Quantitative Aptitude PYQs

Shortcut Trick

Simplify inner values first

24² − 16² + 2² = 576 − 256 + 4 = 324 ⇒ √324 = 18

2²×3² + √169 = 4×9 + 13 = 36 + 13 = 49 ⇒ √49 = 7

Total = √(18 + 7) = √25

∴ Answer = 5

Alternate Method

Given:

√( √(24² − 16² + 2²) + √(2² × 3² + √169) )

Concept Used:

BODMAS rule

Calculation:

⇒ √( √(576 − 256 + 4) + √(4 × 9 + √169) )

⇒ √( √(320 + 4) + √(36 + √169) )

⇒ √( √324 + √(36 + 13) )

⇒ √( 18 + √49 )

⇒ √( 18 + 7 )

⇒ √25

⇒ 5

∴ The correct answer is 5.

Additional Information

BODMAS Rule

Order: Brackets → Orders (powers, roots) → Division → Multiplication → Addition → Subtraction

Perfect Squares

Numbers like 324 and 49 are perfect squares: √324 = 18, √49 = 7

Nested Roots

Solve inner square roots first, then move outward step-by-step

Simplification Trick

Always try to convert expressions into perfect squares for quick evaluation

Shortcut Trick

Let the cost price for Poonam be x.

Using the multiplier method: x × (95 ÷ 100) × (108 ÷ 100) = 1,539.

x × 0.95 × 1.08 = 1,539.

x × 1.026 = 1,539.

x = 1,539 ÷ 1.026 = 1,500.

∴ The correct answer is ₹1,500.

Alternate Method

Given: Loss% of Poonam = 5%, Profit% of Mohini = 8%, Price paid by Rupashi = ₹1,539.

Formulas: SP = CP × (100 ± P/L%) ÷ 100

⇒ Let Poonam cost price = CP₁.

⇒ Mohini cost price (SP of Poonam) = CP₁ × (95 ÷ 100).

⇒ Rupashi cost price (SP of Mohini) = [CP₁ × (95 ÷ 100)] × (108 ÷ 100) = 1,539.

⇒ CP₁ × (19 ÷ 20) × (27 ÷ 25) = 1,539.

⇒ CP₁ × (513 ÷ 500) = 1,539.

⇒ CP₁ = (1,539 × 500) ÷ 513 = 3 × 500 = 1,500.

∴ The correct answer is ₹1,500.

Additional Information

Chain Rules in Profit and Loss

When an item is passed through multiple hands, the final cost price is calculated by multiplying the initial cost price with successive profit/loss factors.

Successive Percentage Change

If there is a loss of x% followed by a gain of y%, the net change is given by [y − x − (xy ÷ 100)]%.

Shortcut Trick

Total Capacity (LCM of 4, 6, 12) = 12 units.

Efficiencies: A = +3, B = +2, C = −1.

In 1 cycle (3 hours), net work done = 3 + 2 − 1 = 4 units.

After 2 cycles (6 hours), work done = 4 × 2 = 8 units.

Remaining 4 units: A fills 3 units in 1 hour; B fills remaining 1 unit in 1/2 hour.

Total time = 6 + 1 + 1/2 = 7 ½ hours.

∴ The correct answer is 7 ½ hr.

Alternate Method

Given: Filling time for Pipe A = 4 hr, Pipe B = 6 hr. Emptying time for Pipe C = 12 hr.

⇒ Total work in 1 cycle (3 hours) = Efficiency (A + B + C)

⇒ Work in 3 hours = 3 + 2 − 1 = 4 units

⇒ Work in 6 hours (2 cycles) = 4 × 2 = 8 units

⇒ Remaining work = 12 − 8 = 4 units

⇒ On 7^th hour, Pipe A fills 3 units. Remaining = 4 − 3 = 1 unit.

⇒ On 8^th hour, Pipe B fills 1 unit. Time taken = 1 unit ÷ 2 unit/hr = 0.5 hr.

⇒ Total time = 6 + 1 + 0.5 = 7.5 hr.

∴ The correct answer is 7 ½ hr.

Additional Information

Alternative Work Pattern

When pipes work alternately, calculate the net work for one full cycle and determine how many full cycles fit into the total work before the final filling stage.

Pipes and Cisterns Convention

Inlet pipes are assigned positive efficiency (+), while outlet/leakage pipes are assigned negative efficiency (−).

