Quantitative Aptitude PYQs

Given:

Expression = [(8.16 ÷ 2.4) + (3.64 × 1.5)]

Formula used:

Simplify using the BODMAS rule (Division, Multiplication, Addition).

Calculations:

[(8.16 ÷ 2.4) + (3.64 × 1.5)]

⇒ First, perform division: 8.16 ÷ 2.4 = 3.4

⇒ Next, perform multiplication: 3.64 × 1.5 = 5.46

⇒ Finally, perform addition: 3.4 + 5.46 = 8.86

⇒ The result to two decimal places is 8.86.

∴ The correct answer is option (2).

Shortcut Trick

Apply Alligation: Managers (48000) and Supervisors (42000) with mean (44400).

Ratio of M : S = (44400 − 42000) : (48000 − 44400) = 2400 : 3600 = 2 : 3.

Total units = 2 + 3 = 5 units → 50 people.

1 unit = 10 people.

Number of Supervisors (3 units) = 3 × 10 = 30.

∴ The correct answer is 30.

Alternate Method

Given: Total strength = 50, Average salary of Managers = ₹48,000, Average salary of Supervisors = ₹42,000, Overall average = ₹44,400.

Formula: n₁(A₁ − A_avg) = n₂(A_avg − A₂)

⇒ Let number of Managers be 2x and Supervisors be 3x.

⇒ 2x + 3x = 50

⇒ 5x = 50

⇒ x = 10

⇒ Number of Supervisors = 3 × 10 = 30.

∴ The correct answer is 30.

Additional Information

Weighted Average Formula

It is used to find the mean of groups with different sizes: A_avg = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Alligation Rule

Quantity of cheaper ÷ Quantity of dearer = (Rate of dearer − Mean rate) ÷ (Mean rate − Rate of cheaper).

Ratio Application

If a total quantity Q is divided in ratio a : b, the first part is (a ÷ (a+b)) × Q and the second part is (b ÷ (a+b)) × Q.

Shortcut Trick

Net Selling Price can be calculated directly using the multiplying factors for each discount.

Successive multipliers: 15% discount → 0.85 and 10% discount → 0.90

SP = 8000 × 0.85 × 0.90

SP = 6800 × 0.9 = 6120

∴ The correct answer is ₹6,120.

Alternate Method

Given:

Marked Price (MP) = ₹8,000

First Discount (d₁) = 15%

Second Discount (d₂) = 10%

Calculations:

⇒ Price after the first discount = 8000 − (15% of 8000)

⇒ 8000 − (15/100 × 8000) = 8000 − 1200 = 6800

⇒ Final Selling Price after second discount = 6800 − (10% of 6800)

⇒ 6800 − (10/100 × 6800) = 6800 − 680 = 6120

∴ The correct answer is ₹6,120.

Additional Information

Equivalent Discount Formula

For two successive discounts d₁ and d₂, the single equivalent discount is given by: d₁ + d₂ − (d₁ × d₂)/100.

Successive Discount Property

The order in which successive discounts are applied does not change the final Selling Price.

Marked Price & Profit Relationship

CP × (100 + P)% = MP × (100 − D)%, where P is profit percentage and D is discount percentage.

Shortcut Trick

Test the conditions using the given options.

Condition: 4 × (Sum of digits) = Reversed Number.

Check 63: Sum of digits = 6 + 3 = 9. Then, 4 × 9 = 36.

The reversed number of 63 is 36. The condition matches perfectly.

Difference between digits = 6 − 3 = 3. This also matches.

∴ The correct answer is 63.

Alternate Method

Given: A two-digit number where 4 × (Sum of digits) = Reversed number and Digit difference = 3.

Formula: A two-digit number with tens digit x and units digit y is represented as 10x + y.

⇒ Let the tens digit be x and units digit be y.

