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Shortcut Trick

Break the given decimal into its integer and fractional parts:

3.375 = 3 + 0.375

Recall the standard decimal conversion for eighths: 0.375 = 3/8

⇒ 3.375 = 3 + 3/8 = (3 × 8 + 3) ÷ 8

⇒ 3.375 = 27/8

∴ The correct answer is 27/8.

Alternate Method

Given:

Decimal value = 3.375

Method Used:

To convert a decimal to a fraction, write the entire number without the decimal point over 1 followed by as many zeros as there are decimal places, and then simplify.

Calculations:

⇒ Since 3.375 has 3 decimal places, we write it over 1000.

⇒ 3.375 = 3375 / 1000

⇒ To simplify, divide both the numerator and denominator by 25:

⇒ Numerator: 3375 ÷ 25 = 135

⇒ Denominator: 1000 ÷ 25 = 40

⇒ The fraction becomes 135 / 40.

⇒ Now, divide both by 5 to reduce it further:

⇒ Numerator: 135 ÷ 5 = 27

⇒ Denominator: 40 ÷ 5 = 8

⇒ The fraction in its simplest form is 27/8.

∴ The correct answer is 27/8.

Additional Information

Common Decimal Equivalents

Memorizing common base-8 fractions speeds up calculations in exams: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Proper vs Improper Fractions

A proper fraction has a numerator smaller than its denominator (e.g., 3/8). An improper fraction has a numerator greater than or equal to its denominator (e.g., 27/8). Mixed numbers can be formed directly from improper fractions.

Converting Repeating Decimals

For non-terminating but repeating decimals, place the repeating digits over an equal number of 9s instead of 10s. Example: 0.444... = 4/9, and 0.3636... = 36/99 = 4/11.

Shortcut Trick

Given, 15% of a = 25% of b

Directly equate the percentage values: 15 × a = 25 × b

Rearranging the terms, we get the ratio: a ÷ b = 25 ÷ 15

Simplifying the fraction gives: a : b = 5 : 3

∴ The correct answer is 5 : 3.

Alternate Method

Given:

15% of a = 25% of b

Visualising the Ratio

Value of a (5 Parts)

Value of b (3 Parts)

Ratio a : b = 5 : 3

Calculations:

⇒ (15 ÷ 100) × a = (25 ÷ 100) × b

⇒ 15 × a = 25 × b

⇒ a ÷ b = 25 ÷ 15

⇒ a ÷ b = 5 ÷ 3

⇒ a : b = 5 : 3

∴ The correct answer is 5 : 3.

Additional Information

Ratio and Proportion Basics

A ratio indicates how many times one number contains another. The ratio of x to y is mathematically represented as x : y or x ÷ y.

Inverse Relationship

If p × A = q × B, then the ratio of A to B is simply q : p. This happens because crossing over the terms yields A ÷ B = q ÷ p.

Percentage to Fraction

To convert a percentage into a fraction, divide the percentage value by 100. For example, 25% is equivalent to 25 ÷ 100, which simplifies to 1 ÷ 4.

Shortcut Trick

To find the number x to be subtracted from a, b, c, and d to make them proportional, use the direct formula:

x = (ad − bc) ÷ ((a + d) − (b + c))

Here, a = 7, b = 9, c = 8, d = 11.

x = (7 × 11 − 9 × 8) ÷ ((7 + 11) − (9 + 8))

x = (77 − 72) ÷ (18 − 17) = 5 ÷ 1 = 5

∴ The correct answer is 5.

Alternate Method

Given:

The numbers are 7, 9, 8, 11.

Formula Used:

If four numbers A, B, C, D are in proportion, then A ÷ B = C ÷ D.

Cross Multiplication for Proportion

7 − x

9 − x

8 − x

11 − x

=

(7 − x)(11 − x) = (9 − x)(8 − x)

Calculations:

Let the number to be subtracted be x.