Total Work Formula

Total Capacity = LCM of individual times. Time taken = Total Capacity ÷ Combined Efficiency.

Shortcut Trick

Let the number be N.

Direct Equation: 4/7 × 2/100 × N = 648

8/700 × N = 648

N = 648 × 700 ÷ 8 = 81 × 700 = 56700

∴ The correct answer is 56700.

Alternate Method

Given: 4/7 of 2% of a number is 648.

Formula Used: Value = (Percentage ÷ 100) × Fraction × Number

⇒ Let the number be x.

⇒ (4/7) × (2/100) × x = 648

⇒ (8/700) × x = 648

⇒ x = (648 × 700) ÷ 8

⇒ x = 81 × 700

⇒ x = 56700

∴ The correct answer is 56700.

Additional Information

Percentage to Fraction Conversion

To convert percentage to fraction, divide by 100. Example: 2% = 2/100 = 1/50.

Product of Fractions

The word 'of' in mathematical expressions represents multiplication (×).

Base Value Calculation

If a/b of c% of N = V, then N = (V × b × 100) ÷ (a × c).

Shortcut Trick

In right-angled ΔBDC, the acute angles are complementary.

∠C = 90° − ∠DBC = 90° − 65° = 25°.

By the Exterior Angle Theorem in ΔAEC, the exterior angle ∠AEB is the sum of interior opposite angles.

∠AEB = ∠CAE + ∠C = 30° + 25° = 55°.

∴ The correct answer is 55°.

Alternate Method

Given: ∠DBC = 65°, ∠CAE = 30°, BD ⊥ AC at D.

Formula Used: Sum of angles in a triangle = 180°; Exterior angle property.

In ΔBDC:

⇒ ∠BDC = 90° (Since BD ⊥ AC)

⇒ ∠BCD = 180° − (∠BDC + ∠DBC)

⇒ ∠BCD = 180° − (90° + 65°) = 25°

Shortcut Trick

The age difference between two people remains constant throughout their lives.

Difference = 25 years. After 15 years, Ratio of Mother : Daughter = 2 :

1. Ratio difference (2 − 1) = 1 unit → 25 years.

Daughter's age after 15 years = 1 unit × 25 = 25 years.

Daughter's current age = 25 − 15 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given: Difference in age = 25 years; Future condition = after 15 years.

Let Daughter's current age = x years.

Mother's current age = x + 25 years.

After 15 years:

⇒ Daughter = x + 15

⇒ Mother = (x + 25) + 15 = x + 40

According to the question:

⇒ Mother = 2 × Daughter

⇒ x + 40 = 2(x + 15)

⇒ x + 40 = 2x + 30

⇒ 2x − x = 40 − 30

⇒ x = 10

∴ The correct answer is 10 years.

Additional Information

Constant Age Difference

The difference between the ages of two persons never changes over time; if A is 5 years older than B today, A will always be 5 years older than B.

Ratio Method for Ages

When ratios at two different times are given, equate the difference in ratio units to the constant actual age difference to find the value of 1 unit.

Linear Equation in One Variable

Most age problems can be solved by assuming the current age of the younger person as 'x' and forming an equation based on the given conditions.

Shortcut Trick

Identify the Free Articles and Total Articles: Free = 16, Paid = 5.

Total Articles = 16 + 5 = 21.

Discount% = (Free Articles ÷ Total Articles) × 100.

Discount% = (16 ÷ 21) × 100 ≈ 76.19%.

∴ The correct answer is 76.19%.

Alternate Method

Given → Buy 5 articles and get 16 articles free.
Formulas: Discount% = (Discount ÷ MP) × 100 | MP = Price of total articles.

→ Let the Marked Price (MP) of 1 article be 1 unit.
→ MP of total articles (5 + 16 = 21) = 21 units.
→ Selling Price (SP) = Price of 5 articles = 5 units.
→ Discount = MP − SP = 21 − 5 = 16 units.
→ Discount% = (16 ÷ 21) × 100.
→ Discount% = 1600 ÷ 21 = 76.1904...% ≈ 76.19%.

∴ The correct answer is 76.19%.

Additional Information

Buy X Get Y Free

In such offers, the discount is always calculated on the total quantity (X + Y) as the Marked Price is representative of all items leaving the store. Discount% = [Y ÷ (X + Y)] × 100.