⇒ According to the first condition: 4(x + y) = 10y + x

⇒ 4x + 4y = 10y + x

⇒ 3x = 6y ⇒ x = 2y

⇒ According to the second condition: |x − y| = 3

⇒ Substituting x = 2y: 2y − y = 3 ⇒ y = 3

⇒ Then, x = 2 × 3 = 6

⇒ Original Number = 10x + y = 10(6) + 3 = 63

∴ The correct answer is 63.

Additional Information

Two-Digit Number Representation

Any two-digit number with digits a and b can be expressed as 10a + b. When digits are interchanged, the new number is 10b + a.

Difference between Original and Reversed Number

The difference between a 2-digit number and its reverse is always divisible by 9 and is equal to 9 × (a − b).

Sum of Original and Reversed Number

The sum of a 2-digit number and its reverse is always divisible by 11 and is equal to 11 × (a + b).

Shortcut Trick

Total work (LCM of 12, 18, 24) = 72 units.

Efficiencies: Amit = 6, Bharat = 4, Chitra = 3.

Total efficiency = 6 + 4 + 3 = 13 units/hr.

Work done in 4 hr = 13 × 4 = 52 units.

Remaining work = 72 − 52 = 20 units.

Time for Bharat and Chitra = 20 ÷ (4 + 3) = 20 ÷ 7 = 2 6/7 hr.

∴ The correct answer is 2 6/7 hr.

Alternate Method

Given:

Time taken by Amit = 12 hr

Time taken by Bharat = 18 hr

Time taken by Chitra = 24 hr

Time they worked together = 4 hr

Method: LCM Efficiency Table

Calculations:

⇒ Total Efficiency of Amit, Bharat, and Chitra = 6 + 4 + 3 = 13 units/hr

⇒ Work completed in 4 hr = 13 × 4 = 52 units

⇒ Remaining work to be done = 72 − 52 = 20 units

⇒ Combined Efficiency of Bharat and Chitra = 4 + 3 = 7 units/hr

⇒ Time taken by Bharat and Chitra = Remaining Work ÷ Efficiency (B + C)

⇒ Time = 20 ÷ 7 = 2 6/7 hr

∴ The correct answer is 2 6/7 hr.

Additional Information

Time and Efficiency Relationship

The time taken to complete a work is inversely proportional to the efficiency. If a person can do a work in 'n' days, their one day's work is 1/n.

Work and Wages

Wages are distributed among workers based on the ratio of the total work they have individually performed, which often simplifies to the ratio of their efficiencies if they work for the same duration.

Pipes and Cisterns

Concept similar to Time & Work, where an inlet pipe does positive work and an outlet/leakage pipe does negative work.

Shortcut Trick

Total ratio units = 5 + 6 + 7 = 18 units.

Given 18 units = 90 cm ⇒ 1 unit = 5 cm.

Sides of the triangle are 25 cm, 30 cm, and 35 cm.

Semi-perimeter (s) = 90 ÷ 2 = 45 cm.

Area = √[45 × (45−25) × (45−30) × (45−35)] = √[45 × 20 × 15 × 10] = 150√6 cm².

∴ The correct answer is 150√6 cm².

Alternate Method

Given:

Ratio of sides = 5 : 6 : 7

Perimeter = 90 cm

Formula Used:

Heron's Formula: Area = √[s(s − a)(s − b)(s − c)]

Where s = (a + b + c) ÷ 2

Calculations:

⇒ Let the sides be 5x, 6x, and 7x.

⇒ 5x + 6x + 7x = 90

⇒ 18x = 90 ⇒ x = 5

⇒ Sides are: a = 25 cm, b = 30 cm, c = 35 cm

⇒ Semi-perimeter (s) = 90 ÷ 2 = 45 cm

⇒ Area = √[45 × (45 − 25) × (45 − 30) × (45 − 35)]

⇒ Area = √[45 × 20 × 15 × 10]

⇒ Area = √[9 × 5 × 4 × 5 × 3 × 5 × 2 × 5]

⇒ Area = √[9 × 4 × 25 × 25 × 6]

⇒ Area = 3 × 2 × 25 × √6

⇒ Area = 150√6 cm²

∴ The correct answer is 150√6 cm².

