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Shortcut Trick

Ratio of marks obtained by students = 68 : (100 − 68) = 68 : 32 = 17 : 8.

Difference in ratio units = 17 − 8 = 9 units.

Given: 9 units = 7 marks ⇒ 1 unit = 7 ÷ 9.

Total sum of marks = (17 + 8) units = 25 units.

Calculation: 25 × (7 ÷ 9) = 175 ÷ 9 = 19.44.

∴ The correct answer is 19.44.

Alternate Method

Given: One student scored 7 marks more than the other. Higher marks = 68% of the sum.

Formula Used: Sum = (Numerical Difference) ÷ (Difference in Percentages) × 100

Let the sum of the marks be S.

⇒ Marks of Student A = 68% of S = 0.68S

⇒ Marks of Student B = 100% − 68% = 32% of S = 0.32S

⇒ Difference = 0.68S − 0.32S = 0.36S

⇒ 0.36S = 7

⇒ S = 7 ÷ 0.36 = 700 ÷ 36

⇒ S = 19.444...

∴ The correct answer is 19.44.

Additional Information

Percentage Difference Principle

When comparing two quantities as percentages of the same base, the difference between the quantities is equal to the product of the base and the difference in their percentages: Diff = (P₁ − P₂) × Total.

Ratio of Marks

If A obtains x% of the total and B obtains the rest, the ratio of their marks is x : (100 − x). This ratio can be used to find any value if the absolute difference is known.

Rounding Decimals

To round to two decimal places, look at the third decimal digit; if it is less than 5, keep the second digit same. If 5 or more, increase the second digit by 1.

Shortcut Trick

Apply the Alligation Method to find the mixing ratio.

Ratio = (Dearer Price − Mean Price) : (Mean Price − Cheaper Price)

Ratio = (95 − 80) : (80 − 68)

Ratio = 15 : 12 = 5 : 4

∴ The correct answer is 5 : 4.

Alternate Method

Given:

Cost of Variety 1 (Cheaper) = ₹68 per kg

Cost of Variety 2 (Dearer) = ₹95 per kg

Required Average Cost (Mean Price) = ₹80 per kg

Formula Used:

Ratio of quantities = (Price of Dearer − Mean Price) : (Mean Price − Price of Cheaper)

Calculations:

⇒ Quantity of Cheaper / Quantity of Dearer = (95 − 80) / (80 − 68)

⇒ Ratio = 15 / 12

⇒ Ratio = 5 / 4

⇒ Required Ratio = 5 : 4

∴ The correct answer is 5 : 4.

Additional Information

Rule of Alligation

It is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a specified price.

Weighted Average Formula

Mean Price = (n₁ × C₁ + n₂ × C₂) / (n₁ + n₂), where n is quantity and C is cost.

Mixture Profit Shortcut

If a mixture is sold at profit P%, always convert the Selling Price to Cost Price [CP = (100 × SP) / (100 + P)] before using alligation.

Shortcut Trick

Use the Weighted Average formula for mixture concentration:

Resulting Concentration = [(Ratio_A × Conc_A) + (Ratio_B × Conc_B)] ÷ (Total Ratio)

Resulting Concentration = [(1 × 30%) + (3 × 70%)] ÷ (1 + 3)

= (30% + 210%) ÷ 4 = 240% ÷ 4 = 60%

∴ The correct answer is 60%.

Alternate Method

Given:

Concentration of Solution A = 30%

Concentration of Solution B = 70%

Mixing Ratio (A : B) = 1 : 3

Calculations:

Using the Rule of Alligation:

⇒ (Quantity of A) ÷ (Quantity of B) = (Conc_B − Conc_Mixture) ÷ (Conc_Mixture − Conc_A)

⇒ 1 ÷ 3 = (70 − x) ÷ (x − 30)

⇒ 1 × (x − 30) = 3 × (70 − x)

⇒ x − 30 = 210 − 3x

⇒ x + 3x = 210 + 30

⇒ 4x = 240

⇒ x = 240 ÷ 4

⇒ x = 60%

∴ The correct answer is 60%.

Additional Information

Rule of Alligation

It states that when two ingredients are mixed, the ratio of the quantities is inversely proportional to the differences of their concentrations from the mean concentration: Q₁/Q₂ = (d − m)/(m − c).

Weighted Average Concept

The resultant value of a mixture is the sum of (weight × value) of each component divided by the total weight: Average = (n₁v₁ + n₂v₂) ÷ (n₁ + n₂).

Mixing Multiple Solutions

If more than two solutions are mixed, the final concentration is calculated by ∑(Volume_i × Concentration_i) ÷ ∑Volume_i.

Shortcut Trick

In Compound Interest, if a sum becomes x times in t years, it becomes xⁿ times in n × t years.