Marked Price vs Selling Price

The Marked Price is the total value of all items (Paid + Free), while the Selling Price is the value the customer actually pays (Paid items only).

Shortcut Trick

Using the Weighted Average formula: Average Speed = (s₁×t₁ + s₂×t₂) ÷ (t₁ + t₂)

58 = (70×1 + p×1.5) ÷ (1 + 1.5)

58 × 2.5 = 70 + 1.5p

145 − 70 = 1.5p ⇒ 75 = 1.5p ⇒ p = 50

∴ The correct answer is 50 km/hr.

Alternate Method

Given:

Speed 1 (s₁) = 70 km/hr, Time 1 (t₁) = 1 hour

Speed 2 (s₂) = p km/hr, Time 2 (t₂) = 1.5 hours

Total Average Speed = 58 km/hr

Formula Used:

Total Distance = (s₁ × t₁) + (s₂ × t₂)

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Time = 1 + 1.5 = 2.5 hours

⇒ Total Distance = 58 × 2.5 = 145 km

⇒ Distance covered in 1^st part = 70 × 1 = 70 km

⇒ Distance covered in 2^nd part = 145 − 70 = 75 km

⇒ Speed p = Distance ÷ Time = 75 ÷ 1.5

⇒ p = 50 km/hr

∴ The correct answer is 50 km/hr.

Additional Information

Average Speed (Equal Distances)

If a body travels equal distances at speeds x and y, the average speed is (2xy) ÷ (x + y).

Weighted Average Concept

Average Speed is the weighted mean of speeds with time as weights: (s₁t₁ + s₂t₂ + ...) ÷ (t₁ + t₂ + ...).

Relative Speed

When two objects move in the same direction, relative speed is |s₁ − s₂|. In opposite directions, it is s₁ + s₂.

Shortcut Trick

Let the total registered voters be 100%.

Voters who cast votes = 70%.

Candidate X got 60% of cast votes → Candidate Y got 40% of cast votes.

Votes for Y = 40% of 70% of Total = 28% of Total.

28% = 8400 → 1% = 300 → 100% = 30,000.

∴ The correct answer is 30,000.

Alternate Method

Given:

Percentage of voters who cast votes = 70%

Percentage of votes for Candidate X = 60% of cast votes

Votes for Candidate Y = 8400

Calculations:

⇒ Percentage of votes for Candidate Y = 100% − 60% = 40% of cast votes

⇒ Let total registered voters be V

⇒ Cast votes = 0.70 × V

⇒ Votes for Y = 40% of (0.70 × V) = 0.40 × 0.70 × V

⇒ 8400 = 0.28 × V

⇒ V = 8400 ÷ 0.28

⇒ V = 30,000

∴ The correct answer is 30,000.

Additional Information

Successive Percentage

When a percentage is applied on a value that has already been modified by another percentage, it is a successive change: Total × a% × b%.

Valid and Invalid Votes

In election problems, always distinguish between Total Voters, Cast Votes, and Valid Votes based on the question constraints.

Base Value Selection

Always identify what the percentage refers to (e.g., % of total vs % of cast votes) to avoid calculation errors.

Shortcut Trick

Effective rate for 2 years at 10% compound interest = 10 + 10 + (10 × 10)/100 = 21%.

Total amount percentage = 100% (Principal) + 21% (Interest) = 121%.

Amount = 121% of ₹1,800 = 1.21 × 1800 = ₹2,178.

∴ The correct answer is ₹2,178.

Alternate Method

Given:

Principal (P) = ₹1,800

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 1800 × (1 + 10/100)²

⇒ A = 1800 × (110/100)²

⇒ A = 1800 × (11/10) × (11/10)

⇒ A = 1800 × (121/100)

⇒ A = 18 × 121

⇒ A = 2178

∴ The correct answer is ₹2,178.

Additional Information

Compound Interest vs Simple Interest

The difference between CI and SI for 2 years is given by P × (R/100)².

Successive Percentage Method

Compound interest is essentially a successive percentage increase on the principal. For two years, the net percentage change is x + y + (xy/100).

Rule of 72

To find the approximate time required to double the money at a given interest rate, divide 72 by the annual rate of interest.

Shortcut Trick

Let the initial population be 100 units.