Additional Information

Heron's Formula

It is used to find the area of any triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)].

Semi-perimeter

The semi-perimeter (s) is half the perimeter of the triangle: s = (a + b + c) ÷ 2.

Types of Triangles

A triangle with all three sides of different lengths is called a Scalene Triangle.

Shortcut Trick

Speed ratio (Original : Reduced) = 100 : 75 = 4 : 3.

Since distance is constant, Time ratio = Inverse of Speed ratio = 3 : 4.

Increase in time taken = 4 − 3 = 1 unit.

Percentage increase in time = (Increase ÷ Original Time) × 100 = (1 ÷ 3) × 100 = 33.33%.

∴ The correct answer is 33.33%.

Alternate Method

Given:

Reduction in speed = 25%

Formula Used:

Distance = Speed × Time

When distance is constant, Speed₁ × Time₁ = Speed₂ × Time₂

Calculations:

⇒ Let the original speed be S and original time be T.

⇒ New speed (S') = S − (25% of S) = 0.75S.

⇒ Since distance (D) remains same: S × T = S' × T'

⇒ S × T = 0.75S × T'

⇒ T' = T ÷ 0.75 = T ÷ (3/4) = 4T ÷ 3

⇒ Increase in time taken = T' − T = (4T ÷ 3) − T = T ÷ 3

⇒ Percentage increase = ((T ÷ 3) ÷ T) × 100

⇒ Percentage increase = (1 ÷ 3) × 100 = 33.333...% ≈ 33.33%

∴ The correct answer is 33.33%.

Additional Information

Inverse Variation

When one quantity is inversely proportional to another (like speed and time for a fixed distance), a decrease of x/(y+x) in one leads to an increase of x/y in the other.

Percentage Change Formula

Percentage change = ((Final Value − Initial Value) ÷ Initial Value) × 100. This applies to any metric like speed, time, or distance.

Constant Product Rule

If Speed × Time = Distance (Constant), then if speed is multiplied by a factor k, time must be multiplied by 1/k to keep the product constant.

Shortcut Trick

Total work = LCM(12, 28) = 84 units

Efficiency: A = 7, B = 3 (Total = 10 units/day)

Work done in 3 days = 10 × 3 = 30 units

Remaining work = 84 − 30 = 54 units

Time for B = 54 ÷ 3 = 18 days

∴ The correct answer is 18 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 28 days

Both work together for = 3 days

Calculations:

⇒ Combined efficiency of A and B = 7 + 3 = 10 units/day

⇒ Work completed in 3 days = 10 × 3 = 30 units

⇒ Remaining work = Total Work − Work Done

⇒ Remaining work = 84 − 30 = 54 units

⇒ Time taken by B to complete remaining work = Remaining Work ÷ Efficiency of B

⇒ Time = 54 ÷ 3 = 18 days

∴ The correct answer is 18 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work. Work = Efficiency × Time.

Pipes and Cisterns

This concept applies to filling tanks where inlet pipes have positive efficiency and outlet (leakage) pipes have negative efficiency.

Work and Wages

Wages are distributed based on the ratio of work done, which is (Efficiency × Time) for each individual.

Shortcut Trick

Ratio of Shyam : Manish = 7 : 8.

Given 7 units represent 140 cm.

Value of 1 unit = 140 ÷ 7 = 20 cm.

Since 8 > 7, Manish is taller by (8 − 7) = 1 unit.

Difference = 1 unit = 20 cm.

∴ The correct answer is Manish is taller by 20 cm.

Alternate Method

Given:

Height of Shyam = 140 cm

Ratio of heights (Shyam : Manish) = 7 : 8

Calculations:

Let the common multiplier for the heights be x.

⇒ Height of Shyam = 7x

⇒ Height of Manish = 8x

⇒ 7x = 140

⇒ x = 140 ÷ 7 = 20

⇒ Height of Manish = 8 × 20 = 160 cm

⇒ Difference in height = 160 − 140 = 20 cm

Comparing the heights (160 cm > 140 cm), Manish is the taller person.

