Quantitative Aptitude PYQs

Shortcut Trick

Calculate both expressions directly using basic simplification techniques:

⇒ 65% of 65 = (65 × 65) / 100

Using the squaring trick for numbers ending in 5: 65² = (6 × 7)25 = 4225. So, 65% of 65 = 42.25

⇒ 2/5 of 25 = 2 × (25 / 5) = 2 × 5 = 10

⇒ Difference = 42.25 − 10 = 32.25

∴ The correct answer is 32.25.

Alternate Method

Given:

Expression 1 = 65% of 65

Expression 2 = 2/5 of 25

Calculations:

⇒ Expression 1 = (65 / 100) × 65

⇒ Expression 1 = 4225 / 100

⇒ Expression 1 = 42.25

⇒ Expression 2 = (2 / 5) × 25

⇒ Expression 2 = 2 × 5

⇒ Expression 2 = 10

⇒ Required Difference = Expression 1 − Expression 2

⇒ Required Difference = 42.25 − 10

⇒ Required Difference = 32.25

∴ The correct answer is 32.25.

Additional Information

Square of Numbers Ending in 5

To find the square of a number ending in 5 (e.g., N5), multiply N by (N+1) and append 25 to the result. Example: 65² → 6 × 7 = 42 → 4225.

BODMAS Rule

Always follow the sequence: Brackets, Of, Division, Multiplication, Addition, and Subtraction. The word 'Of' represents multiplication but must be evaluated before regular multiplication and division.

Percentage to Fraction Conversion

Converting common percentages to fractions makes calculations easier. Since 20% = 1/5, it follows that 40% = 2/5. Hence, finding 2/5 of a number is exactly the same as finding 40% of it.

Shortcut Trick

Total man-days = 4 × 10 = 40; Total woman-days = 3 × 16 = 48.

Equating total work: 40 M = 48 W ⇒ M / W = 6 / 5.

Total work = 40 × 6 = 240 units.

Target efficiency (6 women + 5 men) = 6 × 5 + 5 × 6 = 60 units/day.

Time required = 240 ÷ 60 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given:

⇒ 4 men can complete the work in 10 days.

⇒ 3 women can complete the work in 16 days.

Calculations:

⇒ Let total work be the LCM of 10 and 16.

⇒ Efficiency of 4 men = 8 units/day ⇒ Efficiency of 1 man = 8 ÷ 4 = 2 units/day.

⇒ Efficiency of 3 women = 5 units/day ⇒ Efficiency of 1 woman = 5 ÷ 3 units/day.

⇒ Target group = 6 women + 5 men.

⇒ Combined efficiency = 6 × (5/3) + 5 × 2

⇒ Combined efficiency = 10 + 10 = 20 units/day.

⇒ Total days required = Total Work ÷ Combined efficiency

⇒ Total days = 80 ÷ 20 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Chain Rule (MDH Concept)

If M₁ persons can do W₁ work in D₁ days working H₁ hours/day, then M₁D₁H₁ ÷ W₁ = M₂D₂H₂ ÷ W₂.

Work and Wages

Wages are distributed in direct proportion to the work done. If people work for the same number of days, wages are divided in the ratio of their efficiencies.

Alternate Days Work

Find the total work done in a cycle of days and multiply by the maximum possible full cycles, then assign the remaining work to the person whose turn is next.

Shortcut Trick

Total time taken = (612 ÷ 68) + (312 ÷ 39) + (620 ÷ 62) = 9 + 8 + 10 = 27 hours.

Total distance covered = 612 + 312 + 620 = 1544 km.

Average speed = Total distance ÷ Total time = 1544 ÷ 27

Average speed ≈ 57.19 km/hr.

∴ The correct answer is 57.19 km/hr.