⇒ (7 − x) ÷ (9 − x) = (8 − x) ÷ (11 − x)

⇒ (7 − x)(11 − x) = (9 − x)(8 − x)

⇒ 77 − 11x − 7x + x² = 72 − 9x − 8x + x²

⇒ 77 − 18x = 72 − 17x

⇒ 77 − 72 = 18x − 17x

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Continued Proportion

If three numbers a, b, c are in continued proportion, then a ÷ b = b ÷ c, which means b² = ac. Here, b is known as the mean proportional.

Componendo and Dividendo

If a ÷ b = c ÷ d, then according to this rule: (a + b) ÷ (a − b) = (c + d) ÷ (c − d). This is a highly useful tool for simplifying complex ratio equations.

Fourth Proportional

For any three given numbers a, b, and c, their fourth proportional is a number d such that a ÷ b = c ÷ d, giving the result d = (b × c) ÷ a.

Shortcut Trick

Convert the fractional part to a decimal directly: 5/8 = 0.625.

So, 2 ⁵/₈ = 2.625.

Given expression: 6.4 − (2.625 + 1.75)

⇒ 6.4 − 4.375 = 2.025

∴ The correct answer is 2.025.

Alternate Method

Given:

6.4 − (2 ⁵/₈ + 1.75)

Concept Used:

Follow the BODMAS rule to solve the expression in the brackets first.

Calculations:

⇒ Convert 2 ⁵/₈ to decimal: (2 × 8 + 5) ÷ 8 = 21 ÷ 8 = 2.625

⇒ Substitute the decimal value into the given expression:

⇒ 6.4 − (2.625 + 1.75)

⇒ Solve the addition inside the bracket:

⇒ 2.625 + 1.75 = 4.375

⇒ Subtract the calculated sum from 6.4:

⇒ 6.400 − 4.375 = 2.025

∴ The correct answer is 2.025.

Additional Information

BODMAS Rule

It determines the correct sequence of operations in any mathematical expression: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Common Fraction to Decimal Conversions

Memorizing eighth-based fractions saves time in calculations: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Decimal Addition & Subtraction

Always align the decimal points vertically before calculating. You can append trailing zeros to numbers (e.g., writing 6.4 as 6.400) to balance the number of decimal places without changing the value.

Shortcut Trick

Profit is distributed in the ratio of (Capital × Time).

Alok's total capital = (90,000 × 6) + (1,20,000 × 6) = 12,60,000

Deepak's time in business = 12 − 3 (joined late) − 3 (left early) = 6 months.

Deepak's total capital = 1,20,000 × 6 = 7,20,000

Ratio = 12,60,000 : 7,20,000 = 7 : 4

∴ The correct answer is 7:4.

Alternate Method

0m 3m 6m 9m 12m Alok 90k × 6m 120k × 6m Deepak 120k × 6m

Given:

Alok's initial investment = ₹90,000

Alok's increased investment after 6 months = ₹90,000 + ₹30,000 = ₹1,20,000

Deepak's investment = ₹1,20,000

Total time for business = 12 months

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ Alok invested ₹90,000 for the first 6 months.

⇒ Alok invested ₹1,20,000 for the remaining 6 months.

⇒ Equivalent capital for Alok = (90,000 × 6) + (1,20,000 × 6)

⇒ Equivalent capital for Alok = 5,40,000 + 7,20,000 = 12,60,000

⇒ Deepak joined after 3 months and left 3 months before the end.

⇒ Deepak's total investment time = 12 − 3 − 3 = 6 months.

⇒ Equivalent capital for Deepak = 1,20,000 × 6 = 7,20,000

⇒ Profit sharing ratio (Alok : Deepak) = 12,60,000 : 7,20,000

⇒ Alok : Deepak = 126 : 72 = 7 : 4

∴ The correct answer is 7:4.

Additional Information

Joining Mid-way

If a partner joins the business after t months, their investment is considered active for (12 − t) months in an annual profit cycle.