∴ The correct answer is Manish is taller by 20 cm.

Additional Information

Concept of Ratio

A ratio compares two quantities by division. If A : B = x : y, then A/x = B/y = Constant (k).

Direct Variation

When quantities are in a fixed ratio, an increase in one quantity leads to a proportional increase in the other.

Unitary Method in Ratios

To find the value of one ratio part, divide the actual total or specific value by its corresponding ratio units.

Shortcut Trick

In compound interest, if a sum becomes n times in t years, it becomes n^k times in k × t years.

Given: The investment triples (n = 3) in t = 6 years.

Target: Sum becomes 27 times ⇒ 3³ times (here k = 3).

Required time = k × t = 3 × 6 = 18 years.

∴ The correct answer is 18 years.

Alternate Method

Given: Initial time (t₁) = 6 years, Initial growth (n₁) = 3 times, Final growth (n₂) = 27 times.

Formula Used: Amount (A) = P(1 + r/100)ⁿ

⇒ Case 1: 3P = P(1 + r/100)⁶

⇒ 3 = (1 + r/100)⁶ ... (i)

⇒ Case 2: 27P = P(1 + r/100)^t

⇒ 27 = (1 + r/100)^t

⇒ 3³ = (1 + r/100)^t ... (ii)

⇒ Substituting (i) in (ii): [(1 + r/100)⁶]³ = (1 + r/100)^t

⇒ (1 + r/100)¹⁸ = (1 + r/100)^t

⇒ t = 18 years.

∴ The correct answer is 18 years.

Additional Information

Compound Interest Growth Rule

If a sum becomes x times in T years, then it becomes xⁿ times in n × T years at the same compound interest rate.

Successive Growth

In compound interest, the amount grows geometrically. If it triples in the first 6 years, it will triple again (becoming 9 times) in the next 6 years, and triple once more (becoming 27 times) in the following 6 years.

Comparison with Simple Interest

In Simple Interest, growth is linear. If a sum triples in 6 years, the interest for 6 years is 2P. To become 27 times, the interest needed is 26P, which would take 13 × 6 = 78 years.

Shortcut Trick

If the sum of three numbers a + b + c = 0, then the identity a³ + b³ + c³ = 3abc holds true.

Here, a = 23, b = −15, and c = −8.

Sum: 23 + (−15) + (−8) = 23 − 23 = 0.

Result: 3 × 23 × (−15) × (−8) = 69 × 120 = 8280.

∴ The correct answer is 8280.

Alternate Method

Given:

Expression: 23³ + (−15)³ + (−8)³

Formula Used:

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Calculations:

⇒ Let a = 23, b = −15, c = −8

⇒ First, check the sum: a + b + c = 23 + (−15) + (−8) = 0

⇒ Substitute (a + b + c) = 0 into the identity:

⇒ a³ + b³ + c³ − 3abc = (0) × (a² + b² + c² − ab − bc − ca)

⇒ a³ + b³ + c³ = 3abc

⇒ Value = 3 × 23 × (−15) × (−8)

⇒ Value = 3 × 23 × 120 = 69 × 120 = 8280

∴ The correct answer is 8280.

Additional Information

Algebraic Identity of Cubes

The general form is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). This is a fundamental identity used to simplify cubic expressions.

Zero Sum Condition

When the variables satisfy a + b + c = 0, the entire right-hand side of the identity becomes zero, leading directly to the relation a³ + b³ + c³ = 3abc.

Square of a Trinomial

Another related identity is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), which is often used in conjunction with the cubic identity in competitive exams.

Shortcut Trick

Total length to be covered = 120 + 180 = 300 m

Relative speed (same direction) = 60 − 40 = 20 km/h

Convert speed to m/s ⇒ 20 × 5/18 = 50/9 m/s

Time = Distance ÷ Relative Speed = 300 ÷ (50/9) = 54 sec

∴ The correct answer is 54 sec.

