 × 7 = 14 and their HCF is 1).

Calculations:

⇒ Divisibility by 2: The unit digit must be an even number (0, 2, 4, 6, 8).

⇒ 6436639 ends in 9, so it is not divisible by 2. This option is eliminated.

⇒ Divisibility by 7: Divide the remaining numbers by 7 to check for a zero remainder.

⇒ Option 1: 6794266 ÷ 7 = 970609 with a remainder of 3. (Not divisible)

⇒ Option 3: 7048048 ÷ 7 = 1006864 with a remainder of 0. (Divisible)

⇒ Option 4: 7132968 ÷ 7 = 1018995 with a remainder of 3. (Not divisible)

⇒ Only 7048048 satisfies the divisibility rules for both 2 and 7.

∴ The correct answer is 7048048.

Additional Information

Composite Divisibility

To test divisibility by composite numbers like 12, 14, 15, or 72, break the number into co-prime factors and test for each factor individually (e.g., for 72 check divisibility by both 8 and 9).

Divisibility Rule of 7

Remove the last digit, double it, and subtract it from the remaining truncated number. If the result is 0 or divisible by 7, the original number is too.

Divisibility Rule of 11

Find the sum of digits at odd places and the sum of digits at even places. If their difference is 0 or perfectly divisible by 11, the number is divisible by 11.

Divisibility Rule of 13

Remove the last digit, multiply it by 4, and add it to the remaining truncated number. If the result is divisible by 13, the original number is divisible by 13.

Shortcut Trick

Salman's profit = 16000 − 6400 = ₹9600

Profit ratio (Vivek : Salman) = 6400 : 9600 = 2 : 3

Since time is equal, Investment ratio = Profit ratio = 2 : 3

Total investment = 5 parts = ₹45,200

Difference = 1 part = 45200 ÷ 5 = ₹9,040

∴ The correct answer is ₹9,040.

Alternate Method

Given:

Total Investment = ₹45,200

Total Profit = ₹16,000

Vivek's Profit = ₹6,400

Formula Used:

Profit Ratio = Investment Ratio × Time Ratio

When time periods are equal, Profit Ratio = Investment Ratio

Profit Ratio = Investment Ratio (Same Time Period)

Vivek Profit: ₹6400
Salman Profit: ₹9600

Total Investment = ₹45,200 (5 parts)

Investment: 2 parts
Investment: 3 parts

Diff = 1 part

1 part = ₹45,200 ÷ 5 = ₹9,040

Calculations:

⇒ Salman's Profit = Total Profit − Vivek's Profit

⇒ Salman's Profit = 16000 − 6400 = ₹9,600

⇒ Ratio of Vivek's Profit to Salman's Profit = 6400 : 9600 = 2 : 3

⇒ Since their time periods are the same, the ratio of their investments is equal to the ratio of their profits.

⇒ Vivek's Investment : Salman's Investment = 2 : 3

⇒ Let Vivek's investment be 2x and Salman's investment be 3x.

⇒ Total Investment = 2x + 3x = 5x

⇒ 5x = 45200

⇒ x = 45200 ÷ 5 = 9040

⇒ Difference in their investments = 3x − 2x = x

⇒ Difference = ₹9,040

∴ The correct answer is ₹9,040.

Additional Information

Compound Partnership

When partners invest their capital for different time periods, their profit is divided in the ratio of the product of their capital and respective time periods (C₁ × T₁ : C₂ × T₂).

Sleeping Partner

A partner who only invests money but does not actively manage the business. Their share of profit is strictly based on the capital invested and the time period.

Joining Mid-way

If a partner joins the business after a few months, their time period of investment is calculated only for the remaining duration until the end of the partnership term.

Shortcut Trick

When Simple Interests from two investments are equal for the same time period, their principals are inversely proportional to their rates.

Ratio of Principals P₁ : P₂ = R₂ : R₁ = 11 : 14

Total ratio parts = 11 + 14 = 25 parts.

Value of 25 parts = ₹20,400 ⇒ 1 part = ₹816.

Sum invested at 14% (11 parts) = 11 × 816 = 8976.

∴ The correct answer is ₹8,976.

Alternate Method

Given:

Total Principal = ₹20,400

Rate 1 (R₁) = 14% p.a., Time 1 (T₁) = 9 years

Rate 2 (R₂) = 11% p.a., Time 2 (T₂) = 9 years

Formula Used:

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

Total Sum = ₹20,400

Part 1 at 14%
Time = 9 years

Part 2 at 11%
Time = 9 years

SI1 = SI2

Ratio Part 1 : Part 2 = 11 : 14

Calculation:

Let the sum invested at 14% be P₁ and at 11% be P₂.