Alternate Method

Given:

Distance D₁ = 612 km, Speed S₁ = 68 km/hr

Distance D₂ = 312 km, Speed S₂ = 39 km/hr

Distance D₃ = 620 km, Speed S₃ = 62 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

612 km 312 km 620 km 68 km/hr 39 km/hr 62 km/hr T = 9 hr T = 8 hr T = 10 hr Average Speed = 1544 ÷ 27 ≈ 57.19 km/hr

Calculations:

⇒ Total Distance = 612 + 312 + 620 = 1544 km

⇒ Time for 1^st part (T₁) = D₁ ÷ S₁ = 612 ÷ 68 = 9 hours

⇒ Time for 2^nd part (T₂) = D₂ ÷ S₂ = 312 ÷ 39 = 8 hours

⇒ Time for 3^rd part (T₃) = D₃ ÷ S₃ = 620 ÷ 62 = 10 hours

⇒ Total Time = 9 + 8 + 10 = 27 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = 1544 ÷ 27 = 57.185...

⇒ Average Speed ≈ 57.19 km/hr (rounded off to two decimal places)

∴ The correct answer is 57.19 km/hr.

Additional Information

Average Speed

Average speed is the total distance traveled divided by the total time taken. It provides a single speed value representing the overall journey.

Average Speed for Equal Distances

When a journey consists of two equal distance segments traveled at speeds S₁ and S₂, the average speed is given by the harmonic mean: (2 × S₁ × S₂) ÷ (S₁ + S₂).

Average Speed for Equal Time Intervals

If an object travels at different speeds for exactly equal time intervals, the overall average speed is simply the arithmetic mean of all individual speeds.

Shortcut Trick

By eliminating options using basic divisibility rules:

1. The given number ends in 3 (odd), so it is not even. Hence, it is not divisible by 18.
2. Sum of digits = 7 + 1 + 8 + 7 + 5 + 4 + 0 + 2 + 3 = 37, which is not divisible by 3.
3. Because it is not divisible by 3, it cannot be divisible by 3 or 21.
4. The only remaining option is 13.

∴ The correct answer is 13.

Alternate Method

Given:

Number = 718754023

Method Used:

Divisibility rule of 13: Form alternating blocks of 3 digits from right to left. If the alternating difference of these blocks is divisible by 13, the number is divisible by 13.

Calculation:

⇒ Split 718754023 into blocks of 3 from right to left: 023, 754, 718

⇒ Alternating sum = 718 − 754 + 023

⇒ Alternating sum = 741 − 754

⇒ Alternating sum = −13

⇒ Since −13 is a multiple of 13, the entire number is divisible by 13.

∴ The correct answer is 13.

Additional Information

Divisibility Rule of 3 & 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 of 7, 11 & 13

Group the number into blocks of 3 digits starting from the right. If the alternating sum of these blocks is divisible by 7, 11, or 13, then the original number is divisible by that specific number.

Divisibility Rule of 11 (Digit Method)

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 either 0 or a multiple of 11.

Divisibility of Composite Numbers

To check divisibility by a composite number (like 18 or 21), break it into its co-prime factors (like 2 × 9 or 3 × 7). The number must be individually divisible by all those co-prime factors.

Shortcut Trick

Ratio of investments A : B : C = 8 : 36 : 76 = 2 : 9 : 19

Total profit parts = 2 + 9 + 19 = 30 parts

Value of 30 parts = ₹3,030 ⇒ 1 part = ₹101

Difference between shares of B and C = 19 − 9 = 10 parts

Difference in amount = 10 × 101 = ₹1,010

∴ The correct answer is ₹1,010.

Alternate Method

Given:

Investment ratio of A, B and C = 8 : 36 : 76

Total profit earned at the end of the year = ₹3,030

Formula Used:

Profit Share = (Individual Investment Ratio ÷ Total Investment Ratio) × Total Profit

2
A

9
B

19
C

Diff = 10 parts

Total Profit = 30 parts = ₹3030

Calculations:

⇒ Simplified investment ratio of A, B and C = 8 : 36 : 76 = 2 : 9 : 19

⇒ Total ratio parts = 2 + 9 + 19 = 30 parts

⇒ B's profit share = (9 ÷ 30) × 3030

⇒ B's profit share = 9 × 101 = ₹909

⇒ C's profit share = (19 ÷ 30) × 3030

⇒ C's profit share = 19 × 101 = ₹1,919

⇒ Difference between shares of B and C = 1919 − 909 = ₹1,010

∴ The correct answer is ₹1,010.

