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Shortcut Trick

Apply the successive multiplying factor for the discounts.

Final Price = Initial Price × (100 − d₁)% × (100 − d₂)%

Final Price = 1200 × 80% × 85%

Final Price = 1200 × 0.8 × 0.85 = 816

∴ The correct answer is 816.

Alternate Method

Given:

Initial price of the laptop = 1200

First discount percentage = 20%

Second discount percentage = 15%

Calculations:

⇒ Amount of first discount = 20% of 1200 = (20/100) × 1200 = 240

⇒ Price after first discount = 1200 − 240 = 960

⇒ Amount of second discount = 15% of 960 = (15/100) × 960 = 144

⇒ Final price of the laptop = 960 − 144 = 816

∴ The correct answer is 816.

Additional Information

Single Equivalent Discount Formula

For two successive discounts of a% and b%, the single effective discount is calculated as (a + b − (a × b)/100)%.

Relationship between MP and SP

Selling Price (SP) = Marked Price (MP) × (100 − Discount%)/100. Discounts are always applied to the Marked Price.

Successive Percentage Change

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100), where r represents the percentage increase (+) or decrease (−).

Shortcut Trick

Total capacity (LCM of 20, 30, 60) = 60 units

Efficiency: A = 3, B = 2, C = −1 (outlet)

Work done in 10 minutes (A+B+C) = (3 + 2 − 1) × 10 = 40 units

Time taken by A+B to fill the remaining work (20 units) = 20 ÷ 5 = 4 minutes

Total time = 10 + 4 = 14 minutes

∴ The correct answer is 14 minutes.

Alternate Method

Given:

Time for pipe A = 20 minutes

Time for pipe B = 30 minutes

Time for outlet pipe C = 60 minutes

Formula used:

Total work = Efficiency × Time

Calculation:

⇒ Initially, all three pipes (A, B, and C) are opened together for 10 minutes.

⇒ Part filled in 10 minutes = (3 + 2 − 1) × 10 = 4 × 10 = 40 units

⇒ Remaining part of the tank = 60 − 40 = 20 units

⇒ Now Pipe C is closed, so only A and B will work.

⇒ Combined efficiency of A and B = 3 + 2 = 5 units/minute

⇒ Time taken to fill the remaining part = 20 ÷ 5 = 4 minutes

⇒ Total time taken to fill the tank = 10 minutes + 4 minutes = 14 minutes

∴ The correct answer is 14 minutes.

Additional Information

Efficiency of an outlet pipe (Negative Efficiency)

The efficiency of a pipe that empties a tank is always considered negative (−) because it reduces the total filled volume.

Combined Efficiency

When multiple pipes work together, their total efficiency is the algebraic sum of their individual efficiencies.

Relationship between time and efficiency

For the same amount of work, time and efficiency are inversely proportional to each other: Time ∝ 1/Efficiency.

Shortcut Trick

The age difference between two people always remains constant (10 years).

5 years ago, the ratio of Rahul to Brother was 2 : 1.

Difference in ratio = 2 − 1 = 1 unit → 10 years.

Rahul's age 5 years ago = 2 units = 2 × 10 = 20 years.

Rahul's present age = 20 + 5 = 25 years.

∴ The correct answer is 25 years.

Alternate Method

Given:

Age difference = 10 years

Condition 5 years ago: Rahul = 2 × Brother

Let Brother's present age be x years.

Then, Rahul's present age = x + 10 years.

⇒ 5 years ago, Brother's age = x − 5

⇒ 5 years ago, Rahul's age = (x + 10) − 5 = x + 5

According to the question:

⇒ x + 5 = 2 × (x − 5)

⇒ x + 5 = 2x − 10

⇒ 2x − x = 10 + 5

⇒ x = 15 years (Brother's present age)

⇒ Rahul's present age = 15 + 10 = 25 years.

∴ The correct answer is 25 years.

Additional Information

Age Difference Invariance

The numerical difference between the ages of two individuals remains the same at any point in time (past, present, or future).

Ratio Method for Ages

If the ratio of ages changes over time, calculate the value of 1 unit by equating the constant difference to the difference in ratio parts.

Time Transition

To find age 'n' years ago, subtract 'n' from the present age. To find age 'n' years hence, add 'n' to the present age.

