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Shortcut Trick

Convert the fractions into decimals for faster and easier multiplication.

⇒ 2 ²/₅ = 2.4

⇒ ³/₄ = 0.75

⇒ Product = 1.2 × 2.4 × 0.75

⇒ Rearranging for ease: (1.2 × 0.75) × 2.4 = 0.9 × 2.4 = 2.16

∴ The correct answer is 2.16.

Alternate Method

Given:

Expression: 1.2 × 2 ²/₅ × ³/₄

Calculation:

⇒ Convert 1.2 to a decimal fraction: 1.2 = ¹²/₁₀ = ⁶/₅

⇒ Convert the mixed fraction into an improper fraction: 2 ²/₅ = (2 × 5 + 2) / 5 = ¹²/₅

⇒ Substitute these back into the expression:

⇒ ⁶/₅ × ¹²/₅ × ³/₄

⇒ Divide 12 by 4 to simplify:

⇒ (6 × 3 × 3) / (5 × 5)

⇒ ⁵⁴/₂₅

⇒ Multiply the numerator and denominator by 4 to get a base of 100 for immediate decimal conversion:

⇒ (54 × 4) / (25 × 4) = ²¹⁶/₁₀₀

⇒ 2.16

∴ The correct answer is 2.16.

Additional Information

Fraction to Decimal Conversion

Memorising standard fractions significantly speeds up calculation in exams: ¹/₄ = 0.25, ³/₄ = 0.75, ¹/₅ = 0.2, and ²/₅ = 0.4.

Simplification Trick for Base 100

When dividing any number by 25, simply multiply the numerator by 4 and place the decimal two spots from the right (effectively dividing by 100).

Mixed Fractions Rule

A mixed fraction represented as a(^b/_c) is converted into an improper fraction using the mathematical rule: (a × c + b) ÷ c.

Shortcut Trick

Observe the relationship between the numerator and the denominator.

Each term in the denominator is exactly 3 times the corresponding term in the numerator.

Expression = (0.1³ + 0.03³) ÷ [(3 × 0.1)³ + (3 × 0.03)³]

⇒ 1 ÷ 3³ = 1 ÷ 27

⇒ 1 ÷ 27 ≈ 0.037

∴ The correct answer is 0.037.

Alternate Method

Given:

Expression = (0.1 × 0.1 × 0.1 + 0.03 × 0.03 × 0.03) ÷ (0.3 × 0.3 × 0.3 + 0.09 × 0.09 × 0.09)

Formula Used:

(x × y)ⁿ = xⁿ × yⁿ

Calculations:

⇒ Numerator = 0.1³ + 0.03³

⇒ Denominator = 0.3³ + 0.09³

⇒ Denominator = (3 × 0.1)³ + (3 × 0.03)³

⇒ Denominator = 3³ × 0.1³ + 3³ × 0.03³

⇒ Denominator = 27 × (0.1³ + 0.03³)

⇒ Expression = (0.1³ + 0.03³) ÷ [27 × (0.1³ + 0.03³)]

⇒ Expression = 1 ÷ 27

⇒ 1 ÷ 27 = 0.037037...

∴ The correct answer is 0.037.

Additional Information

Laws of Exponents

When terms are multiplied and raised to a common power, the power distributes directly to both numbers: (xy)ⁿ = xⁿ × yⁿ. This allows easy extraction of common multiples from complex decimals.

Algebraic Simplification

In simplification problems involving decimals that look similar, always check if the denominator is a direct multiple of the numerator before calculating the actual cubes or squares.

Recurring Decimals

Fractions like 1 ÷ 27 result in non-terminating, repeating decimals. The value 1 ÷ 27 equals 0.037037037..., which is generally rounded off to 0.037 as given in the exam options.

Shortcut Trick Heights strictly less than 1.65 m are 1.55 m, 1.60 m, and 1.62 m. Students in Section A = 5 + 9 + 11 = 25 Students in Section B = 4 + 8 + 13 = 25 Required Ratio = 25 : 25 = 1 : 1 ∴ The correct answer is 1:1.

Alternate Method Given: A table containing the heights of students in two sections, A and B. We need the ratio of students with heights less than 1.65 metres in Section A to Section B. Calculations: ⇒ From the table, heights less than 1.65 m correspond to 1.55 m, 1.60 m, and 1.62 m. ⇒ Total number of students in Section A with height < 1.65 m = 5 + 9 + 11 ⇒ Total number of students in Section A = 25 ⇒ Total number of students in Section B with height < 1.65 m = 4 + 8 + 13 ⇒ Total number of students in Section B = 25 ⇒ Required Ratio = (Total students in Section A) : (Total students in Section B) ⇒ Required Ratio = 25 : 25 ⇒ Required Ratio = 1 : 1 ∴ The correct answer is 1:1.