Additional Information

Partnership Basic Rule

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time (Profit Ratio = C₁×T₁ : C₂×T₂).

Sleeping vs Working Partner

A sleeping partner only invests capital, while a working partner also manages the business and usually receives a fixed salary or a certain percentage of the profit before the remainder is split according to the investment ratio.

Calculating Unknown Capitals or Time

If the profit ratio and time ratio are given, the ratio of investments can be found by dividing the profit parts by the time parts (Capital Ratio = P₁÷T₁ : P₂÷T₂).

Shortcut Trick

Given 8A = 4B = 3C.

Ratio of A : B : C can be found directly by hiding one variable and multiplying the coefficients of the other two.

A : B : C = (4 × 3) : (8 × 3) : (8 × 4) = 12 : 24 : 32 = 3 : 6 : 8

Total parts = 3 + 6 + 8 = 17 parts.

17 parts = 30,600 ⇒ 1 part = 30,600 ÷ 17 = 1,800

A's share = 3 parts = 3 × 1,800 = 5,400

∴ The correct answer is ₹5,400.

Alternate Method

Total Amount = ₹30,600 (17 Parts)

A (3)

B (6)

C (8)

₹5,400
₹10,800
₹14,400

Given:

Total amount to be divided = ₹30,600

8 × (Share of A) = 4 × (Share of B) = 3 × (Share of C)

Calculations:

⇒ Let 8A = 4B = 3C = k

⇒ A = k/8

⇒ B = k/4

⇒ C = k/3

⇒ The ratio of shares of A, B, and C is A : B : C = 1/8 : 1/4 : 1/3

⇒ LCM of denominators (8, 4, 3) is 24.

⇒ Multiplying the ratio by 24, we get:

⇒ A : B : C = 24/8 : 24/4 : 24/3

⇒ A : B : C = 3 : 6 : 8

⇒ Sum of the ratio terms = 3 + 6 + 8 = 17 units

⇒ 17 units = 30,600

⇒ 1 unit = 30,600 ÷ 17 = 1,800

⇒ Share of A = 3 units = 3 × 1,800 = 5,400

∴ The correct answer is ₹5,400.

Additional Information

Inverse Ratio

The inverse ratio of a : b : c is 1/a : 1/b : 1/c. It can be simplified by taking the LCM of a, b, and c and multiplying each term, yielding bc : ac : ab.

Distribution of Amount

If an amount S is to be divided among individuals in the ratio of x : y : z, the share of the first individual is given by (x ÷ (x + y + z)) × S.

Proportion Rule

When two ratios are equal (a : b = c : d), they are said to be in proportion. The fundamental rule states that the product of means equals the product of extremes, meaning a × d = b × c.

Shortcut Trick

Volume of a cylinder is proportional to r² × h.

New radius factor = 0.82 (decreased by 18%)

New height factor = 2.41 (increased by 141%)

Net volume factor = 0.82 × 0.82 × 2.41 = 1.620484

Percentage increase = (1.620484 − 1) × 100% = 62.0484% ≈ 62%

∴ The correct answer is 62%.

Alternate Method

Given:

⇒ Decrease in the base radius of the cylinder = 18%

⇒ Increase in the height of the cylinder = 141%

Formula Used:

⇒ Volume of a cylinder (V) = πr²h

Original Cylinder
V₁ = 100%

r

h

New Cylinder
V₂ ≈ 162%

0.82r

2.41h

r − 18%
h + 141%

Calculations:

⇒ Let the original radius of the cylinder be r and the original height be h.

