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Shortcut Trick

Net percentage growth for 2 years at 6% rate = 6 + 6 + (6 × 6)/100 = 12.36%.

Current Population = 50,000.

Total Growth = 12.36% of 50,000 = 0.1236 × 50,000 = 6,180.

Population after 2 years = 50,000 + 6,180 = 56,180.

∴ The correct answer is 56180.

Alternate Method

Given:

Current Population (P) = 50,000

Growth Rate (R) = 6% per annum

Time Period (n) = 2 years

Formula Used:

Population after n years = P × (1 + R/100)ⁿ

⇒ Population after 1 year = 50,000 × (1 + 6/100)

⇒ Population after 1 year = 50,000 × 1.06 = 53,000

⇒ Population after 2 years = 53,000 × (1 + 6/100)

⇒ Population after 2 years = 53,000 × 1.06

⇒ Population after 2 years = 56,180

∴ The correct answer is 56180.

Additional Information

Successive Percentage Change

When a value increases by x% and then y% successively, the net percentage change is given by [x + y + (xy/100)]%.

Depreciation Formula

If the population decreases at a rate R, the population after n years is calculated as P × (1 − R/100)ⁿ.

Compound Growth Concept

Growth in population, bacteria, or compound interest follows the same mathematical model of exponential growth.

Given:

The marked price of article = Rs. 1200

Discounts are 30% and 40%

Concept Used:

Discount is always given on the marked price.

Formula used:

SP = MP × (100 - D)/100

Here, SP → Selling price, D → Discount

Calculation:

The selling price of article = \(1200 × {70 \ \over 100} × {60 \ \over 100} = 504\)

∴ The selling price of an article after two successive discounts is Rs. 504

Shortcut Trick

Effective Selling Price after first discount = Final SP ÷ (1 − Second Discount %)

Second discount = 12.5% = 1/8. So, Price after 1^st discount = 105 × (8/7) = ₹120.

First discount = List Price − Intermediate Price = 150 − 120 = ₹30.

First discount % = (30/150) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: List Price (MP) = ₹150, Final SP = ₹105, D₂ = 12.5%.

Formula: Final SP = MP × (1 − D₁/100) × (1 − D₂/100)

⇒ 105 = 150 × (1 − D₁/100) × (1 − 12.5/100)

⇒ 105 = 150 × (1 − D₁/100) × (87.5/100)

⇒ 105 = 150 × (1 − D₁/100) × (7/8)

⇒ 105 ÷ 150 = (1 − D₁/100) × (7/8)

⇒ 0.7 = (1 − D₁/100) × 0.875

⇒ (1 − D₁/100) = 0.7 ÷ 0.875 = 0.8

⇒ D₁/100 = 1 − 0.8 = 0.2

⇒ D₁ = 20%

∴ The correct answer is 20%.

Additional Information

Successive Discount Formula

The single equivalent discount for two successive discounts a% and b% is given by [a + b − (ab/100)]%.

Marked Price and Selling Price Relationship

Selling Price = Marked Price × (100 − Discount%)/100. Discount is always calculated on the Marked Price.

Fraction Equivalents

Using fractions like 12.5% = 1/8 or 20% = 1/5 simplifies multi-step discount calculations significantly.

Formula Used:

1. Relative speed: If two objects travel with speeds u & v respectively

• In the same direction, Relative speed = u - v
• In the opposite direction, Relative speed = u + v

2. Time = Distance/speed

3. 1 km/hr = 5/18 m/s

Calculation:

Using above formula

Relative speed = (50 - 5) = 45 km/hr

Relative speed = 45 × \(\frac{5}{18}\) = 12.5 m/s

Time taken = \(\frac{125}{12.5}\) = 10 sec

∴ The correct answer is 10 sec

Shortcut Trick

Speed of Stream = Speed in still water − Upstream speed

⇒ Speed of Stream = 10 − 8 = 2 km/hr

∴ The correct answer is 2 km/hr.

Alternate Method

Given:

Speed of boat in still water (u) = 10 km/hr

Speed of boat in upstream (U_s) = 8 km/hr

Formula Used:

Upstream Speed (U_s) = Speed in still water (u) − Speed of stream (v)

Calculations:

⇒ 8 = 10 − v

⇒ v = 10 − 8

⇒ v = 2 km/hr

∴ The correct answer is 2 km/hr.

Additional Information

Downstream Speed

It is the speed of the boat in the direction of the stream. Formula: Downstream Speed = Speed in still water + Speed of stream.

Still Water Speed Calculation

If downstream (D) and upstream (U) speeds are given, speed in still water = (D + U) ÷ 2.

Stream Speed Calculation

If downstream (D) and upstream (U) speeds are given, speed of stream = (D − U) ÷ 2.

Given

Amount = Rs. 13,200

Time = 4 years

Rate = 5% per an um

Formula used

Simple interest = (p × r × t)/100

Where p, r and t are principal, rate and time respectively.