Additional Information Ratio Comparison A ratio compares two quantities of the same kind. It is typically expressed in its simplest form by dividing both components by their highest common factor. Tabular Data Interpretation When verifying conditions such as 'less than a value', you only aggregate the frequencies of the categories strictly falling under that boundary condition.

Shortcut Trick

Profit Ratio = Ratio of (Capital × Time)

Since the time is the same (1 year), Profit Ratio = Capital Ratio

A : B : C = 40,000 : 60,000 : 80,000 = 2 : 3 : 4

B's share = [3 ÷ (2 + 3 + 4)] × 55,000 = (3 ÷ 9) × 55,000

B's share = 55,000 ÷ 3 = 18,333.33

∴ The correct answer is ₹18,333.33.

Alternate Method

Given:

A's investment = ₹40,000

B's investment = ₹60,000

C's investment = ₹80,000

Total time for each = 1 year

Total profit = ₹55,000

Formula Used:

Profit Ratio = (Investment of A × Time) : (Investment of B × Time) : (Investment of C × Time)

Share of a partner = (Partner's ratio part ÷ Total ratio parts) × Total Profit

₹40k A

₹60k B

₹80k C

Profit Ratio = 2 : 3 : 4

Calculations:

⇒ Ratio of profits of A, B, and C = 40000 : 60000 : 80000

⇒ Ratio of profits = 4 : 6 : 8

⇒ Simplifying the ratio further, we get A : B : C = 2 : 3 : 4

⇒ Sum of ratio parts = 2 + 3 + 4 = 9 parts

⇒ B's share in profit = (3 parts ÷ 9 parts) × Total Profit

⇒ B's share = (1 ÷ 3) × 55,000

⇒ B's share = 55,000 ÷ 3 = 18,333.33

∴ The correct answer is ₹18,333.33.

Additional Information

Simple Partnership

When all partners invest their capital for the same time period, the profits or losses are distributed exactly in the ratio of their original investments.

Compound Partnership

When partners invest their capital for different time periods, the profit sharing ratio is determined by multiplying each partner's capital by their respective time period (C₁T₁ : C₂T₂).

Sleeping Partner

A partner who only invests capital but does not actively participate in the management of the business. They still receive a share of the profit proportional to their investment unless explicitly stated otherwise.

Shortcut Trick

Total Capital = 60,000 + 90,000 = 1,50,000

Total Interest = 10% of 1,50,000 = 15,000

Remaining Profit = 25,000 − 15,000 = 10,000

Profit Ratio = 60,000 : 90,000 = 2 : 3

A's Total Share = (10% of 60,000) + (2/5 × 10,000) = 6,000 + 4,000 = 10,000

∴ The correct answer is ₹10,000.

Alternate Method

₹60,000 Ratio: 2 A

₹90,000 Ratio: 3 B

Given:

A's investment = ₹60,000

B's investment = ₹90,000

Total profit including interest = ₹25,000

Rate of interest on capital = 10% per annum

Formula Used:

Total Interest = Interest on A's capital + Interest on B's capital

Profit Ratio = (A's Investment × Time) : (B's Investment × Time)

Calculation:

⇒ Total investment = 60,000 + 90,000 = ₹1,50,000

⇒ Total interest to be paid = 10% of 1,50,000 = ₹15,000

⇒ Remaining profit = 25,000 − 15,000 = ₹10,000

⇒ Ratio of investments A:B = 60,000 : 90,000 = 2 : 3

⇒ A's share in remaining profit = (2 ÷ 5) × 10,000 = ₹4,000

⇒ A's interest amount = 10% of 60,000 = ₹6,000

⇒ A's total share = A's interest + A's share in profit

⇒ A's total share = 6,000 + 4,000 = ₹10,000

∴ The correct answer is ₹10,000.

Additional Information

Partnership Profit Division

Profits in a partnership are divided in the ratio of the product of capital invested and the time period of investment. Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An active partner manages the business and usually receives a fixed salary or commission from the profit before the remaining profit is distributed. A sleeping partner only invests capital.

Interest on Capital in Partnership

If the partnership deed specifies interest on capital, it is paid out of the total profit first. The leftover profit is then distributed in the standard capital ratio.