⇒ Original Volume (V₁) = πr²h

⇒ New radius (r₂) = r − (18 ÷ 100)r = 0.82r

⇒ New height (h₂) = h + (141 ÷ 100)h = 2.41h

⇒ New Volume (V₂) = π × (0.82r)² × (2.41h)

⇒ V₂ = π × 0.6724r² × 2.41h = 1.620484πr²h

⇒ Increase in Volume = V₂ − V₁ = 1.620484πr²h − πr²h = 0.620484πr²h

⇒ Percentage Increase = (Increase in Volume ÷ Original Volume) × 100%

⇒ Percentage Increase = (0.620484πr²h ÷ πr²h) × 100%

⇒ Percentage Increase = 62.0484% ≈ 62%

∴ The correct answer is 62%.

Additional Information

Successive Percentage Change

When a value is subjected to multiple percentage changes consecutively (such as changing r twice due to squaring, and h once), the most efficient approach is to use decimal multiplying factors instead of the standard formula A + B + (A × B ÷ 100).

Volume of a Right Circular Cone

The volume of a right circular cone is exactly one-third of the volume of a cylinder with the same base radius and height. Formula: V = (1 ÷ 3)πr²h.

Curved Surface Area of a Cylinder

The lateral or curved surface area is determined by the perimeter of the base multiplied by the height. Formula: CSA = 2πrh.

Shortcut Trick

Replacing the walking journey with a drive (driving both ways) saves 2 hours.

Conversely, replacing the driving journey with a walk (walking both ways) will add 2 hours to the total time.

Given time to walk one way and drive back = 4 hours 45 minutes.

Time to walk both ways = 4 hours 45 minutes + 2 hours = 6 hours 45 minutes.

∴ The correct answer is 6 hours 45 minutes.

Alternate Method

Start

Viewpoint

W + C = 4h 45m

Walk (W)

Drive (C)

Drive (C)

Drive (C)

2C = 4h 45m − 2h = 2h 45m

Given:

Time to walk one way and drive back (W + C) = 4 hours 45 minutes = 285 minutes.

Time saved by driving both ways = 2 hours = 120 minutes.

Time to drive both ways (2C) = 285 − 120 = 165 minutes.

Formula Used:

Total time for a round trip logic: 2W + 2C = 2 × (W + C)

Calculations:

⇒ 2 × (W + C) = 2 × 285 = 570 minutes

⇒ 2W + 2C = 570 minutes

⇒ 2W + 165 = 570

⇒ 2W = 570 − 165 = 405 minutes

⇒ 405 minutes = 6 hours 45 minutes

∴ The correct answer is 6 hours 45 minutes.

Additional Information

Average Speed (Equal Distances)

When a person travels a certain distance at speed x and returns at speed y, the average speed for the whole journey is calculated using the formula 2xy ÷ (x + y).

Relative Speed Concept

When two objects move in the same direction, their relative speed is the difference of their individual speeds (S₁ − S₂). When moving in opposite directions, it is the sum of their speeds (S₁ + S₂).

Inverse Relation of Speed and Time

For a constant distance, speed and time are inversely proportional. If the ratio of speeds of two vehicles is a : b, then the ratio of the time taken by them to cover the same distance will be b : a.

Shortcut Trick

Assume the base average is 14 (from the 10 matches).

Deviation of the first 23 matches = 43 − 14 = +29 runs per match.

Total net deviation = 29 × 23 = +667 runs.

Change in overall average = 667 ÷ 33 matches ≈ +20.21.

Combined average = Base average + Net change = 14 + 20.21 = 34.21.

∴ The correct answer is 34.21.

Alternate Method

Given:

Average runs scored in the first 23 matches = 43

Average runs scored in the next 10 matches = 14

Formula Used:

Combined Average = (N₁A₁ + N₂A₂) ÷ (N₁ + N₂)

Total 33 Matches (Combined Average)

23 Matches
Average: 43 runs

10 Matches
Average: 14 runs

Total Runs = (23 × 43) + (10 × 14)

Calculations:

⇒ Total runs scored in first 23 matches = 23 × 43 = 989

⇒ Total runs scored in next 10 matches = 10 × 14 = 140

⇒ Total matches played = 23 + 10 = 33

⇒ Total runs scored in all 33 matches = 989 + 140 = 1129

⇒ Combined average of 33 matches = 1129 ÷ 33

⇒ Combined average ≈ 34.2121...