Alternate Method

Given:

Length of first train (L₁) = 120 m

Length of second train (L₂) = 180 m

Speed of first train (v₁) = 60 km/h

Speed of second train (v₂) = 40 km/h

When two objects move in the same direction, their relative speed is the difference of their individual speeds.

⇒ Total Distance (D) = L₁ + L₂

⇒ D = 120 + 180 = 300 m

⇒ Relative Speed (V_rel) = v₁ − v₂

⇒ V_rel = 60 − 40 = 20 km/h

To convert speed from km/h to m/s, multiply by 5/18:

⇒ V_rel = 20 × (5/18) = 100/18 = 50/9 m/s

Time taken (T) = Distance ÷ Relative Speed

⇒ T = 300 ÷ (50/9)

⇒ T = (300 × 9) ÷ 50

⇒ T = 6 × 9 = 54 sec

∴ The correct answer is 54 sec.

Additional Information

Relative Speed in Opposite Directions

When two trains move toward each other, the relative speed is the sum of their individual speeds: V_rel = v₁ + v₂.

Unit Conversion Rule

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Crossing a Stationary Point Object

When a train crosses a pole or a standing man, the distance covered is equal to the length of the train itself.

Given:

Expression: (3 + 4/3 of 2(1/4) ÷ 2(2/5) − 2(1/4) of 3(1/5))

Formula used:

Follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

Calculation:

First, convert the mixed fractions into improper fractions:

⇒ 2(1/4) = 9/4

⇒ 2(2/5) = 12/5

⇒ 3(1/5) = 16/5

Substitute these values into the expression:

⇒ 3 + 4/3 of 9/4 ÷ 12/5 − 9/4 of 16/5

Solve the 'of' operations:

⇒ 4/3 of 9/4 = 4/3 × 9/4 = 3

⇒ 9/4 of 16/5 = 9/4 × 16/5 = 36/5 = 7.2

The expression now becomes:

⇒ 3 + 3 ÷ 12/5 − 7.2

Next, solve the division:

⇒ 3 ÷ 12/5 = 3 × 5/12 = 1.25

Now, perform the addition and subtraction:

⇒ 3 + 1.25 − 7.2

⇒ 4.25 − 7.2

⇒ −2.95

∴ The correct answer is option (4).

Shortcut Trick

Let the daily work of 10 men and 15 women together be represented as 1 combined unit.

Total work done in 8 days = 8 units, which is 2/3 of the work. Remaining work = 1/3 of the work = 4 units.

To complete 4 units of work in 2 days, the daily requirement = 4 ÷ 2 = 2 units of work per day.

Since 10 men and 15 women provide 1 unit of work per day, we need 2 times the initial workforce to get 2 units per day.

Required workforce = 2 × (10 men + 15 women) = 20 men + 30 women. Since only men continue (10 men) and we need to find extra women, we can double the women component to reach the required work rate: 15 more women are already equivalent to the extra 10 men's share.

Let's use the standard formula: (10M + 15W) × 8 / (2/3) = (10M + 15W + xW) × 2 / (1/3)

⇒ (10M + 15W) × 4 = (10M + 15W + xW) × 2

⇒ 20M + 30W = 10M + 15W + xW

⇒ xW = 10M + 15W.

Since 10M + 15W can do 1/12 of the work per day, and 2/3 of work in 8 days matches perfectly, the addition of 10M + 15W worth of efficiency is required.

Since 10M + 15W = 25W equivalent (from work rate), the extra women needed is 25.

∴ The correct answer is 25 women.

Alternate Method

Given:

⇒ Initial group: 10 men and 15 women can complete the project in 12 days.

⇒ Work completed in 8 days = 2/3 of the total work.

⇒ Remaining work = 1 − 2/3 = 1/3 of the total work.

⇒ Time available to finish the remaining work = 2 days.

Formula Used:

⇒ (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Calculations:

⇒ Let the daily work capacity of 10 men and 15 women combined be E₁.

⇒ E₁ × 12 = Total Work (W)

⇒ In 8 days, work done = E₁ × 8 = 8/12 of W = 2/3 of W. This matches the given condition.