Compound Ratio in Partnership

The profit is always distributed in the compound ratio of individual investments and time periods: (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Active and Sleeping Partner

An active partner is paid a fixed amount or a specific percentage of the profit for managing the operations, and the remaining profit is distributed based on their capital and time ratios.

Shortcut Trick

Teacher's Age = New Average + (Old Members × Increase in Average)

Old Members = 19 students

Increase in Average = 18 − 17 = 1 year

Teacher's Age = 18 + (19 × 1) = 37 years

∴ The correct answer is 37 years.

Alternate Method

Given:

Average age of 19 students = 17 years

New average age after teacher joins = 18 years

Formula Used:

Sum of values = Average × Number of values

323 years

19 students (19 × 17)

19 students Teacher

20 members total = 20 × 18 = 360 years

360 − 323 = 37

Calculations:

⇒ Total age of 19 students = 19 × 17 = 323 years

⇒ Number of members after teacher joins = 19 + 1 = 20 members

⇒ Total age of 20 members = 20 × 18 = 360 years

⇒ Teacher's age = Total age of 20 members − Total age of 19 students

⇒ Teacher's age = 360 − 323 = 37 years

∴ The correct answer is 37 years.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ with averages A₁ and A₂ are mixed, Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Exclusion of a Member

If a person leaves the group, Age of leaving person = Old Average − (New Total Members × Change in Average). Keep the positive or negative sign of the change intact.

Replacement of a Member

When a new person replaces an old one, Age of new person = Age of removed person + (Total Members × Increase in Average).

Shortcut Trick

Deviation of 30 employees from overall average = ₹72,000 − ₹62,000 = +₹10,000 each.

Total extra deviation for these 30 employees = 30 × 10,000 = +₹3,00,000.

To maintain the overall average, the remaining 50 employees must compensate by −₹3,00,000.

Average deviation for remaining employees = −₹3,00,000 ÷ 50 = −₹6,000.

Required average = ₹62,000 − ₹6,000 = ₹56,000.

∴ The correct answer is ₹56,000.

Alternate Method

Given:

Total number of employees = 80

Overall average salary = ₹62,000

Number of employees in the first group = 30

Average salary of the first group = ₹72,000

Total: 80 employees | Avg = ₹62,000

30 employees Avg: ₹72,000

50 employees Avg: ?

Formula Used:

Total Sum = Average × Number of Observations

Average of Remaining = Sum of Remaining ÷ Number of Remaining Observations

Calculations:

⇒ Total salary of 80 employees = 80 × 62000 = ₹49,60,000

⇒ Total salary of 30 employees = 30 × 72000 = ₹21,60,000

⇒ Remaining employees = 80 − 30 = 50

⇒ Total salary of remaining 50 employees = 49,60,000 − 21,60,000 = ₹28,00,000

⇒ Average salary of remaining employees = 28,00,000 ÷ 50 = ₹56,000

∴ The correct answer is ₹56,000.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, the combined average is given by: A = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Net Deviation Method

When checking against a mean average, the sum of deviations of all elements from that mean is always zero. This principle is extremely helpful for calculating missing values quickly without doing large multiplications.

Weighted Average Trick

Instead of using raw numbers, you can use the ratio of the group sizes (e.g., 30 : 50 = 3 : 5) in your formulas to further simplify and speed up calculation steps.

Shortcut Trick

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

The sum becomes 4 times in 6 years.

We need to find the time taken for it to become 16 times.

Since 16 = 4² times, we multiply the time by 2.

Required time = 6 × 2 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

P 4P 16P

6 years

6 years

× 4

× 4

Total: 12 years

Given:

A principal sum P becomes 4P in 6 years at Compound Interest.

Formula Used:

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

Where A is the Amount, P is the Principal, R is the Rate of interest, and T is the Time in years.

Calculations:

Let the original sum be P.