Simple interest = Amount - Principal

Calculation

Let the principal is Rs. x

Simple interest = 13,200 - x

13,200 - x = (x × 5 × 4)/100

⇒ 1320000 - 100x = 20x

⇒ 1320000 = 120x

⇒ x = 1320000/120

⇒ x = Rs. 11,000

∴ Rs. 11,000 is a sum borrowed by Mr Peter.

Shortcut Trick

Total mixture = 400 liters, Quantity A = 240 liters.

Quantity B = 400 − 240 = 160 liters.

Ratio A : B = 240 : 160.

Divide both by HCF (80): 240 ÷ 80 : 160 ÷ 80 = 3 : 2.

∴ The correct answer is 3 : 2.

Alternate Method

Given:

Total quantity of mixture = 400 liters

Quantity of component A = 240 liters

Formula used: Quantity B = Total Mixture − Quantity A

Total Mixture = 400 L A = 240 L B = ? Mixture Components Distribution

Calculations:

⇒ Total volume = 400 liters

⇒ Quantity of A = 240 liters

⇒ Quantity of B = 400 − 240 = 160 liters

⇒ Required Ratio = A : B

⇒ Ratio = 240 : 160

⇒ Ratio = 24 : 16

⇒ Ratio = 3 : 2

∴ The correct answer is 3 : 2.

Additional Information

Ratio and Proportion

A ratio is a comparison of two quantities by division, expressed as a : b. It must always be simplified to its lowest terms by dividing with the HCF.

Percentage in Mixture

To find the percentage of a component: (Quantity of component ÷ Total Mixture) × 100. For A, it is (240 ÷ 400) × 100 = 60%.

Quantity from Ratio

If ratio A:B is x:y and total is T, then A = [x ÷ (x + y)] × T and B = [y ÷ (x + y)] × T.

Shortcut Trick

If x% of the first number = y% of the second number, the ratio of the first to the second is directly y : x.

First Number : Second Number = 64 : 28

⇒ First Number : Second Number = 16 : 7

∴ The correct answer is 16 : 7.

Alternate Method

Given:

28% of the first number = 64% of the second number

Calculations:

Let the first number be A and the second number be B.

⇒ 28% of A = 64% of B

⇒ (28 ÷ 100) × A = (64 ÷ 100) × B

⇒ 28 × A = 64 × B

⇒ A ÷ B = 64 ÷ 28

Dividing numerator and denominator by their highest common factor (4):

⇒ A ÷ B = 16 ÷ 7

⇒ A : B = 16 : 7

∴ The correct answer is 16 : 7.

Additional Information

Percentage to Ratio Conversion

If a% of x = b% of y, then the ratio of x to y is strictly the inverse ratio of their percentages, which directly means x : y = b : a.

Properties of Proportions

If a : b = c : d, then a × d = b × c. This fundamental rule states that the product of the extremes equals the product of the means.

Simplifying Ratios

A ratio a : b is in its simplest form when the highest common factor (HCF) of a and b is exactly 1. In the given question, 64 and 28 are divided by their HCF, which is 4.

Shortcut Trick

Difference in ratio units between B and A = 5 − 2 = 3 units.

Given income difference = Rs. 291

1 unit = 291 ÷ 3 = Rs. 97

Income of C = 11 units = 11 × 97 = Rs. 1067

∴ The correct answer is Rs. 1067.

Alternate Method

Given:

Ratio of daily income of A, B and C = 2 : 5 : 11

Income of B is Rs. 291 more than that of A.

A

2x

B

2x

3x

3x = Rs. 291

C

11x

11x = 11 × 97 = Rs. 1067

Calculation:

Let the daily incomes of A, B and C be 2x, 5x and 11x respectively.

⇒ Income of B − Income of A = 291

⇒ 5x − 2x = 291

⇒ 3x = 291

⇒ x = 291 ÷ 3

⇒ x = 97

Now, the income of C is 11x.

⇒ Income of C = 11 × 97

⇒ Income of C = 1067

∴ The correct answer is Rs. 1067.

Additional Information

Ratio Multiplier

When quantities are given in a ratio a : b : c, their actual values can be assumed as ax, bx, and cx where x is a common multiplying factor.

Direct Proportion

If two quantities are in direct proportion, their ratio remains constant. If one quantity increases, the other increases at exactly the same rate.

Difference in Ratio

The real-world difference between two absolute values corresponds directly to the difference in their ratio parts. This is the fastest way to find the base multiplier.

We know;

Area of square = (a)² where a = side of square

And,  Area of rectangle = (l × b); where l = length and b = breadth

Given: a = 12 cm, l = 18 cm

And sum of area of square and area of rectangle = 252 cm² 

Calculation:

(a)² + (l × b) = 252

Firstly, , we need to find the value of b. Put value of 'a' and 'l';

⇒ (12)² + (18 × b) = 252

⇒ 144 + 18 × b = 252 

⇒ 18 × b = 252 - 144 

⇒ 18 × b = 108

⇒ b = 108 ÷ 18

⇒ b = 6 cm

Now,

Perimeter of rectangle = 2 (l + b)

⇒ 2 (18 + 6)

⇒ 2 × 24

⇒ 48 cm

Hence, the correct answer is "48 cm".