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Shortcut Trick

Convert the fractions into percentages to easily compare their values.

5/9 ≈ 55.55%, 4/7 ≈ 57.14%, 3/5 = 60%, 2/3 ≈ 66.67%

Arranging the percentages in ascending order: 55.55% < 57.14% < 60% < 66.67%

This exactly corresponds to the fractions: 5/9 < 4/7 < 3/5 < 2/3

∴ The correct answer is 5/9, 4/7, 3/5, 2/3.

Alternate Method

Given:

Fractions to arrange: 5/9, 4/7, 3/5, 2/3

Method Used:

LCM Equalization Method

Comparing Fractions (Percentage Value) 5/9 55.55% 4/7 57.14% 3/5 60.00% 2/3 66.67%

Calculations:

⇒ LCM of denominators (9, 7, 5, 3) = 9 × 7 × 5 = 315

⇒ Convert each fraction so that the denominator becomes 315:

⇒ 5/9 = (5 × 35) ÷ (9 × 35) = 175/315

⇒ 4/7 = (4 × 45) ÷ (7 × 45) = 180/315

⇒ 3/5 = (3 × 63) ÷ (5 × 63) = 189/315

⇒ 2/3 = (2 × 105) ÷ (3 × 105) = 210/315

⇒ Comparing the numerators: 175 < 180 < 189 < 210

⇒ This implies: 175/315 < 180/315 < 189/315 < 210/315

⇒ Resulting ascending order: 5/9 < 4/7 < 3/5 < 2/3

∴ The correct answer is 5/9, 4/7, 3/5, 2/3.

Additional Information

Cross-Multiplication Method

To compare two fractions a/b and c/d, cross-multiply to get a×d and b×c. If a×d < b×c, then the first fraction is smaller (a/b < c/d).

Decimal Conversion Method

Convert each fraction to its decimal equivalent by simply dividing the numerator by the denominator. Compare the resulting decimal numbers up to 2 to 3 decimal places for accurate results.

Difference Method

When the difference between the numerator and denominator is the exact same for a set of proper fractions, the fraction with the largest numerator is strictly the greatest.

Shortcut Trick

Apply the BODMAS rule and convert the fraction to a decimal directly.

Expression: (2½ + 3.6) − 1.9

⇒ (2.5 + 3.6) − 1.9

⇒ 6.1 − 1.9 = 4.2

∴ The correct answer is 4.2.

Alternate Method

Given:

Expression: (2½ + 3.6) − 1.9

Method Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ First, convert the mixed fraction into a decimal:

⇒ 2½ = 2.5

⇒ Substitute this value back into the original expression:

⇒ (2.5 + 3.6) − 1.9

⇒ According to the BODMAS rule, perform the addition inside the brackets first:

⇒ 2.5 + 3.6 = 6.1

⇒ Finally, subtract 1.9 from the sum obtained:

⇒ 6.1 − 1.9 = 4.2

∴ The correct answer is 4.2.

Additional Information

BODMAS Rule

It defines the correct order of operations to solve an expression: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.

Fraction to Decimal Conversion

Common fractions should be converted to decimals to make addition and subtraction easier (e.g., 1/2 = 0.5, 1/4 = 0.25).

Decimal Arithmetic

Always align the decimal points vertically when adding or subtracting numbers to ensure correct place value calculation.

Shortcut Trick

Using the BODMAS rule, evaluate from the innermost brackets outwards.

Inner terms: (3/4 − 1/2) = 1/4 ⇒ 1/2 × 1/4 = 1/8

Curly brackets: {1/3 − 1/8} = 8/24 − 3/24 = 5/24

Square brackets: [5/6 ÷ 5/24] = 5/6 × 24/5 = 4

Final step: 7 1/4 − 4 = 3 1/4

∴ The correct answer is 3 1/4.

Alternate Method

Given:

Expression: 7 1/4 − [ 5/6 ÷ { 1/3 − ( 1/2 × ( 3/4 − 1/2 ) ) } ]

Concept Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ 29/4 − [ 5/6 ÷ { 1/3 − ( 1/2 × ( 3/4 − 2/4 ) ) } ]

⇒ 29/4 − [ 5/6 ÷ { 1/3 − ( 1/2 × 1/4 ) } ]

⇒ 29/4 − [ 5/6 ÷ { 1/3 − 1/8 } ]

⇒ 29/4 − [ 5/6 ÷ { 8/24 − 3/24 } ]

⇒ 29/4 − [ 5/6 ÷ 5/24 ]

⇒ 29/4 − [ 5/6 × 24/5 ]

⇒ 29/4 − 24/6

⇒ 29/4 − 4

⇒ 29/4 − 16/4

⇒ 13/4 = 3 1/4

∴ The correct answer is 3 1/4.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. It dictates the proper sequence to solve any mathematical expression systematically.