First reduction (28% left): 100 × (100 − 28)/100 = 72 units.

Second reduction (50% left): 72 × (100 − 50)/100 = 36 units.

Ratio = Initial : Present = 100 : 36 = 25 : 9.

∴ The correct answer is 25:9.

Alternate Method

Given:

⇒ Initial Population (Let) = P

⇒ First reduction = 28%

⇒ Second reduction = 50% of remaining

⇒ Present population = 4450

Calculations:

⇒ Remaining after 28% left = P × (1 − 28/100) = P × 72/100

⇒ Present population = (P × 72/100) × (1 − 50/100)

⇒ 4450 = P × 72/100 × 50/100

⇒ 4450 = P × 18/50

⇒ P = (4450 × 50) / 18

⇒ Initial Population (P) = 12361.11 (approx)

⇒ Ratio = P : 4450

⇒ Ratio = [P × (72/100) × (50/100)]⁻¹ is not needed, use simplified units:

⇒ Initial : Present = 1 : (72/100 × 50/100)

⇒ Initial : Present = 1 : 0.36 = 100 : 36 = 25 : 9

∴ The correct answer is 25:9.

Additional Information

Successive Percentage Change

When a value changes by a% and then b%, the net effective change is [a + b + (ab/100)]%.

Base Value Independence

Ratio calculations involving percentages do not require the actual numerical values (like 4450) if only the final ratio is asked.

Fraction Equivalents

Using fractions like 1/2 for 50% and 18/25 for 72% can significantly speed up calculations in population problems.

Shortcut Trick

Equate the volumes of both cylinders: Volume_Solid = Volume_Hollow

r_s² × H = (R² − r²) × h

2² × H = (6² − 5.5²) × 15

4 × H = (36 − 30.25) × 15 ⇒ 4 × H = 5.75 × 15

H = 86.25 ÷ 4 = 21.5625 cm

∴ The correct answer is 21.56 cm.

Alternate Method

Given:

Base diameter of solid cylinder = 4 cm ⇒ Radius (r_s) = 2 cm

External diameter of hollow cylinder = 12 cm ⇒ External radius (R) = 6 cm

Thickness (t) = 0.5 cm ⇒ Internal radius (r) = 6 − 0.5 = 5.5 cm

Height of hollow cylinder (h) = 15 cm

Formula Used:

Volume of solid cylinder = πr²h

Volume of hollow cylinder = π(R² − r²)h

Calculations:

⇒ Volume of solid cylinder = Volume of hollow cylinder

⇒ π × (2)² × H = π × (6² − 5.5²) × 15

⇒ 4 × H = (36 − 30.25) × 15

⇒ 4 × H = 5.75 × 15

⇒ 4 × H = 86.25

⇒ H = 86.25 ÷ 4

⇒ H = 21.5625 cm

Rounding off to 2 decimal places, H ≈ 21.56 cm.

∴ The correct answer is 21.56 cm.

Additional Information

Recasting Concept

When a solid object is melted and recast into another shape, the total volume remains constant, provided there is no wastage.

Hollow Cylinder Parameters

Internal Radius (r) = External Radius (R) − Thickness. Curved Surface Area (CSA) = 2πh(R + r).

Volume of Cylinder

For a solid cylinder, Volume = Base Area × Height = πr²h.

Shortcut Trick

Simplify the given equation: (x² − 1)/x = 4 → x − 1/x = 4.

Simplify the required expression: (x⁶ − 1)/x³ = x³ − 1/x³.

Apply the standard algebraic trick: If x − 1/x = k, then x³ − 1/x³ = k³ + 3k.

Substitute k = 4: 4³ + 3 × 4 = 64 + 12 = 76.

∴ The correct answer is 76.

Alternate Method

Given:

(x² − 1) / x = 4

Formula Used:

(a − b)³ = a³ − b³ − 3ab(a − b)

Calculations:

⇒ Split the given fraction: (x² / x) − (1 / x) = 4

⇒ x − 1/x = 4

⇒ Now, split the expression to be found: (x⁶ / x³) − (1 / x³)

⇒ x³ − 1/x³

⇒ Cubing both sides of the equation x − 1/x = 4:

⇒ (x − 1/x)³ = 4³

⇒ x³ − 1/x³ − 3 × x × (1/x) × (x − 1/x) = 64

⇒ x³ − 1/x³ − 3 × (4) = 64

⇒ x³ − 1/x³ = 64 + 12

⇒ x³ − 1/x³ = 76

∴ The correct answer is 76.