Given: 3 times → 5 years

We need: 27 times, which is 3³ times (here n = 3).

Time required = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Alternate Method

Given:

Initial Sum = P

Time (t₁) = 5 years

Amount after 5 years (A₁) = 3P

Final Amount (A₂) = 27P

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ 3P = P(1 + R/100)⁵

⇒ 3 = (1 + R/100)⁵ −−− (Eq. 1)

⇒ Let the required time be t years.

⇒ 27P = P(1 + R/100)^t

⇒ 27 = (1 + R/100)^t

⇒ 3³ = (1 + R/100)^t

⇒ [(1 + R/100)⁵]³ = (1 + R/100)^t [From Eq. 1]

⇒ (1 + R/100)^{5 × 3} = (1 + R/100)^t

⇒ (1 + R/100)¹⁵ = (1 + R/100)^t

⇒ t = 15 years

∴ The correct answer is 15 years.

Additional Information

Standard Compound Interest Formula

The amount A after n years is given by A = P(1 + r/100)ⁿ, where P is the principal and r is the annual interest rate.

Rule of Multiples in CI

In Compound Interest, growth is geometric. If money becomes m times in T years, it will become m^k times in k × T years.

Comparison with Simple Interest

Unlike CI, in Simple Interest (SI), growth is arithmetic. If a sum triples (gains 2P) in 5 years, it will take 5 + 5 + ... years to gain subsequent multiples of P.

Shortcut Trick

Total Simple Interest = ₹684 − ₹380 = ₹304.

Interest for 1 year = 8% of ₹380 = ₹30.4.

Time = Total Interest ÷ Interest per year = 304 ÷ 30.4 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given:

Principal (P) = ₹380

Amount (A) = ₹684

Rate (R) = 8% per annum

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Amount (A) = Principal (P) + Simple Interest (SI)

Calculations:

⇒ SI = A − P

⇒ SI = 684 − 380 = 304

⇒ 304 = (380 × 8 × T) ÷ 100

⇒ 304 × 100 = 3040 × T

⇒ 30400 = 3040 × T

⇒ T = 30400 ÷ 3040

⇒ T = 10 years

∴ The correct answer is 10 years.

Additional Information

Simple Interest Concept

In Simple Interest, the Principal remains constant throughout the entire Time period. The interest earned each year is always the same.

Amount in SI

The total Amount can also be calculated directly using A = P(1 + RT/100).

Unit of Time

In the SI formula, Time (T) must always be in years. If time is given in months, divide by 12; if in days, divide by 365.

Shortcut Trick

Effective Rate for 2 years = x + y + (x × y)/100

Effective Rate = 8 + 8 + (8 × 8)/100 = 16.64%

Total Amount % = 100% (Principal) + 16.64% (Interest) = 116.64%

Amount = 6000 × 116.64% = 6000 × 1.1664 = 6998.40

∴ The correct answer is ₹6,998.40.

Alternate Method

Given:

Principal (P) = ₹6,000

Rate (R) = 8% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 6000(1 + 8/100)²

⇒ A = 6000(108/100)²

⇒ A = 6000 × (1.08) × (1.08)

⇒ A = 6000 × 1.1664

⇒ A = 6998.40

∴ The correct answer is ₹6,998.40.

Additional Information

Compound Interest Formula

Interest alone can be calculated as CI = P[(1 + R/100)ⁿ − 1].

Difference between SI and CI

For 2 years, the difference (D) is given by D = P(R/100)².

Rule of 72

To find the approximate time to double an investment at CI, divide 72 by the annual interest rate.

Shortcut Trick

Age of new member = New Average + (Initial Number of Members × Increase in Average)

⇒ Increase in Average = 30 − 25 = 5 years

⇒ Age of new member = 30 + (11 × 5)

⇒ Age of new member = 30 + 55 = 85 years

∴ The correct answer is 85 years.

Alternate Method

Given:

Initial number of persons = 11

Initial average age = 25 years

New number of persons (including new member) = 12

New average age = 30 years

Formula Used:

Total Age = Average Age × Number of Persons

Calculations:

⇒ Total age of 11 persons = 11 × 25 = 275 years

⇒ Total age of 12 persons = 12 × 30 = 360 years

⇒ Age of new member = Total age of 12 persons − Total age of 11 persons

⇒ Age of new member = 360 − 275

⇒ Age of new member = 85 years

∴ The correct answer is 85 years.

Additional Information

Average Concept

Average = (Sum of all observations) ÷ (Total number of observations).

Inclusion and Exclusion

When a person joins/leaves, the age of the person is calculated by finding the difference between the Total Sum before and after the change.

Net Deviation Rule

The sum of deviations of all items from their average is always zero. New age = Old Average + (New count × Change in average).