Fraction Division

To divide a number by a fraction, multiply the number by the reciprocal of the fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Types of Brackets

Brackets are resolved in a specific order: first Line brackets (vinculum), then Round brackets ( ), followed by Curly brackets { }, and finally Square brackets [ ].

Shortcut Trick

Total engineers proficient in Python = Python only + Both = 70 + 50 = 120

Total engineers proficient in Java = Java only + Both = 60 + 50 = 110

Required Ratio = 120 : 110 = 12 : 11

∴ The correct answer is 12:11.

Alternate Method

Given:

Engineers proficient in both Python and Java = 50

Engineers proficient in Python only = 70

Engineers proficient in Java only = 60

Formula Used:

Total elements in a set = Elements exclusively in that set + Elements in intersection

Calculations:

⇒ Total engineers proficient in Python = Proficient in Python only + Proficient in both

⇒ Total engineers proficient in Python = 70 + 50 = 120

⇒ Total engineers proficient in Java = Proficient in Java only + Proficient in both

⇒ Total engineers proficient in Java = 60 + 50 = 110

⇒ Required Ratio = Total Python : Total Java

⇒ Required Ratio = 120 : 110 = 12 : 11

∴ The correct answer is 12:11.

Additional Information

Set Theory (Venn Diagram) Formula

For any two overlapping sets A and B, the total number of unique elements is given by: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Total Elements vs 'Only' Elements

The total number of elements in Set A is the sum of elements exclusively in Set A and the elements shared with Set B: n(A) = n(Only A) + n(A ∩ B).

Simplifying Ratios

A ratio indicates how many times one number contains another. It is simplified by dividing both quantities by their Highest Common Factor (HCF).

Shortcut Trick

Profit Ratio = (Capital × Time)

Arvind : Bhavin : Chandan = (80 × 12) : (120 × 8) : (160 × 4)

⇒ Profit Ratio = 960 : 960 : 640 = 3 : 3 : 2

Total parts = 3 + 3 + 2 = 8 parts ≡ ₹1,05,000

Chandan's share (2 parts) = ₹1,05,000 × (2 ÷ 8) = ₹26,250.

∴ The correct answer is ₹26,250.

Alternate Method

Given:

Arvind's investment = ₹80,000 for 12 months

Bhavin's investment = ₹1,20,000 for (12 − 4) = 8 months

Chandan's investment = ₹1,60,000 for (12 − 8) = 4 months

Total profit = ₹1,05,000

Arvind ₹80k × 12m = 960k (3 parts) Bhavin ₹120k × 8m = 960k (3 parts) Chandan ₹160k × 4m = 640k (2 parts) Total Profit = ₹1,05,000 (8 parts)

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃)

Calculations:

⇒ Profit Ratio of A : B : C = (80,000 × 12) : (1,20,000 × 8) : (1,60,000 × 4)

⇒ Ratio = 9,60,000 : 9,60,000 : 6,40,000

⇒ Dividing by 3,20,000, we get the simplified Ratio = 3 : 3 : 2

⇒ Sum of profit ratios = 3 + 3 + 2 = 8 parts

⇒ Chandan's share = (2 ÷ 8) × Total Profit

⇒ Chandan's share = (1 ÷ 4) × 1,05,000

⇒ Chandan's share = ₹26,250

∴ The correct answer is ₹26,250.

Additional Information

Profit Ratio in Partnership

The profit share of any partner is directly proportional to the product of their capital invested and the time duration of the investment.

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or percentage of total profit before the remaining profit is distributed among all partners based on their investments.

Joining Mid-way

When a partner joins a business mid-way, their effective time period is calculated from the date of joining to the end of the business cycle (typically 12 months).

Shortcut Trick

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Profit Ratio = (80000 × 9) : (120000 × 6)

Profit Ratio = 720000 : 720000 = 1 : 1

B's share = (1 ÷ 2) × 45000 = 22500

∴ The correct answer is ₹22,500.