Shortcut Trick Since the number of months is the same (3) for all products, the product with the highest total sales will have the highest average. By simple observation of the table values: Sales of Product C (150, 160, 155) are strictly greater than those of Product A (100, 120, 110) and Product B (90, 85, 95) in every month. Thus, Product C clearly has the maximum sum and therefore the highest average without needing full calculations. ∴ The correct answer is Product C.

Alternate Method Given: Sales of Product A for Jan, Feb, Mar are 100, 120, and 110 respectively. Sales of Product B for Jan, Feb, Mar are 90, 85, and 95 respectively. Sales of Product C for Jan, Feb, Mar are 150, 160, and 155 respectively. Formula Used: Average = Total sum of values ÷ Number of values Calculations: ⇒ Average units sold for Product A = (100 + 120 + 110) ÷ 3 ⇒ Average for Product A = 330 ÷ 3 = 110 ⇒ Average units sold for Product B = (90 + 85 + 95) ÷ 3 ⇒ Average for Product B = 270 ÷ 3 = 90 ⇒ Average units sold for Product C = (150 + 160 + 155) ÷ 3 ⇒ Average for Product C = 465 ÷ 3 = 155 ⇒ Comparing the averages: 155 > 110 > 90 Product C clearly has the highest average. ∴ The correct answer is Product C.

Additional Information

1. Average and Sum Relationship For datasets with the same number of elements, the dataset with the largest sum will always have the largest arithmetic mean. This saves valuable calculation time in exams.
2. Data Comparison by Observation In data interpretation problems, if every element of Set X is greater than the corresponding elements of Sets Y and Z, no calculation is required to deduce that Set X has the highest sum and average.
3. Percentage Change A concept often requested along with averages in tabular data is percentage growth: % Change = ((New Value − Old Value) ÷ Old Value) × 100.

Shortcut Trick

Summing the three given results evaluates to 2 times the sum of the three numbers.

⇒ 2 × (Sum of three numbers) = 180 + 168 + 150 = 498

⇒ Sum of three numbers = 498 ÷ 2 = 249

⇒ Average of three numbers = 249 ÷ 3 = 83

∴ The correct answer is 83.

Alternate Method

Given:

Three numbers where the average of any two added to the third gives 180, 168, and 150 respectively.

Formula Used:

Average = Sum of all observations ÷ Total number of observations

a + b + 2c = 360

2a + b + c = 336

a + 2b + c = 300

4(a + b + c) = 996

Average = 83

Calculations:

Let the original three numbers be a, b, and c.

⇒ (a + b) ÷ 2 + c = 180

⇒ a + b + 2c = 360     ... (Equation 1)

⇒ (b + c) ÷ 2 + a = 168

⇒ 2a + b + c = 336     ... (Equation 2)

⇒ (c + a) ÷ 2 + b = 150

⇒ a + 2b + c = 300     ... (Equation 3)

Adding Equation 1, Equation 2, and Equation 3 together:

⇒ 4a + 4b + 4c = 360 + 336 + 300

⇒ 4(a + b + c) = 996

⇒ a + b + c = 249

The average of the three numbers is their sum divided by 3:

⇒ Average = (a + b + c) ÷ 3

⇒ Average = 249 ÷ 3 = 83

∴ The correct answer is 83.

Additional Information

Properties of Average

If a constant value is added, subtracted, multiplied, or divided by each observation in a dataset, the new average also changes exactly by the same arithmetic operation and constant.

Weighted Average

If quantities are divided into multiple groups with respective averages, the combined group average is evaluated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Average of Arithmetic Progression

For any sequence of numbers operating with a common difference, the average is strictly derived as (First Term + Last Term) ÷ 2, which also represents the exact middle term.

Shortcut Trick

Total items for students = 100 × (15% of 100) = 100 × 15 = 1500

Total items for chaperones = 10 × (25% of 100) = 10 × 25 = 250

Total items distributed = 1500 + 250 = 1750

∴ The correct answer is 1750 items.

Alternate Method

Given:

Total number of students = 100

Total number of chaperones = 10

Items per student's snack pack = 15% of total students

Items per chaperone's snack pack = 25% of total students

Formula Used:

Value = (Percentage ÷ 100) × Base Value

Students Count = 100 Items/pack = 15% of 100 = 15 Total = 100 × 15 = 1500

Chaperones Count = 10 Items/pack = 25% of 100 = 25 Total = 10 × 25 = 250

Total Snack Items = 1500 + 250 = 1750

Calculations:

⇒ Items in each student's pack = 15% of 100 = (15 ÷ 100) × 100 = 15 items

⇒ Total items given to 100 students = 100 × 15 = 1500 items

⇒ Items in each chaperone's pack = 25% of 100 = (25 ÷ 100) × 100 = 25 items

⇒ Total items given to 10 chaperones = 10 × 25 = 250 items

⇒ Total items distributed = Total student items + Total chaperone items

⇒ Total items distributed = 1500 + 250 = 1750

∴ The correct answer is 1750 items.

