Search Previous Year Questions

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.