Alternate Method

Given:

Investment by A = ₹80,000

Time period of A = 9 months

Investment by B = ₹1,20,000

Time period of B = 6 months

Total Profit = ₹45,000

Formula Used:

Profit Ratio = (Investment × Time)

Partner A80k × 9mPartner B120k × 6mProfit: 720kProfit: 720kRatio A : B = 1 : 1

Calculations:

⇒ Equivalent Capital of A = 80000 × 9 = 7,20,000

⇒ Equivalent Capital of B = 120000 × 6 = 7,20,000

⇒ Profit Ratio (A : B) = 720000 : 720000 = 1 : 1

⇒ Total ratio parts = 1 + 1 = 2

⇒ B's share in profit = (1 ÷ 2) × Total Profit

⇒ B's share = (1 ÷ 2) × 45000 = 22500

∴ The correct answer is ₹22,500.

Additional Information

Simple Partnership

When partners invest their capital for the exact same duration, profits or losses are distributed directly in the ratio of their original investments.

Compound Partnership

When capital is invested for different time periods, the actual profit sharing ratio is derived by multiplying each partner's invested capital with the respective time period.

Working vs Sleeping Partner

A working partner actively manages the business and often draws a fixed salary from the profit before it is divided. A sleeping partner strictly provides capital and gets a pure proportional share based on investment and time.

Shortcut Trick

For an Arithmetic Progression, Average = (First term + Last term) ÷ 2

First multiple of 11 after 100 = 110

Last multiple of 11 before 250 = 242

Average = (110 + 242) ÷ 2 = 352 ÷ 2 = 176

∴ The correct answer is 176.

Alternate Method

100110121176242250Average = 176Arithmetic Progression (Common Difference = 11)

Given:

⇒ Range = Integers between 100 and 250.

⇒ The numbers must be exactly divisible by 11.

Calculations:

⇒ First integer divisible by 11 = 110 (since 11 × 10 = 110).

⇒ Last integer divisible by 11 = 242 (since 11 × 22 = 242).

⇒ The series is 110, 121, 132, ..., 242.

⇒ Total number of terms (n) = [(Last term − First term) ÷ Common Difference] + 1

⇒ n = [(242 − 110) ÷ 11] + 1

⇒ n = [132 ÷ 11] + 1 = 12 + 1 = 13 terms

⇒ Sum of an AP = (n ÷ 2) × (First term + Last term)

⇒ Sum = (13 ÷ 2) × (110 + 242)

⇒ Sum = (13 ÷ 2) × 352 = 13 × 176 = 2288

⇒ Average = Total Sum ÷ n

⇒ Average = 2288 ÷ 13 = 176

∴ The correct answer is 176.

Additional Information

Average of an Arithmetic Progression

If a series has a constant common difference (like multiples of a number), it forms an Arithmetic Progression (AP). Its average is always the exact mean of the first and last terms: Average = (First Term + Last Term) ÷ 2.

Sum of an Arithmetic Progression

The sum of n terms in an AP is calculated as: S_n = (n ÷ 2) × (First Term + Last Term).

Finding the Number of Terms

To find the total number of terms (n) in any given range for a specific sequence: n = [(Last Term − First Term) ÷ Common Difference] + 1.

Shortcut Trick

Net deviation of first 12 numbers = 6 × (72 − 80) + 6 × (90 − 80) = −48 + 60 = +12

Sum of the 13^th, 14^th, and 15^th numbers = (3 × 80) − 12 = 228

Assuming the 15^th number is x: (x + 8) + (x − 10) + x = 228 ⇒ 3x = 230

Average of 13^th and 14^th numbers = [(x + 8) + (x − 10)] ÷ 2 = x − 1

Required Average = 230/3 − 1 = 227/3 ≈ 75.67

∴ The correct answer is 75.67.

Alternate Method

Given:

Total numbers = 15 (Average = 80)

Average of first 6 numbers = 72

Average of next 6 numbers = 72 + 25% of 72 = 90

13^th number = 15^th number + 8

14^th number = 15^th number − 10

Formula Used:

Total Sum = Average × Total Observations

Total 15 numbers (Sum = 1200)First 6 (Avg 72)Sum: 432Next 6 (Avg 90)Sum: 540Last 3Sum: 22813^th + 14^th + 15^th = 1200 − (432 + 540) = 228

Calculations:

⇒ Total sum of 15 numbers = 15 × 80 = 1200

⇒ Sum of the first 6 numbers = 6 × 72 = 432

⇒ Sum of the next 6 numbers = 6 × 90 = 540

⇒ Sum of remaining 3 numbers (13^th, 14^th, 15^th) = 1200 − (432 + 540) = 228

⇒ Let the 15^th number be x.