∴ The correct answer is 34.21.

Additional Information

Combined Average Formula

When two groups of sizes N₁ and N₂ have averages A₁ and A₂ respectively, the overall combined average is calculated as (N₁A₁ + N₂A₂) ÷ (N₁ + N₂).

Deviation Method for Averages

Instead of multiplying large numbers, you can assume a base average and find the net deviation. The final average is simply the assumed base average plus the net deviation divided by the total number of entities.

Weighted Average

It represents an average where each component has a specific weight based on its frequency. Here, the number of matches acts as the weights for their respective average scores.

Shortcut Trick

Apply the BODMAS rule directly to the expression.

135 ÷ {25 + 5 × (7 − 3)}

⇒ 135 ÷ {25 + (5 × 4)}

⇒ 135 ÷ (25 + 20) = 135 ÷ 45 = 3

∴ The correct answer is 3.

Alternate Method

Given:

Expression: [135 ÷ {25 + 5 × (7 − 3)}]

Formula Used:

BODMAS Rule: Brackets (B), Of (O), Division (D), Multiplication (M), Addition (A), Subtraction (S).

Calculations:

⇒ 135 ÷ {25 + 5 × (7 − 3)}

Solve the innermost bracket ( ) first:

⇒ 135 ÷ {25 + 5 × 4}

Solve the multiplication inside the curly bracket { }:

⇒ 135 ÷ {25 + 20}

Solve the addition inside the curly bracket { }:

⇒ 135 ÷ 45

Perform the final division:

⇒ 3

∴ The correct answer is 3.

Additional Information

BODMAS Rule

It dictates the mathematical order of operations: Brackets first, then Of (powers/roots), followed by Division, Multiplication, Addition, and Subtraction.

Order of Brackets

When simplifying expressions with multiple brackets, always follow this order from inside to outside: Bar/Vinculum, Round/Small ( ), Curly { }, and finally Square [ ].

Continuous Division

When dealing with multiple continuous division signs, proceed systematically from left to right, or convert each division to multiplication by writing the reciprocal: a ÷ b ÷ c = a × (1/b) × (1/c).

Shortcut Trick

When two objects are on opposite sides of a tower of height h, the distance between them is directly given by:

Distance = h × (cot θ₁ + cot θ₂)

⇒ Distance = 150 × (cot 30° + cot 60°)

⇒ Distance = 150 × (√3 + 1/√3)

⇒ Distance = 150 × (4 / √3) = 200√3 m

∴ The correct answer is 200√3 m.

Alternate Method

Given:

Height of the lighthouse (AB) = 150 m

Angle of depression of Boat 1 (at C) = 30°

Angle of depression of Boat 2 (at D) = 60°

30°
60°
30°
60°

A
B
C (Boat 1)
D (Boat 2)

150 m

Formula Used:

tan θ = Perpendicular ÷ Base

Since alternate interior angles are equal, the angle of elevation of the top of the lighthouse from Boat 1 (C) is 30° and from Boat 2 (D) is 60°.

Calculations:

In right-angled triangle ABC:

⇒ tan 30° = AB ÷ BC

⇒ 1 / √3 = 150 ÷ BC

⇒ BC = 150√3 m

In right-angled triangle ABD:

⇒ tan 60° = AB ÷ BD

⇒ √3 = 150 ÷ BD

⇒ BD = 150 ÷ √3

⇒ BD = (150 × √3) ÷ 3 = 50√3 m

Total distance between the two boats (CD) = BC + BD

⇒ CD = 150√3 + 50√3

⇒ CD = 200√3 m

∴ The correct answer is 200√3 m.

Additional Information

Distance Formula for Same Side

If two objects are on the same side of a vertical tower of height h, the distance between them is given by h × |cot θ₁ − cot θ₂|.

Angle of Elevation vs Depression

The angle of elevation refers to looking upward from the horizontal line of sight, while the angle of depression refers to looking downward. Geometrically, both angles are equivalent due to alternate interior properties along parallel horizontal lines.