Shortcut Trick

Ratio of investment for Amit and Raghav = 5 : 7

Total ratio units = 5 + 7 = 12 units

12 units → ₹48,000 (total profit)

1 unit → ₹4,000

Raghav's share = 7 units = 7 × ₹4,000 = ₹28,000

∴ The correct answer is ₹28,000.

Alternate Method

Given:

Investment ratio (Amit : Raghav) = 5 : 7

Total profit = ₹48,000

Formula Used:

Individual profit share = (Individual ratio part ÷ Total ratio parts) × Total profit

Calculations:

⇒ Total ratio parts = 5 + 7 = 12

⇒ Raghav's share = (7 ÷ 12) × 48,000

⇒ Raghav's share = 7 × 4,000

⇒ Raghav's share = ₹28,000

∴ The correct answer is ₹28,000.

Additional Information

Partnership Profit Rule

When the investment time is the same for all partners, the profit is distributed strictly in the ratio of their capital investments.

Compound Partnership

If partners invest for different time periods, the profit ratio is (C₁ × T₁) : (C₂ × T₂), where C is capital and T is time.

Ratio to Value Conversion

To find an individual share from a total value V and ratio a:b, the share of the first person is [a ÷ (a + b)] × V.

Shortcut Trick

The Mode is defined as the value that appears most frequently in a dataset.

Scan the data for repetitions: 53 appears 3 times (at positions 1, 4, and 7).

Other values like 55, 54, 47, and 43 appear only 2 times each.

Since 53 has the maximum frequency, it is the mode.

∴ The correct answer is 53.

Alternate Method

Given Data: 53, 46, 55, 53, 40, 48, 53, 47, 41, 54, 43, 55, 54, 51, 43, 42, 44, 47

To find the Mode, we construct a frequency table for the repeating values:

⇒ Frequency of 43: 2

⇒ Frequency of 47: 2

⇒ Frequency of 53: 3

⇒ Frequency of 54: 2

⇒ Frequency of 55: 2

⇒ Frequency of other values (40, 41, 42, 44, 46, 48, 51): 1 each

⇒ The value with the highest frequency (3) is 53.

∴ The correct answer is 53.

Additional Information

Mean

The arithmetic average calculated as the sum of all observations divided by the total number of observations (Sum ÷ n).

Median

The middle-most value when the data is arranged in ascending or descending order. For even n, it is the average of the two middle terms.

Empirical Relationship

For a moderately skewed distribution, the relationship between measures of central tendency is: Mode = 3 × Median − 2 × Mean.

Types of Mode

A dataset can be unimodal (one mode), bimodal (two modes), trimodal (three modes), or multimodal.

Given:

Equation: 2^x − 2^x−1 = 32

Formula used:

a^m − a^m−n = a^m−n(aⁿ − 1)

Calculations:

2^x − 2^x−1 = 32

⇒ 2^x−1(2¹ − 1) = 32

⇒ 2^x−1(2 − 1) = 32

⇒ 2^x−1(1) = 32

⇒ 2^x−1 = 2⁵

⇒ x − 1 = 5

⇒ x = 6

Now, finding the value of (2x − 1) / (2x + 3) by substituting x = 6:

⇒ (2 × 6 − 1) / (2 × 6 + 3)

⇒ (12 − 1) / (12 + 3)

⇒ 11 / 15

∴ The correct answer is option (1).

Shortcut Trick

Total Interest Percentage = Rate × Time = 8% × 3 = 24%.

Given, 24% of the Principal = ₹1,920.

Principal (100%) = (1,920 ÷ 24) × 100 = 80 × 100 = ₹8,000.

∴ The correct answer is ₹8,000.

Alternate Method

Given: Simple Interest (SI) = ₹1,920, Rate (R) = 8% p.a., Time (T) = 3 years.

Formula Used: SI = (P × R × T) ÷ 100

⇒ 1,920 = (P × 8 × 3) ÷ 100

⇒ 1,920 = (P × 24) ÷ 100

⇒ P × 24 = 1,920 × 100

⇒ P = (1,920 × 100) ÷ 24

⇒ P = 80 × 100

⇒ P = 8,000

∴ The correct answer is ₹8,000.

Additional Information

Simple Interest Principle

Simple Interest is always calculated on the original Principal amount for every time period, unlike compound interest where interest is added back.

Total Amount Formula

The total Amount (A) received at the end of the term is the sum of the Principal and the interest earned: A = P + SI.