Additional Information

Algebraic Identity for Addition

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Relationship between Sum and Difference

(x + 1/x)² − (x − 1/x)² = 4. This is useful for converting between x + 1/x and x − 1/x.

Squaring Identity

If x − 1/x = k, then x² + 1/x² = k² + 2. If x + 1/x = k, then x² + 1/x² = k² − 2.

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

Required percentage = (1/30) ÷ (2/40) × 100

Required percentage = (1/30) × (40/2) × 100 = (1/30) × 20 × 100

Required percentage = 2000 ÷ 30 = 200 ÷ 3 = 66.666...

∴ The correct answer is 66.67%.

Alternate Method

Given: First fraction = 1/30, Second fraction = 2/40 (which simplifies to 1/20).

Formula Used: Percentage = (Part / Total) × 100

⇒ Required Percentage = [(1/30) / (2/40)] × 100

⇒ Required Percentage = (1/30) × (40/2) × 100

⇒ Required Percentage = (40 / 60) × 100

⇒ Required Percentage = (2 / 3) × 100

⇒ Required Percentage = 200 / 3 = 66.666...

⇒ Rounding off to two decimal places = 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

Percentage change = [(New Value − Old Value) / Old Value] × 100.

Fraction to Percentage

To convert any fraction a/b into a percentage, multiply it by 100 and add the % symbol.

Rounding Rule

When rounding to two decimal places, if the third decimal digit is 5 or more, increase the second decimal digit by 1.

Shortcut Trick

Ratio method: Let CP be 100. Then MP = 100 + 75 = 175.

Given Profit is 33%, so SP = 100 + 33 = 133.

Discount = MP − SP = 175 − 133 = 42.

X% (Discount%) = (Discount ÷ MP) × 100 = (42 ÷ 175) × 100.

X = (6 ÷ 25) × 100 = 6 × 4 = 24.

∴ The correct answer is 24.

Alternate Method

Given: Markup% = 75%, Profit% = 33%, Discount% = X%.

Formulas: SP = CP × (100 + Profit%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | MP = CP × (100 + Markup%) ÷ 100.

⇒ Let Cost Price (CP) = 100.

⇒ Marked Price (MP) = 100 × (100 + 75) ÷ 100 = 175.

⇒ Selling Price (SP) = 100 × (100 + 33) ÷ 100 = 133.

⇒ Using SP = MP × (100 − X) ÷ 100:

⇒ 133 = 175 × (100 − X) ÷ 100.

⇒ 133 = 1.75 × (100 − X) ⇒ 133 ÷ 1.75 = 100 − X.

⇒ 76 = 100 − X ⇒ X = 100 − 76 = 24.

∴ The correct answer is 24.

Additional Information

Relationship between CP and MP

The direct formula relating CP and MP when profit and discount are known is: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Markup Percentage

Markup% is always calculated on the Cost Price: Markup% = [(MP − CP) ÷ CP] × 100.

Shortcut Trick

To find the greatest number that exactly divides given numbers, find the HCF.

Check the differences between numbers: 74 − 46 = 28 and 98 − 74 = 24.

The HCF must be a factor of the differences (28, 24).

The factors of 24 and 28 are 1, 2, 4. Testing 4: 46 is not divisible by 4. Testing 2: 46, 74, and 98 are all even.

∴ The correct answer is 2.

Alternate Method

Given: Numbers are 46, 98, and 74.

Formula Used: The greatest number that exactly divides the given numbers is their Highest Common Factor (HCF).

⇒ Prime factorization of 46 = 2 × 23

⇒ Prime factorization of 74 = 2 × 37

⇒ Prime factorization of 98 = 2 × 7 × 7

⇒ Common factor in all three numbers = 2

⇒ HCF(46, 74, 98) = 2

∴ The correct answer is 2.

Additional Information

Highest Common Factor (HCF)

HCF is the largest positive integer that divides each of the integers without a remainder.

Least Common Multiple (LCM)

LCM is the smallest positive integer that is divisible by each of the given numbers.

HCF of Prime Numbers

The HCF of any two or more prime numbers is always 1.

Product Rule

For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b.