Key Trigonometric Standard Values

Memorise standard tangent ratios for exams: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Cotangent values are simply their reciprocals.

Shortcut Trick

Successive percentage change can be calculated directly using the formula: x + y + (x × y) / 100

Here, reduction x = −25% and raise y = +12%

⇒ Net Change = −25 + 12 + (−25 × 12) / 100

⇒ Net Change = −13 − (300 / 100)

⇒ Net Change = −13 − 3 = −16%

The negative sign indicates an overall decrease of 16%.

∴ The correct answer is 16%.

Alternate Method

Given:

First percentage change = 25% reduction

Second percentage change = 12% raise

Formula Used:

Final Value = Initial Value × (1 − Reduction / 100) × (1 + Raise / 100)

100
Initial

−25%

75
Reduced

+12%

84
Final

Net Decrease: 100 − 84 = 16%

Calculations:

⇒ Let Salil's initial salary be 100 units.

⇒ Salary after 25% reduction = 100 × (1 − 25 / 100) = 100 × 0.75 = 75 units

⇒ Salary after subsequent 12% raise = 75 × (1 + 12 / 100) = 75 × 1.12

⇒ Final Salary = 75 + (12% of 75) = 75 + 9 = 84 units

⇒ Overall decrease = Initial Salary − Final Salary = 100 − 84 = 16 units

⇒ Percentage decrease = (16 / 100) × 100% = 16%

∴ The correct answer is 16%.

Additional Information

Successive Percentage Change

When a number is changed successively by x% and y%, the net percentage change is given by x + y + (x × y) / 100. Use positive signs for an increase and negative signs for a decrease.

Base Change Principle

In consecutive percentage changes, the second percentage change is always calculated on the new updated base (the intermediate value), not on the original starting value.

Multiplying Factor Method

For quick calculation, convert percentages to decimal multipliers. A 25% decrease means multiplying the value by 0.75, and a 12% increase means multiplying the value by 1.12. The net multiplier is 0.75 × 1.12 = 0.84, which directly indicates a 16% overall drop from the base 1.00.

Shortcut Trick

Make the number of decimal places equal: 4.70 and 0.04

Remove the decimal points to treat them as whole numbers: 470 and 4

Find the LCM of 470 and 4: LCM(470, 4) = 940

Place the decimal point back exactly 2 places from the right: 9.40

∴ The correct answer is 9.4.

Alternate Method

Given:

Numbers to find LCM: 4.7 and 0.04

Formula Used:

LCM of fractions = LCM of Numerators ÷ HCF of Denominators

Calculations:

⇒ First, express each decimal as a fraction in its simplest form.

⇒ 4.7 = 47/10

⇒ 0.04 = 4/100 = 1/25

⇒ The simplified fractions are 47/10 and 1/25.

⇒ Numerators: 47 and 1

⇒ Denominators: 10 and 25

⇒ LCM of numerators (47, 1) = 47

⇒ HCF of denominators (10, 25) = 5

⇒ LCM of the fractions = 47 ÷ 5 = 9.4

∴ The correct answer is 9.4.

Additional Information

LCM of Decimals

To find the LCM of decimal numbers without fractions, equalize the decimal places, treat them as integers to find the LCM, and put the decimal back at the same place.

HCF of Fractions

HCF = (HCF of Numerators) ÷ (LCM of Denominators). Always reduce fractions to their simplest form before applying this formula.

Relation between LCM and HCF

For any two given numbers 'a' and 'b', Product of the numbers (a × b) = LCM(a, b) × HCF(a, b).

Shortcut Trick

If x is the third proportional to a and b, it is given directly by the formula x = b² ÷ a.

Here, a = 16 and b = 40.

x = (40 × 40) ÷ 16

x = 1600 ÷ 16 = 100

∴ The correct answer is 100.

Alternate Method

Given:

The numbers are 16 and 40.