⇒ 13^th number = x + 8

⇒ 14^th number = x − 10

⇒ (x + 8) + (x − 10) + x = 228

⇒ 3x − 2 = 228

⇒ 3x = 230 ⇒ x = 230/3

⇒ Average of 13^th and 14^th = [(x + 8) + (x − 10)] ÷ 2

⇒ Average = (2x − 2) ÷ 2 = x − 1

⇒ Average = 230/3 − 1 = 227/3 = 75.666... ≈ 75.67

∴ The correct answer is 75.67.

Additional Information

Net Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. The deviation trick helps compute missing elements by finding how much the known parts differ from the combined average.

Weighted Average

When combining separate groups with varying numbers of elements, the new average is calculated as (N₁A₁ + N₂A₂ + ...) ÷ (N₁ + N₂ + ...).

Successive Substitution

When elements relate to one unknown variable directly, convert the sum into an algebraic expression and solve for the unknown to simplify calculation of required parts directly.

Shortcut Trick

Monthly Return = (9% of ₹8,00,000) ÷ 12 = ₹72,000 ÷ 12 = ₹6,000

Monthly Rent = Monthly Return + Maintenance

Monthly Rent = ₹6,000 + ₹2,000 = ₹8,000

∴ The correct answer is ₹8,000.

Alternate Method

Given:

Investment (Cost of flat) = ₹8,00,000

Required annual return = 9%

Monthly maintenance = ₹2,000

Formula Used:

Monthly Rent = Monthly Net Return + Monthly Maintenance

Total Monthly Rent = ₹8,000

Maintenance ₹2,000

Net Return (9% p.a.) ₹6,000

9% of ₹8,00,000 ÷ 12 months

Calculations:

⇒ Annual Net Return = 9% of ₹8,00,000

⇒ Annual Net Return = (9 ÷ 100) × 8,00,000 = ₹72,000

⇒ Monthly Net Return = 72,000 ÷ 12 = ₹6,000

⇒ Total Monthly Rent = Monthly Net Return + Monthly Maintenance

⇒ Total Monthly Rent = 6,000 + 2,000 = ₹8,000

∴ The correct answer is ₹8,000.

Additional Information

Return on Investment (ROI)

ROI is a financial metric used to evaluate the profitability of an investment. It is calculated as the ratio of net profit to the initial investment cost, often expressed as an annual percentage.

Gross vs Net Income

Gross income refers to the total revenue generated (e.g., total rent collected), while net income is the profit remaining after deducting all operating expenses such as maintenance and taxes.

Annual to Monthly Equivalence

When cash flows or returns are uniform throughout the year, a monthly equivalent is obtained by simply dividing the annual expected value by 12. Monthly Return = Annual Return ÷ 12.

Shortcut Trick

If an amount becomes n times in t years at compound interest, it becomes n^m times in m × t years.

Given: The amount becomes 2 times (n = 2) in 5 years (t = 5).

We need it to become 8 times, which is 2³ times, so m = 3.

Required time = m × t = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Alternate Method

P

2P

4P

8P

5 years × 2

5 years × 2

5 years × 2

Total Time = 5 + 5 + 5 = 15 years

Given:

Amount doubles in 5 years at compound interest.

Let the initial principal be P.

Amount (A) = 2P, Time (t) = 5 years.

Formula Used:

A = P(1 + r/100)^t

Calculations:

⇒ 2P = P(1 + r/100)⁵

⇒ 2 = (1 + r/100)⁵

We need to find the time T when the amount becomes 8 times (A = 8P).

⇒ 8P = P(1 + r/100)^T

⇒ 8 = (1 + r/100)^T

Since 8 can be written as 2³:

⇒ 2³ = (1 + r/100)^T

Substituting the value of 2 from the first equation:

⇒ [(1 + r/100)⁵]³ = (1 + r/100)^T

⇒ (1 + r/100)¹⁵ = (1 + r/100)^T

Comparing the exponents on both sides:

⇒ T = 15

∴ The correct answer is 15 years.

Additional Information

Compound Interest Multiple Concept

If an amount becomes n times in t years at CI, it will become n^m times in m × t years.

Simple Interest Multiple Concept

If an amount becomes n times in t years at SI, it will become m times in [(m−1)/(n−1)] × t years.

Rule of 72

A quick formula to approximate the number of years required to double the principal at a fixed annual CI rate (Time ≈ 72 ÷ r).