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Calculation:

Principal = 2500 & Amount = 8100

As we know, A = P (1 + r/100)²

⇒ (1 + r/100)² = 81/25

⇒ 1 + r/100 = 9/5

⇒ r = 80%

Amount after 4 years

A = 2500 (1 + 80/100)⁴

⇒ A = 2500 × 9/5 × 9/5 × 9/5 × 9/5

∴ A = 26244

Given: The simplified value of the equation is 143.

1980 ÷ 9 + [-77 + {-1980 + (1/4 of 7920)}]

Concept used:

BODMAS rule

Calculations:

1980 ÷ 9 + [-77 + {-1980 + (1/4 × 7920)}]

⇒ 1980 ÷ 9 + [-77 + {-1980 + (7920/4)}]

⇒ 1980 ÷ 9+ [-77 + {-1980 + (1980)}]

⇒ 1980 ÷ 9+ [-77 + {-1980 + 1980}]

⇒ 1980 ÷ 9 + [-77 + {0}]

⇒ 220 + [-77]

⇒ 220 - 77 = 143

∴ The simplified value of the equation is 143.

Shortcut Trick

Time taken by both pipes to fill the whole tank = (12 × 36) ÷ (12 + 36)

Time taken for whole tank = 432 ÷ 48 = 9 hours

To fill one-third of the tank, the time required is simply 1/3 of the total time.

Required time = 9 × (1/3) = 3 hours

∴ The correct answer is 3 hours.

Alternate Method

Given:

Time taken by Pipe X to fill the tank = 12 hours

Time taken by Pipe Y to fill the tank = 36 hours

Calculations:

⇒ Total capacity of the tank (Total Work) = 36 units

⇒ Combined efficiency of Pipe X and Pipe Y = 3 + 1 = 4 units/hour

⇒ We need to find the time to fill one-third part of the tank.

⇒ Required work = 36 × (1/3) = 12 units

⇒ Time required = Required work ÷ Combined efficiency

⇒ Time required = 12 ÷ 4 = 3 hours

∴ The correct answer is 3 hours.

Additional Information

Two Pipes Filling a Tank

If pipe A fills a tank in x hours and pipe B fills it in y hours, total time taken together = (x × y) ÷ (x + y) hours.

One Filling and One Emptying Pipe

If pipe A fills in x hours and pipe B empties in y hours (where y > x), time taken to fill = (x × y) ÷ (y − x) hours.

Part of Tank Filled

To fill a specific fraction of the tank, simply multiply the total time taken to fill the entire tank by that fraction.

(Formula Used:

1. SI = \(P × R× T\over100\)

For CI

2. A = P × (1 + \(R\over 100\))T

3. CI = A - P

Where, P = Principal, R = Rate, T = Time, A = Amount

4. Effective rate = r1 + r2 + \(r_1 × r_2 \over 100\)

Where “r1” and “r2” are rates of interest during successive years

Concept Used:

SI and CI for the first year is the same. The difference in the CI and SI for 2 years is due to the compounding part of the interest for the first year.

Calculation:

According to the question

SI for 2 years = ₹ 1,080

⇒ SI for 1 year = ₹ 1,080 ÷ 2 = ₹ 540

Difference between CI and SI for 2 years

⇒ ₹1,128.6 -  ₹1,080 =  ₹48.6

Using SI formula

⇒ R = \(48.6× 100\over540× 1\) = 9%

Calculating principal using the SI formula

⇒ P = \(1080× 100\over9× 2\) = ₹ 6000

Effective compound interest rate for 2 years

⇒ 9 + 9 +\(9 × 9\over100\) = 18.81%

Effective compound interest rate for 3 years

⇒ 18.81 + 9 + \(18.81 × 9\over100\) = 27.81 + 1.6929 = 29.5029%

Effective simple interest rate for 3 years = 3 × 9 = 27%

Since the difference in the interest would be due to the difference in the rate component

⇒ Difference in the interest rate = 29.5029% - 27% = 2.5029%

⇒ Difference in the interest at 2.5029% = \(6000× 2.5029× 1\over100\) = ₹150.174

∴ The difference in the interest is ₹ 150.174.

Alternate Method Formula Used:

For 2 years, CI - SI = P × (R/100)²

For 3 years, CI - SI = P × (R/100)² × (300 + R)/100

Calculation:

SI for 2 years = ₹ 1,080

⇒ SI for 1 year = ₹ 1,080 ÷ 2 = ₹ 540

Difference between CI and SI for 2 years

⇒ ₹1,128.6 -  ₹1,080 =  ₹48.6

Using SI formula

⇒ R = \(48.6× 100\over540× 1\) = 9%

Calculating principal using the SI formula

⇒ P = \(1080× 100\over9× 2\) = ₹ 6000

Now, according to the formula for 3 years

CI - SI = 6000 × (9/100)² × (300 + 9)/100

⇒ CI - SI = 6000 × (81/10000) × 309/100

⇒ CI - SI = 150.174

∴ The correct answer is option 3

given: 

Total number of persons 8. Is

Average weight increased by 2.5 kg

Calculation:

total weight increased,

8 x 2.5 kg = 20 kg

Replaced person's weight = 45 kg

new person's weight,

(45 + 20) kg  = 65 kg

The weight of the new person is 65 kg.

Shortcut Trick

Age difference remains constant at 8 years.

4 years ago, Ratio A : B = 1 : 2. Difference = 2 − 1 = 1 unit.

1 unit = 8 years → B's age 4 years ago = 2 × 8 = 16 years.

Present age of B = 16 + 4 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given: Present age difference = 8 years. Ratio 4 years ago = 1 : 2.