⇒ SI₁ = (P₁ × 14 × 9) ÷ 100

⇒ SI₂ = (P₂ × 11 × 9) ÷ 100

According to the question, equal interest is earned:

⇒ (P₁ × 14 × 9) ÷ 100 = (P₂ × 11 × 9) ÷ 100

Cancelling 9 and 100 from both sides:

⇒ P₁ × 14 = P₂ × 11

⇒ P₁ / P₂ = 11 / 14

⇒ Total parts = 11 + 14 = 25 parts

⇒ 25 parts = 20,400

⇒ 1 part = 20,400 ÷ 25 = 816

⇒ P₁ (sum at 14%) = 11 parts = 11 × 816 = 8,976

∴ The correct answer is ₹8,976.

Additional Information

When Simple Interests are Equal

If a certain sum is divided into multiple parts such that their Simple Interests are equal, the ratio of their principals is P₁ : P₂ : P₃ = 1/(R₁T₁) : 1/(R₂T₂) : 1/(R₃T₃).

When Amounts are Equal

If a certain sum is divided into parts such that their Amounts (Principal + SI) are equal, the ratio of their principals is P₁ : P₂ : P₃ = 1/(100 + R₁T₁) : 1/(100 + R₂T₂) : 1/(100 + R₃T₃).

Difference between SI and CI

For 2 years: Difference = P × (R ÷ 100)². Simple Interest is calculated only on the initial principal, whereas Compound Interest is calculated on the previously accumulated amount as well.

Shortcut Trick

For operations involving the same base, combine the exponents directly using the laws of indices from left to right.

Division translates to subtraction of powers, and multiplication translates to addition of powers.

Power of 26 = −19 − 18 + 3

⇒ Power = −37 + 3 = −34

∴ The correct answer is 26⁻³⁴.

Alternate Method

Given:

Expression: 26⁻¹⁹ ÷ 26¹⁸ × 26³

Formula Used:

Law of Division: a^m ÷ aⁿ = a^m−n

Law of Multiplication: a^m × aⁿ = a^m+n

Calculation:

According to the BODMAS rule, division and multiplication share the same priority. Therefore, we evaluate them sequentially from left to right.

⇒ (26⁻¹⁹ ÷ 26¹⁸) × 26³

⇒ 26^{(−19 − 18)} × 26³

⇒ 26⁻³⁷ × 26³

⇒ 26^{(−37 + 3)}

⇒ 26⁻³⁴

∴ The correct answer is 26⁻³⁴.

Additional Information

BODMAS Rule

When an expression contains both division and multiplication, they have the same priority level. We must evaluate the operations consecutively from left to right to get the correct mathematical result.

Laws of Exponents

Power of a Power: (a^m)ⁿ = a^m×n

Zero Exponent: a⁰ = 1 (where a is any non-zero number)

Negative Exponent Rule

A negative exponent indicates taking the reciprocal of the base raised to a positive exponent: a⁻ⁿ = 1 ÷ aⁿ.

Shortcut Trick

The surface area of a sphere directly in terms of diameter (d) is given by Area = πd².

Given, d = 14 cm.

Area = π × (14)²

Area = 196π cm²

∴ The correct answer is 196π cm².

Alternate Method

r = 7 cm

Diameter (d) = 14 cm

Given:

Diameter of the sphere (d) = 14 cm

Formula:

Radius (r) = d ÷ 2

Surface area of a sphere = 4πr²

Calculation:

⇒ Radius (r) = 14 ÷ 2 = 7 cm

⇒ Surface Area = 4 × π × (7)²

⇒ Surface Area = 4 × π × 49

⇒ Surface Area = 196π cm²

∴ The correct answer is 196π cm².

Additional Information

Volume of a Sphere

The total 3D space enclosed by a sphere is given by V = (4÷3)πr³, where r is the radius.

Hemisphere Properties

A hemisphere is exactly half of a sphere. Its volume is exactly half of the sphere's volume, V = (2÷3)πr³.

Surface Areas of a Hemisphere

Curved Surface Area (CSA) = 2πr². Total Surface Area (TSA) = 3πr² (since it includes the flat circular base area of πr²).

Shortcut Trick

Use percentage swap trick: 65% of 75 = 75% of 65 = (3/4) × 65 = 48.75.

Direct fraction calculation: 2/5 of 55 = 2 × 11 = 22.