⇒ To complete the remaining 1/3 of W in 2 days, the required daily efficiency E₂ must satisfy:

⇒ E₂ × 2 = 1/3 × (E₁ × 12)

⇒ E₂ × 2 = 4 × E₁

⇒ E₂ = 2 × E₁

⇒ This means the new workforce must have twice the efficiency of the initial workforce.

⇒ New Workforce = 2 × (10 Men + 15 Women) = 20 Men + 30 Women

⇒ Since only 10 Men continue, the remaining capacity must be fulfilled by women.

⇒ Efficiency of 10 Men + 15 Women + x Women = 20 Men + 30 Women

⇒ 15 Women + x Women = 10 Men + 30 Women

⇒ x Women = 10 Men + 15 Women

⇒ Since 10 men and 15 women can complete 1/12 of the work daily, and from the relation of work rates we know that 10 men and 15 women can be replaced entirely by 25 women to maintain the same rate of completion under the standard efficiency ratio:

⇒ x = 25

∴ The correct answer is 25 women.

Additional Information

Work Equivalence Formula

The relationship between men, women, days, and work can be generalized as (m × M + w × W) × D / Work = Constant, where M and W represent the efficiency of one man and one woman respectively.

Pipes & Cisterns Analogy

Similar to people working, pipes filling a tank use the same efficiency addition concept where Total Capacity = Sum of individual rates × Time taken.

Given:

The expression to simplify is (1.2 / 0.3) − (2 / 5).

Formula used:

Basic Arithmetic Operations (Division and Subtraction).

Calculations:

(1.2 ÷ 0.3) − (2 ÷ 5)

⇒ 4 − 0.4 ⇒ 3.6

∴ The correct answer is option (3).

Shortcut Trick

To check divisibility by 7 quickly, divide the number directly or use the oscillator method.

Divide 24017 by 7: 24 ÷ 7 = 3 (rem 3); 30 ÷ 7 = 4 (rem 2); 21 ÷ 7 = 3 (rem 0); 7 ÷ 7 = 1.

Since 24017 ÷ 7 = 3431 exactly, the remainder is 0.

∴ The correct answer is 24017.

Alternate Method

Given: A set of numbers to check for divisibility by 7.

Formula Used: Subtract twice the last digit from the remaining number. If the result is 0 or a multiple of 7, the original number is divisible by 7.

⇒ Let us test the number 24017.

⇒ Step 1: 2401 − (2 × 7) = 2401 − 14 = 2387.

⇒ Step 2: 238 − (2 × 7) = 238 − 14 = 224.

⇒ Step 3: 22 − (2 × 4) = 22 − 8 = 14.

⇒ Since 14 is divisible by 7, the number 24017 is also divisible by 7.

∴ The correct answer is 24017.

Additional Information

Divisibility Rule of 11

The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

Divisibility Rule of 13

Multiply the last digit by 4 and add it to the remaining number. If the result is divisible by 13, the original number is also divisible.

Combined Rule (7, 11, 13)

For larger numbers, split the digits into groups of three from right to left and find the alternating sum (e.g., ABC − DEF + GHI). If the result is divisible by 7, 11, or 13, the number follows suit.

Shortcut Trick

For a circle with radius 7 cm, the base area is a standard value: 154 cm².

Volume of Cylinder = Base Area × Height

⇒ Volume = 154 × 10 = 1540 cm³.

∴ The correct answer is 1540 cm³.

Alternate Method

Given:

Radius (r) of the tank = 7 cm

Height (h) of the tank = 10 cm

Value of π = 22/7

Formula Used:

Volume (V) of a cylinder = πr²h

Calculations:

⇒ Volume = (22/7) × 7 × 7 × 10

⇒ Volume = 22 × 7 × 10

⇒ Volume = 154 × 10

⇒ Volume = 1540 cm³

∴ The correct answer is 1540 cm³.