Formula: Age = Ratio unit × Value + Time elapsed.

⇒ Let ages 4 years ago be x and 2x.

⇒ Difference = 2x − x = 8

⇒ x = 8

⇒ B's age 4 years ago = 2 × 8 = 16 years.

⇒ B's Present age = 16 + 4 = 20 years.

∴ The correct answer is 20 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two persons never changes regardless of the number of years passed or gone by.

Ratio Proportionality

If ages are in ratio a : b and the difference is D, then one unit equals D ÷ (b − a).

Shortcut Trick

Profit % = [Error ÷ (True Weight − Error)] × 100

Here, Error = 250 gm − 200 gm = 50 gm

Profit % = (50 ÷ 200) × 100

Profit % = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

True Weight (Promised) = 250 gm

False Weight (Actual) = 200 gm

Formula Used:

Profit % = [ (True Weight − False Weight) ÷ False Weight ] × 100

Calculations:

⇒ Profit = 250 gm − 200 gm = 50 gm

⇒ Profit % = (50 ÷ 200) × 100

⇒ Profit % = (1 ÷ 4) × 100

⇒ Profit % = 25%

The dishonest merchant gains 25% by using a false weight.

∴ The correct answer is 25%.

Additional Information

Dishonest Dealer Concept

When a dealer sells goods at cost price but uses less weight, the profit is calculated on the actual weight delivered to the customer.

Combined Profit (Cheating at both ends)

If a dealer cheats by x% while buying and y% while selling, the total profit % = [ (100 + x)(100 + y) ÷ 100 ] − 100.

Weight and Price Relationship

Price is inversely proportional to weight for a fixed cost. Ratio of Weights = 250:200 = 5:4, then Ratio of Profit (Price) = 4:5, giving a 1/4 = 25% gain.

Shortcut Trick

List all prime numbers in the range 60 to 90: 61, 67, 71, 73, 79, 83, 89.

Apply grouping method for faster addition:

(61 + 89) + (67 + 83) + (71 + 79) + 73

⇒ 150 + 150 + 150 + 73 ⇒ 450 + 73 = 523.

∴ The correct answer is 523.

Alternate Method

Given: Numbers in the range (60, 90).

A prime number is a natural number greater than 1 that is divisible only by 1 and itself. We check every odd number in the range (since even numbers > 2 are composite):

⇒ Identify primes between 60 and 70: 61, 67

⇒ Identify primes between 70 and 80: 71, 73, 79

⇒ Identify primes between 80 and 90: 83, 89

⇒ Sum = 61 + 67 + 71 + 73 + 79 + 83 + 89

⇒ Sum = (61 + 67) + (71 + 73 + 79) + (83 + 89)

⇒ Sum = 128 + 223 + 172

⇒ Sum = 523.

∴ The correct answer is 523.

Additional Information

Prime Numbers

Numbers that have exactly two factors: 1 and the number itself. 2 is the only even prime number.

Composite Numbers

Numbers having more than two factors. 1 is neither prime nor composite.

Twin Primes

Pairs of prime numbers that have a difference of 2 (e.g., 71 and 73 in this range).

Testing Primality

To check if a number N is prime, divide it by primes up to √N. If no prime divides N, then N is prime.

Given : 

Ratio of a chemical X and water is 15 ∶ 9.

48 litres of this mixture is replaced with water, the ratio becomes 11 ∶ 13.

Calculation : 

48 liters of water is taken out in which chemical X is 30 and water is 18.

⇒ 15x - 30/(9x - 18 + 48) = 11/13

⇒ 15x - 30/(9x + 30) = 11/13

⇒ 195x - 390 = 99x + 330

⇒ 96x = 720

⇒ x = 720/96 = 7.5

Quantity of water after replacement = 9 × 7.5 + 30 = 97.5

∴ The correct answer is 97.5.

Shortcut Trick

Ratio of investments = 1000 : 1200 : 1500 = 10 : 12 : 15.

Since the time period is identical, profit is distributed in the ratio of their investments.

Share of Q = [12 ÷ (10 + 12 + 15)] × Total Profit

Share of Q = (12 ÷ 37) × 666 = 12 × 18 = 216.

∴ The correct answer is ₹216.

Alternate Method

Given:

Investment of P = ₹1000

Investment of Q = ₹1200

Investment of R = ₹1500

Total profit = ₹666

Formula Used:

When the investment time is the same for all partners, Profit Ratio = Capital Ratio.

Investment and Profit Share Ratio

P 10 parts ₹1000

Q 12 parts ₹1200

R 15 parts ₹1500

Total Profit = ₹666 (37 parts)

Calculation:

⇒ Ratio of profits (P : Q : R) = 1000 : 1200 : 1500

⇒ Simplifying the ratio, we get 10 : 12 : 15

⇒ Sum of ratio parts = 10 + 12 + 15 = 37 parts

⇒ Value of 37 parts = ₹666

⇒ Value of 1 part = 666 ÷ 37 = ₹18

⇒ Profit share of Q = 12 parts

⇒ Share of Q = 12 × 18 = ₹216

∴ The correct answer is ₹216.

Additional Information

Different Time Periods

If partners invest for different durations, their profit is shared in the ratio of the product of their Capital and Time (C₁ × T₁ : C₂ × T₂).

Working & Sleeping Partners

A working partner receives a fixed salary or a specific percentage of the total profit for managing the business. The remaining profit is distributed among all partners according to their investment ratios.

Compound Partnership

When a partner withdraws or adds money during the year, their capital is broken into time segments. The equivalent capital is calculated as (Amount₁ × Months₁) + (Amount₂ × Months₂).