Third proportional = x

Formula Used:

For three numbers to be in continuous proportion a, b, and x:

a : b :: b : x ⇒ a × x = b × b (Product of Extremes = Product of Means)

16

:

40

::

40

:

x

Product of Extremes (16 × x)

Product of Means (40 × 40)

Calculations:

⇒ 16 : 40 :: 40 : x

⇒ 16 × x = 40 × 40

⇒ 16x = 1600

⇒ x = 1600 ÷ 16

⇒ x = 100

∴ The correct answer is 100.

Additional Information

Mean Proportional

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

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional to a, b, and c. It is calculated as d = (b × c) ÷ a.

Continued Proportion

Three quantities a, b, and c are in continued proportion if a : b = b : c, where b is the mean proportional and c is the third proportional.

Shortcut Trick

The number 1.0201 has 4 decimal places, so its square root will have 4 ÷ 2 = 2 decimal places.

Ignoring the decimal point, we find the square root of 10201.

Since 100² = 10000, we can easily estimate and confirm that 101² = 10201.

Placing the decimal point 2 positions from the right gives 1.01.

∴ The correct answer is 1.01.

Alternate Method

Given:

Expression = √1.0201

Calculation:

⇒ √1.0201

⇒ Convert the decimal into a fraction

⇒ √(10201 ÷ 10000)

⇒ Separate the square roots for the numerator and the denominator

⇒ √10201 ÷ √10000

⇒ We know that 101 × 101 = 10201 and 100 × 100 = 10000

⇒ 101 ÷ 100

⇒ Convert the fraction back to a decimal

⇒ 1.01

∴ The correct answer is 1.01.

Additional Information

Square Root of Decimals

If a perfect square has an even number of decimal places (e.g., 2n), its square root will always have exactly half that number of decimal places (n).

Squares of Numbers near 100

For a number expressed as (100 + x), its square is calculated using the formula 10000 + 200x + x². For example, 101² = 10000 + 200(1) + 1² = 10201.

Fraction to Decimal Conversion

When dividing any whole number by a power of 10 (such as 10, 100, 1000), simply shift the decimal point to the left by the number of zeros present in the denominator.

Shortcut Trick

Ratio of numbers = 4 : 3

LCM of ratio parts = 4 × 3 = 12 units

Given 12 units = 84 ⇒ 1 unit (HCF) = 7

Sum of numbers = (4 + 3) × 7 = 7 × 7 = 49

∴ The correct answer is 49.

Alternate Method

Given:LCM = 84, Ratio = 4 : 3

Formula Used: LCM = HCF × Product of ratio parts (for coprime ratios)

⇒ Let the numbers be 4x and 3x (where x is the HCF).

⇒ LCM of 4x and 3x = 12x

⇒ 12x = 84

⇒ x = 84 ÷ 12 = 7

⇒ First number = 4 × 7 = 28

⇒ Second number = 3 × 7 = 21

⇒ Sum = 28 + 21 = 49

∴ The correct answer is 49.

Additional Information

Fundamental Property of Two Numbers

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

Ratio and LCM Relation

If two numbers are in the ratio a : b (where a and b are coprime), then LCM = a × b × HCF.

HCF of Fractions

HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators).

Shortcut Trick

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

⇒ (1/20 ÷ 10/60) × 100

⇒ (1/20 × 60/10) × 100

⇒ (3/10) × 100 = 30%

∴ The correct answer is 30%.

Alternate Method

Given: First fraction = 1/20, Second fraction = 10/60.

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

Calculation:

⇒ Let 1/20 be x% of 10/60.

⇒ 1/20 = (x/100) × (10/60)

⇒ 1/20 = (x/100) × (1/6)

⇒ 1/20 = x/600

⇒ x = 600/20

⇒ x = 30

∴ The correct answer is 30%.

Additional Information

Fraction to Percentage

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

Percentage Change

Percentage change = (Difference / Original Value) × 100.

Reciprocal Percentages

Common values: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%, 1/20 = 5%.

Shortcut Trick

Convert fractional ratio to integer ratio by multiplying with LCM of 6, 9, and 4 (LCM = 36).

Ratio = (1/6 × 36) : (1/9 × 36) : (1/4 × 36) = 6 : 4 : 9.