Rate and Time Calculations

To find the Rate (R), use R = (SI × 100) ÷ (P × T). To find the Time (T), use T = (SI × 100) ÷ (P × R).

Shortcut Trick

First Number = 3 × 7 = 21

Second Number = (LCM × HCF) ÷ First Number

Second Number = (210 × 3) ÷ 21 = 10 × 3 = 30

∴ The correct answer is 30.

Alternate Method

Given:

L.C.M. = 210

H.C.F. = 3

For the first number: Divisor = 3, Quotient = 7, Remainder = 0

Formula Used:

1. Dividend = (Divisor × Quotient) + Remainder
2. Product of two numbers = L.C.M. × H.C.F.

Calculations:

⇒ First number = (3 × 7) + 0

⇒ First number = 21

⇒ Let the second number be x.

⇒ 21 × x = 210 × 3

⇒ 21 × x = 630

⇒ x = 630 ÷ 21

⇒ x = 30

∴ The correct answer is 30.

Additional Information

Fundamental Property of LCM & HCF

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This property only applies to exactly two numbers.

HCF as a Factor of LCM

The HCF of a set of numbers is always a factor of the LCM of those same numbers.

HCF and LCM of Fractions

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

Shortcut Trick

Ratio of Speed = 40 : 30 = 4 : 3

Since Distance is constant, Time Ratio = Inverse of Speed Ratio = 3 : 4

Difference in time units = 4 − 3 = 1 unit → 20 minutes

Original time (3 units) = 3 × 20 = 60 minutes = 1 hour

Distance = Usual Speed × Usual Time = 40 km/hr × 1 hr = 40 km

∴ The correct answer is 40 km.

Alternate Method

Given:

Usual Speed (S₁) = 40 km/hr

New Speed (S₂) = 30 km/hr

Time Delay (Δt) = 20 minutes = 20/60 hour = 1/3 hour

Formula Used:

Distance (D) = [ (S₁ × S₂) ÷ (S₁ − S₂) ] × Δt

Calculations:

⇒ D = [ (40 × 30) ÷ (40 − 30) ] × (1/3)

⇒ D = [ 1200 ÷ 10 ] × (1/3)

⇒ D = 120 × (1/3)

⇒ D = 40 km

∴ The correct answer is 40 km.

Additional Information

Average Speed Formula

When a person covers two equal distances at speeds x and y, the Average Speed = (2 × x × y) ÷ (x + y).

Speed Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert from m/s to km/hr, multiply by 18/5.

Relative Speed

If two objects move in the same direction at speeds u and v, relative speed = u − v. In opposite directions, relative speed = u + v.

Shortcut Trick

Number of smaller cylinders (n) = (Radius_large / Radius_small)² × (Height_large / Height_small)

n = (35 / 3.5)² × (50 / 15) = 10² × (10 / 3)

n = 100 × 3.333 ≈ 333.33

Since we need the number of complete cylinders, we take the integer part.

∴ The correct answer is 333 cylinders.

Alternate Method

Given:

Radius of large cylinder (R) = 35 cm

Height of large cylinder (H) = 50 cm

Radius of small cylinder (r) = 3.5 cm

Height of small cylinder (h) = 15 cm

Formula Used:

Volume of a cylinder = πr²h

Calculations:

⇒ Volume of large cylinder (V_L) = π × 35 × 35 × 50

⇒ Volume of one small cylinder (V_s) = π × 3.5 × 3.5 × 15

⇒ Let 'n' be the number of smaller cylinders formed.

⇒ n × V_s = V_L

⇒ n = (π × 35 × 35 × 50) ÷ (π × 3.5 × 3.5 × 15)

⇒ n = (35/3.5) × (35/3.5) × (50/15)

⇒ n = 10 × 10 × (10/3)

⇒ n = 1000 / 3 = 333.33...

⇒ Complete cylinders = 333

∴ The correct answer is 333 cylinders.

Additional Information

Volume of Cylinder

The volume of a solid cylinder is calculated as πr²h, where r is the radius of the circular base and h is the vertical height.

Conservation of Volume

When one solid object is melted and recast into other objects, the total volume remains constant: Initial Volume = Total Volume of New Objects.

Surface Area of Cylinder

Curved Surface Area = 2πrh and Total Surface Area = 2πr(r + h).

Complete Objects Calculation

When calculating the number of complete objects, always round down the decimal result to the nearest whole number (Floor value).