Shortcut Trick

To find the divisor of 33124, first check the sum of digits for divisibility by 3: 3+3+1+2+4 = 13. Since 13 is not divisible by 3, eliminate 87 (3 × 29).

Now, test 91 (7 × 13) using the last digit rule or direct division.

Direct division: 33124 ÷ 91 = 364.

Since the division leaves a remainder of 0, the number is exactly divisible by 91.

∴ The correct answer is 91.

Alternate Method

Given:

Dividend = 33124

Divisibility Testing:

⇒ Test 87 (3 × 29): Sum of digits of 33124 = 3 + 3 + 1 + 2 + 4 = 13. Since 13 is not divisible by 3, 33124 is not divisible by 87.

⇒ Test 91 (7 × 13):

⇒ Divisibility by 7: 3312 − (4 × 2) = 3304 → 330 − (4 × 2) = 322 → 32 − (2 × 2) = 28. (28 is divisible by 7).

⇒ Divisibility by 13: 3312 + (4 × 4) = 3328 → 332 + (8 × 4) = 364 → 36 + (4 × 4) = 52. (52 is divisible by 13).

⇒ Since 33124 is divisible by both 7 and 13, it is divisible by 91.

⇒ Test 94 (2 × 47): 33124 ÷ 94 ≈ 352.38 (Not divisible).

⇒ Test 86 (2 × 43): 33124 ÷ 86 ≈ 385.16 (Not divisible).

∴ The correct answer is 91.

Additional Information

Divisibility by 7

Subtract twice the last digit from the number formed by the remaining digits; if the result is divisible by 7, the original number is also divisible by 7.

Divisibility by 13

Add four times the last digit to the number formed by the remaining digits; if the result is divisible by 13, the original number is also divisible by 13.

Divisibility by 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Composite Divisibility

If a number is divisible by two co-prime numbers a and b, it is also divisible by their product (a × b).

Shortcut Trick Apply the Alligation Rule directly using the cost prices and mean price. Ratio of cheaper sugar to dearer sugar = (CP of dearer − Mean price) : (Mean price − CP of cheaper) Ratio = (45 − 35) : (35 − 25) = 10 : 10 Simplified Ratio = 1 : 1 ∴ The correct answer is 1 : 1. Alternate Method Given: Cost of Type 1 sugar = ₹45 per kg Cost of Type 2 sugar = ₹25 per kg Selling Price of mixture (at no profit/loss) = Mean Price = ₹35 per kg

Calculations: ⇒ Let the quantity of Type 1 sugar be q1 and Type 2 sugar be q2. ⇒ Total Cost Price = 45 × q1 + 25 × q2 ⇒ Total Selling Price = 35 × (q1 + q2) ⇒ Since there is no profit and no loss, Cost Price = Selling Price ⇒ 45q1 + 25q2 = 35q1 + 35q2 ⇒ 45q1 − 35q1 = 35q2 − 25q2 ⇒ 10q1 = 10q2 ⇒ q1 / q2 = 10 / 10 = 1 / 1 ∴ The correct answer is 1 : 1. Additional Information Alligation Rule It is used to find the ratio in which two or more ingredients at their respective prices must be mixed to produce a mixture at a given price. Mean Price The mean price (M) always lies between the cost price of the cheaper (C) and the cost price of the dearer (D) ingredient: C < M < D. Weighted Average Formula The mean price can also be calculated as M = (n1c1 + n2c2) ÷ (n1 + n2), where n represents quantity and c represents cost.

Given:

30^x−1 = 125 × 15 × 2⁴ × 27

Formula used:

1. a^m × aⁿ = a^m+n
2. aⁿ × bⁿ × cⁿ = (a × b × c)ⁿ
3. If a^m = aⁿ, then m = n

Calculations:

First, we factorize the numbers on the right-hand side into their prime factors:

⇒ 125 = 5³

⇒ 15 = 3 × 5

⇒ 27 = 3³

Now, substitute these values into the equation:

⇒ 30^x−1 = 5³ × (3 × 5) × 2⁴ × 3³

Group the same bases together:

⇒ 30^x−1 = 2⁴ × (3¹ × 3³) × (5³ × 5¹)

⇒ 30^x−1 = 2⁴ × 3¹⁺³ × 5³⁺¹

⇒ 30^x−1 = 2⁴ × 3⁴ × 5⁴

Since the exponents are the same, we can combine the bases:

⇒ 30^x−1 = (2 × 3 × 5)⁴

⇒ 30^x−1 = 30⁴

Comparing the exponents of the same base (30):

⇒ x − 1 = 4

⇒ x = 4 + 1

⇒ x = 5

∴ The correct answer is option 4 (5).