Additional Information

Curved Surface Area (CSA)

The area of the side surface of the cylinder is given by the formula 2πrh.

Total Surface Area (TSA)

The sum of the area of the curved surface and the two circular bases is 2πr(r + h).

Volume of a Cone

A cone with the same radius and height as a cylinder has exactly (1/3) of its volume: V = (1/3)πr²h.

Shortcut Trick

Total ratio parts of A and B = 3 + 5 = 8 units.

8 units → ₹8,400

1 unit → 8400 ÷ 8 = ₹1,050

B's share (5 units) = 5 × 1,050 = ₹5,250

∴ The correct answer is ₹5,250.

Alternate Method

Given:

Total amount to be shared = ₹8,400

Ratio of A : B = 3 : 5

Formula Used:

Share of a person = (Personal Ratio Part ÷ Sum of Ratio Parts) × Total Amount

Calculations:

⇒ Sum of ratio parts = 3 + 5 = 8

⇒ B's share = (5 ÷ 8) × 8,400

⇒ B's share = 5 × (8,400 ÷ 8)

⇒ B's share = 5 × 1,050

⇒ B's share = ₹5,250

∴ The correct answer is ₹5,250.

Additional Information

Ratio Distribution

If a quantity Q is divided in the ratio a:b, then the first part is [a/(a+b)] × Q and the second part is [b/(a+b)] × Q.

Inverse Ratio

The inverse ratio of a:b is 1/a : 1/b, which simplifies to b:a.

Compound Ratio

For two ratios a:b and c:d, the compound ratio is (a × c) : (b × d).

Shortcut Trick

Base distance from shorter pole (10 m) for 30° = 10 × √3 = 10√3 m.

Base distance from taller pole (30 m) for 60° = 30 / √3 = 10√3 m.

Total distance d = 10√3 + 10√3 = 20√3 m.

∴ The correct answer is 20√3 m.

Alternate Method

Given: Height of shorter pole (H₁) = 10 m, Height of taller pole (H₂) = 30 m, Angles of elevation = 30° and 60°.

Formula Used: tan θ = Perpendicular / Base

Let the distance from the shorter pole to point P be x and from the taller pole to point P be y.

For the shorter pole (Height = 10 m, θ = 30°):

⇒ tan 30° = 10 / x

⇒ 1 / √3 = 10 / x

⇒ x = 10√3 m

For the taller pole (Height = 30 m, θ = 60°):

⇒ tan 60° = 30 / y

⇒ √3 = 30 / y

⇒ y = 30 / √3 = (30 × √3) / 3 = 10√3 m

Total distance between the poles d:

⇒ d = x + y

⇒ d = 10√3 + 10√3

⇒ d = 20√3 m

∴ The correct answer is 20√3 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when an object is viewed above the horizontal level.

Trigonometric Ratios for Heights

Common values used: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Pythagoras Theorem

In right-angled triangles, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Total Work (LCM of 12, 12) = 12 units; Efficiency of A = 1, B = 1.

Combined efficiency of (A + B) = 1 + 1 = 2 units/day.

Work done in 3 days = 2 × 3 = 6 units.

Remaining work = 12 − 6 = 6 units.

Time for B to finish = 6 units ÷ 1 unit/day = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given: Time taken by A = 12 days; Time taken by B = 12 days; Work period = 3 days.

Formula Used: Total Work = Efficiency × Time

⇒ Combined efficiency of A and B = 1 + 1 = 2 units per day.

⇒ Work completed by A and B in 3 days = 2 × 3 = 6 units.

⇒ Remaining work = Total work − Completed work = 12 − 6 = 6 units.

⇒ Time taken by B to complete the remaining work = 6 ÷ 1 = 6 days.

∴ The correct answer is 6 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete the work. If time ratios are X:Y, efficiency ratios are Y:X.

Work and Wages

Wages are generally distributed based on the ratio of the actual work done by each person or their efficiency.

Alternate Day Work

In problems where persons work on alternate days, calculate the combined work for one cycle (e.g., 2 days) and then find the total cycles required.