Additional Information

Percentage Calculation Rule

Percentage is fundamentally a fraction based on 100. To find x% of a quantity Y, the formula is (x ÷ 100) × Y. Always identify the correct base from the question text carefully before calculating.

Successive Distribution

When distinct items are given to multiple groups, compute the quantities for each group individually according to their independent rules, and then sum up the results.

Common Percentage Ratios

Memorizing basic percentage-to-fraction conversions helps solve equations faster in exams: 25% = 1/4, 20% = 1/5, 15% = 3/20, and 10% = 1/10.

Shortcut Trick

Effective CI rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%

For the next 6 months (half-year), the rate becomes = 10% ÷ 2 = 5%

Net effective rate for 2.5 years = 21 + 5 + (21 × 5)/100 = 26 + 1.05 = 27.05%

Compound Interest = 27.05% of 10000 = 2705

∴ The correct answer is ₹2,705.

Alternate Method

Given:

Principal (P) = ₹10,000

Rate (R) = 10% per annum

Time (T) = 2 years 6 months = 2.5 years

Formula Used:

Amount (A) = P × [1 + R/100]ⁿ × [1 + (R × fraction of year)/100]

Compound Interest (CI) = A − P

Start ₹10k

Year 1 ₹11k

Year 2 ₹12.1k

Year 2.5 ₹12705

+10% (₹1000)

+10% (₹1100)

+5% (₹605)

Total CI = 1000 + 1100 + 605 = 2705

Calculations:

⇒ Full years (n) = 2, Fraction of year = 6/12 = 1/2 year

⇒ A = 10000 × [1 + 10/100]² × [1 + (10 × 1/2)/100]

⇒ A = 10000 × (11/10) × (11/10) × (105/100)

⇒ A = 10000 × (121/100) × (21/20)

⇒ A = 100 × 121 × (21/20)

⇒ A = 5 × 121 × 21

⇒ A = 121 × 105 = 12705

⇒ CI = 12705 − 10000 = 2705

∴ The correct answer is ₹2,705.

Additional Information

Effective Rate Method

For 2 years, the effective compound interest rate is calculated directly as A + B + (A × B)/100.

Half-Yearly Compounding

When interest is compounded half-yearly, the applied rate is halved (R/2) and the time periods are doubled (2T).

SI vs CI Difference

The exact difference between Compound Interest and Simple Interest for a 2-year period is given by the formula P(R/100)².

Shortcut Trick

Fraction value of 12% = 3/25 ⇒ Multiplying factor = 28/25.

For 2 years, Amount / Principal = (28/25)² = 784/625.

Given Amount = 784 parts = ₹3136 ⇒ 1 part = 4.

Principal = 625 parts = 625 × 4 = ₹2500.

140% of Principal = 1.4 × 2500 = ₹3500.

∴ The correct answer is ₹3500.

Alternate Method

Given:

Rate (R) = 12% per annum

Time (T) = 2 years

Amount (A) = ₹3136

Formula Used:

A = P(1 + R/100)^T

P = ? Year 1 ₹3136

+12% CI +12% CI

Equivalent to 25.44% Total Interest (Multiplier: 1.2544)

Calculation:

⇒ 3136 = P(1 + 12/100)²

⇒ 3136 = P(112/100)²

⇒ 3136 = P(28/25)²

⇒ 3136 = P × (784/625)

⇒ P = (3136 × 625) ÷ 784

⇒ P = 4 × 625

⇒ P = ₹2500

⇒ Required value = 140% of P

⇒ Required value = (140 ÷ 100) × 2500

⇒ Required value = 1.4 × 2500 = 3500

∴ The correct answer is ₹3500.

Additional Information

Effective Rate of Compound Interest

For 2 years at R%, the effective CI rate is given by R + R + (R × R)/100. For 12%, it is 12 + 12 + 1.44 = 25.44%.

Difference between CI and SI for 2 years

The difference between compound interest and simple interest for 2 years is calculated using the formula: Difference = P(R/100)².

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R ÷ 2) and the time periods are doubled (2T).