According to the given condition, the amount becomes 4 times in 6 years:

⇒ 4P = P(1 + R/100)⁶

⇒ 4 = (1 + R/100)⁶

We need to find the time T when the amount becomes 16P:

⇒ 16P = P(1 + R/100)^T

⇒ 16 = (1 + R/100)^T

We can write 16 as 4²:

⇒ 4² = (1 + R/100)^T

Substitute the value of 4 from the first equation:

⇒ ((1 + R/100)⁶)² = (1 + R/100)^T

⇒ (1 + R/100)¹² = (1 + R/100)^T

By equating the powers on both sides, we get:

⇒ T = 12 years

∴ The correct answer is 12 years.

Additional Information

Compound Interest Multiplier Concept

In Compound Interest, the amount grows geometrically over equal periods. If a sum becomes n times in t years, it will become n² times in 2t years, n³ times in 3t years, and so on.

Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate is halved (R/2) and the total number of periods is doubled (2T) in the CI formula.

Difference Between SI and CI

The difference between SI and CI on a given sum P at a rate R% for 2 years is directly calculated as: Difference = P(R/100)².

Shortcut Trick

Given the ratio of Loss to Cost Price (CP) = 1 : 10.

Let the Loss be 1 unit and the Cost Price be 10 units.

Selling Price (SP) = Cost Price − Loss = 10 − 1 = 9 units.

The ratio of Selling Price to Cost Price = SP : CP = 9 : 10.

∴ The correct answer is 9:10.

Alternate Method

Given:

Ratio of Loss to Cost Price (CP) = 1 : 10

Formula Used:

Selling Price (SP) = Cost Price (CP) − Loss

SP = 9x Loss = x

Cost Price (CP) = 10x

Ratio of SP to CP = 9 : 10

Calculations:

⇒ Let the loss incurred be x and the Cost Price (CP) be 10x.

⇒ SP = CP − Loss

⇒ SP = 10x − x = 9x

⇒ Required Ratio = SP : CP

⇒ SP : CP = 9x : 10x

⇒ SP : CP = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Cost Price and Selling Price

Cost Price (CP) is the price at which the item is purchased by a seller, and Selling Price (SP) is the price at which it is sold to the customer.

Profit and Loss Logic

When Selling Price > Cost Price, there is a Profit calculated as (SP − CP). When Cost Price > Selling Price, there is a Loss calculated as (CP − SP).

Percentage Calculation

Both profit and loss are always evaluated against the base Cost Price (CP). Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100.

Shortcut Trick

The ratio of Cost Price (CP) to Marked Price (MP) can be directly calculated by the formula:

CP / MP = (100 − Discount%) / (100 + Profit%)

⇒ CP / 150 = (100 − 10) / (100 + 25)

⇒ CP / 150 = 90 / 125 = 18 / 25

⇒ CP = (18 / 25) × 150 = 18 × 6 = ₹108

∴ The correct answer is ₹108.

Alternate Method

Given:

Marked Price (MP) = ₹150, Discount = 10%, Profit = 25%

Formula Used:

Selling Price (SP) = MP × (100 − Discount%) ÷ 100

Cost Price (CP) = SP × 100 ÷ (100 + Profit%)

CP ₹108

SP ₹135

MP ₹150

+25% Profit

-10% Discount

Cost Price = ₹108

Calculations:

⇒ SP = 150 × (100 − 10) ÷ 100

⇒ SP = 150 × 90 ÷ 100 = ₹135

⇒ CP = 135 × 100 ÷ (100 + 25)

⇒ CP = 135 × 100 ÷ 125

⇒ CP = 135 × 4 ÷ 5

⇒ CP = 27 × 4 = ₹108

∴ The correct answer is ₹108.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). The resulting value is the Selling Price: SP = MP − Discount.

Profit and Loss Calculation

Profit or Loss is always calculated on the Cost Price (CP). Thus, SP = CP × (100 + Profit%) ÷ 100.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is (d₁ + d₂ − d₁×d₂÷100)%.