Shortcut Trick

Ratio of Father age to Son age is 3 :

1. Total ratio parts = 3 + 1 = 4 units.

4 units = 48 years.

1 unit (Son's age) = 48 ÷ 4 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given:

Father's current age = 3 × Son's current age

Sum of their ages = 48 years

Calculations:

Let the Son's age be x.

⇒ Father's age = 3x

⇒ x + 3x = 48

⇒ 4x = 48

⇒ x = 48 ÷ 4

⇒ x = 12

∴ The correct answer is 12 years.

Additional Information

Ratio and Proportion in Ages

Age problems often use linear equations or ratios. If the ratio of ages is a:b, their ages can be taken as ax and bx.

Age Difference Constant

The difference between the ages of two people always remains constant regardless of the number of years passed or preceding.

Future and Past Ages

If current age is x, then age n years ago was (x − n) and age n years hence will be (x + n).

Shortcut Trick

Proportion of chemical Y = Efficiency of B ÷ Total efficiency of (A + B + C).

LCM(30, 40, 45) = 360 units (Total Capacity).

Efficiencies: A = 12, B = 9, C = 8.

Proportion of Y = 9 ÷ (12 + 9 + 8) = 9/29.

∴ The correct answer is 9/29.

Alternate Method

Given: Filling time for Pipe A = 30 min, Pipe B = 40 min, Pipe C = 45 min.

Formula Used: Work = Efficiency × Time; Proportion = Part ÷ Whole.

⇒ Total units filled in 1 minute = 12 + 9 + 8 = 29 units.

⇒ Chemical Y is filled by Pipe B.

⇒ Amount of X in 5 min = 12 × 5 = 60 units.

⇒ Amount of Y in 5 min = 9 × 5 = 45 units.

⇒ Amount of Z in 5 min = 8 × 5 = 40 units.

⇒ Total liquid in 5 min = 60 + 45 + 40 = 145 units.

⇒ Proportion of Y = 45 ÷ 145 = 9/29.

∴ The correct answer is 9/29.

Additional Information

Proportion in Mixtures

The ratio of components in a tank filled by multiple pipes is always equal to the ratio of their individual efficiencies, regardless of time, provided all pipes run for the same duration.

Pipes and Cisterns

If a pipe fills a tank in 'x' hours, its 1-hour work is 1/x. For an outlet pipe (leak), the efficiency is taken as negative.

Shortcut Trick

Mean Proportional between a and b is given by √(a × b).

⇒ √(17 × 272)

⇒ √(17 × 17 × 16)

⇒ 17 × 4 = 68

∴ The correct answer is 68.

Alternate Method

Given: First term (a) = 17, Second term (b) = 272

Formula Used: If x is the mean proportional between a and b, then a : x = x : b ⇒ x² = ab

⇒ x² = 17 × 272

⇒ x² = 17 × (17 × 16)

⇒ x = √(17² × 4²)

⇒ x = 17 × 4

⇒ x = 68

∴ The correct answer is 68.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is calculated such that a : b = b : c, which gives c = b²/a.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is calculated such that a : b = c : d, which gives d = (b × c)/a.

Continued Proportion

Three quantities a, b, c are in continued proportion if a/b = b/c. Here b is the geometric mean of a and c.

Shortcut Trick

List all numbers between 445 and 456: 446, 447, 448, 449, 450, 451, 452, 453, 454, 455.

Eliminate even numbers (446, 448, 450, 452, 454) and multiples of 5 (455).

Check remaining odd numbers: 447 (sum=15, div by 3), 449 (Prime), 451 (11 × 41), 453 (sum=12, div by 3).

Only 449 is a prime number.

∴ The correct answer is 1.

Alternate Method

Given: Range is 445 < n < 456.

Method: Test divisibility for odd numbers within the range.

⇒ 447: 4 + 4 + 7 = 15 (Divisible by 3) → Composite

⇒ 449: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 (since √449 ≈ 21.1) → Prime

⇒ 451: 451 ÷ 11 = 41 → Composite

⇒ 453: 4 + 5 + 3 = 12 (Divisible by 3) → Composite

⇒ 455: Ends in 5 (Divisible by 5) → Composite

∴ The correct answer is 1.

Additional Information

Definition of Prime Numbers

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Primality Test

To check if n is prime, test divisibility by primes up to √n.

Divisibility Rule of 11

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