Total units = 6 + 4 + 9 = 19 units.

19 units = 646 ⇒ 1 unit = 34.

First monkey (6 units) = 6 × 34 = 204.

∴ The correct answer is 204 bananas.

Alternate Method

Given: Total bananas = 646, Ratio = 1/6 : 1/9 : 1/4

⇒ Simplified Ratio = 1/6 : 1/9 : 1/4

⇒ Multiply by 36: (36/6) : (36/9) : (36/4) = 6 : 4 : 9

⇒ Sum of ratio terms = 6 + 4 + 9 = 19

⇒ Share of first monkey = (6/19) × 646

⇒ Share = 6 × 34 = 204

∴ The correct answer is 204 bananas.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d to an integer ratio, multiply both terms by the Least Common Multiple (LCM) of the denominators b and d.

Proportional Division

If a quantity X is divided in the ratio a:b:c, the share of the first part is calculated as [a / (a + b + c)] × X.

Shortcut Trick

Efficiency ratio for Mahi : (Neha + Asha) = 1 : 2 ⇒ Mahi = 1/3 of total.

Efficiency ratio for Asha : (Neha + Mahi) = 1 : 3 ⇒ Asha = 1/4 of total.

Neha efficiency = 1 − (1/3 + 1/4) = 1 − 7/12 = 5/12 of total.

Total time × Total Efficiency = Neha time × Neha efficiency ⇒ 20 × 1 = Days × 5/12.

Days = 20 × 12/5 = 4 × 12 = 48 days.

∴ The correct answer is 48 days.

Alternate Method

Given: Time taken by Mahi = 2 × (Neha + Asha)

⇒ Efficiency ratio (Mahi) : (Neha + Asha) = 1 : 2

⇒ Mahi efficiency = (1/3) × 12 = 4 units/day

Given: Time taken by Asha = 3 × (Neha + Mahi)

⇒ Efficiency ratio (Asha) : (Neha + Mahi) = 1 : 3

⇒ Asha efficiency = (1/4) × 12 = 3 units/day

Calculations:

⇒ Neha efficiency = Total efficiency − Mahi efficiency − Asha efficiency

⇒ Neha efficiency = 12 − 4 − 3 = 5 units/day

⇒ Days taken by Neha = Total Work ÷ Neha efficiency

⇒ Days = 240 ÷ 5 = 48 days

∴ The correct answer is 48 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a task. If time ratio is a:b, then efficiency ratio is b:a.

Total Work Formula

Total Work = Efficiency × Time. For multiple people, Total Work = (Sum of individual efficiencies) × Combined Time.

Ratio Balancing

When comparing partial groups to a whole, ensure the sum of ratios (parts) matches the total efficiency common to all equations.

Shortcut Trick

Ratio of Marked Price (MP) to Cost Price (CP) is given by: MP : CP = (100 + Gain%) : (100 − Discount%).

MP : CP = (100 + 83) : (100 − 77) = 183 : 23.

Given CP = 23 units = ₹92,000.

Multiplier = 92,000 ÷ 23 = 4,000.

MP = 183 × 4,000 = ₹7,32,000.

∴ The correct answer is ₹7,32,000.

Alternate Method

Given → CP = ₹92,000, Discount = 77%, Gain = 83%.
Formulas: SP = CP × (100 + Gain%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%).

⇒ Calculate Selling Price (SP): SP = 92,000 × (100 + 83) ÷ 100 = 92,000 × 1.83.
⇒ SP = ₹1,68,360.
⇒ Since Discount = 77%, SP is 23% of MP: 0.23 × MP = 1,68,360.
⇒ MP = 1,68,360 ÷ 0.23 = ₹7,32,000.

∴ The correct answer is ₹7,32,000.

Additional Information

Relationship between MP and CP

The direct ratio MP/CP = (100 + Profit%)/(100 − Discount%) is the most efficient way to solve problems involving both discount and profit on the same article.

Marked Price and Discount

Discount is always calculated on the Marked Price. SP = MP − Discount. If no discount is offered, MP becomes the SP.