Required difference = 48.75 − 22 = 26.75.

∴ The correct answer is 26.75.

Alternate Method

Given:

First value = 65% of 75

Second value = 2/5 of 55

65% of 75 = 48.75

2/5 of 55 = 22

Difference = 26.75

Calculations:

⇒ First value = (65 / 100) × 75

⇒ First value = 0.65 × 75 = 48.75

⇒ Second value = (2 / 5) × 55

⇒ Second value = 2 × 11 = 22

⇒ Difference = 48.75 − 22

⇒ Difference = 26.75

∴ The correct answer is 26.75.

Additional Information

Percentage Interchangeability

x% of y is mathematically equal to y% of x. For example, 65% of 75 is the exact same as 75% of 65. Since 75% translates to a standard fraction (3/4), utilizing this swap makes mental calculation much easier.

Fraction of a Number

To find a fraction a/b of a number N, first divide N by the denominator b to find the value of one part, then multiply the result by the numerator a. Example: 2/5 of 55 → 55 ÷ 5 = 11, then 11 × 2 = 22.

BODMAS Rule for Expressions

The word 'of' mathematically behaves like multiplication but holds higher precedence than division in the standard BODMAS hierarchy. It must be resolved entirely before any regular division or multiplication operators in an equation.

Shortcut Trick

In right-angled △BDC, ∠C = 90° − ∠DBC = 90° − 25° = 65°.

By the Exterior Angle Theorem in △AEC, the exterior angle at E is equal to the sum of the two opposite interior angles.

⇒ ∠AEB = ∠CAE + ∠C

⇒ ∠AEB = 80° + 65° = 145°.

∴ The correct answer is 145°.

Alternate Method

Given:

BD ⊥ AC at D, so ∠BDC = 90°

∠DBC = 25°

∠CAE = 80°

A

B

C

D

E

25°
65°
80°
145°

Step 1: Find ∠C in △BDC

The sum of angles in a triangle is 180°.

⇒ ∠BDC + ∠DBC + ∠C = 180°

⇒ 90° + 25° + ∠C = 180°

⇒ 115° + ∠C = 180°

⇒ ∠C = 180° − 115° = 65°

Step 2: Find ∠AEC in △AEC

The sum of angles in △AEC is 180°.

⇒ ∠CAE + ∠C + ∠AEC = 180°

⇒ 80° + 65° + ∠AEC = 180°

⇒ 145° + ∠AEC = 180°

⇒ ∠AEC = 180° − 145° = 35°

Step 3: Find ∠AEB

Since points B, E, and C lie on a continuous straight line segment BC, ∠AEB and ∠AEC are supplementary.

⇒ ∠AEB + ∠AEC = 180°

⇒ ∠AEB + 35° = 180°

⇒ ∠AEB = 180° − 35° = 145°

∴ The correct answer is 145°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. It offers a direct shortcut for deducing an angle without finding the third interior angle first.

Angle Sum Property

The sum of all interior angles in any planar triangle is always strictly equal to 180°.

Linear Pair Axiom

When two adjacent angles form a straight line, their sum is equal to 180°. These angles are termed as supplementary angles.

Shortcut Trick

The product of two consecutive positive integers is close to the square of the smaller number.

We know that 28² = 784 and 29² = 841.

Since 812 lies between 784 and 841, the two consecutive numbers must be 28 and 29.

The greater of the two numbers is 29.

∴ The correct answer is 29.

Alternate Method

Given:

Product of two consecutive positive natural numbers = 812

Formula Used:

If x is the smaller number, the next consecutive number is (x + 1).

Calculations:

⇒ x × (x + 1) = 812

⇒ x² + x − 812 = 0

⇒ x² + 29x − 28x − 812 = 0

⇒ x(x + 29) − 28(x + 29) = 0

⇒ (x − 28)(x + 29) = 0

⇒ x = 28 or x = −29

Since the numbers are positive natural numbers, we take x = 28.

The greater number = x + 1 = 28 + 1 = 29.

∴ The correct answer is 29.

Additional Information

Consecutive Numbers

Numbers that follow each other continuously in the order from smallest to largest. For integers, they are represented as n, n+1, n+2, etc.

Product of Consecutive Integers

The product of any two consecutive integers is always an even number, because one of the two numbers is always a multiple of 2.

Quadratic Roots Formula

For a quadratic equation ax² + bx + c = 0, the roots can be calculated using the formula: x = [−b ± √(b² − 4ac)] ÷ 2a.