Shortcut Trick

Apply the BODMAS rule and use fraction equivalents for faster decimal multiplication.

⇒ (2.4 + 3.125) × 0.25 − (0.75 ÷ 0.3)

⇒ 5.525 × (1/4) − 2.5

⇒ 1.38125 − 2.5 = −1.11875

∴ The correct answer is -1.11875.

Alternate Method

Given:

Expression: (2.4 + 3.125) × 0.25 − (0.75 ÷ 0.3)

Formula Used:

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

Calculation:

⇒ Step 1: Solve the terms inside the parentheses first.

⇒ First bracket: (2.4 + 3.125) = 5.525

⇒ Second bracket: (0.75 ÷ 0.3) = 7.5 ÷ 3 = 2.5

⇒ Step 2: Substitute the solved bracket values back into the main expression.

⇒ 5.525 × 0.25 − 2.5

⇒ Step 3: Perform the multiplication operation.

⇒ 5.525 × 0.25 = 1.38125

⇒ Step 4: Perform the final subtraction operation.

⇒ 1.38125 − 2.5 = −1.11875

∴ The correct answer is -1.11875.

Additional Information

BODMAS Rule

It defines the sequence of operations for mathematical expressions: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Expressions inside brackets must always be evaluated first.

Decimal Division Technique

When dividing by a decimal number, shift the decimal point to the right in both the dividend and divisor until the divisor becomes a whole number. For instance, 0.75 ÷ 0.3 simplifies to 7.5 ÷ 3.

Fraction Equivalents of Decimals

Converting decimals to fractions often speeds up calculations. For example, multiplying any value by 0.25 is mathematically equivalent to dividing it by 4, and multiplying by 0.5 is equivalent to dividing by 2.

Shortcut Trick

Given 20% of (P + Q) = 50% of (P − Q).

Write as a ratio: (P + Q) ÷ (P − Q) = 50 ÷ 20 = 5 ÷ 2.

Apply Componendo and Dividendo directly:

P ÷ Q = (5 + 2) ÷ (5 − 2) = 7 ÷ 3.

∴ The correct answer is 7 : 3.

Alternate Method

Given:

20% of (P + Q) = 50% of (P − Q)

20(P + Q)

50(P − Q)

P : Q = 7 : 3

Calculations:

⇒ (20 ÷ 100) × (P + Q) = (50 ÷ 100) × (P − Q)

⇒ 20(P + Q) = 50(P − Q)

⇒ 2(P + Q) = 5(P − Q)

⇒ 2P + 2Q = 5P − 5Q

⇒ 2Q + 5Q = 5P − 2P

⇒ 7Q = 3P

⇒ P ÷ Q = 7 ÷ 3

⇒ P : Q = 7 : 3

∴ The correct answer is 7 : 3.

Additional Information

Componendo and Dividendo Rule

If a ÷ b = c ÷ d, then (a + b) ÷ (a − b) = (c + d) ÷ (c − d). This rule is very useful to directly extract the individual ratio values from the sum and difference equations without lengthy cross-multiplication.

Percentage Equality

When x% of A = y% of B, the relationship can be instantly simplified to A ÷ B = y ÷ x by cancelling out the common percentage base (100) on both sides.

Compound Ratio

When multiple ratios are combined by multiplication, the ratios a:b and c:d become ac:bd. This is often used in proportional reasoning.

Shortcut Trick

Assume the quantities of the two varieties are 3 kg and 1 kg.

Total Cost Price = (80 × 3) + (100 × 1) = ₹340 for 4 kg.

Mean Cost Price = 340 ÷ 4 = ₹85 per kg.

Given Selling Price (SP) = ₹91.80 per kg.

Gain % = [(91.80 − 85) ÷ 85] × 100 = (6.80 ÷ 85) × 100 = 8%.

∴ The correct answer is 8% Gain.

Alternate Method

Given:

Cost of 1^st variety = ₹80/kg

Cost of 2^nd variety = ₹100/kg

Ratio of mixing = 3 : 1

Selling Price (SP) of mixture = ₹91.80/kg

₹80 (CP1) ₹100 (CP2)

₹85 Mean CP

3 1 Ratio

(100 − 85 = 15) (85 − 80 = 5)

Formula Used:

Average CP = (CP₁ × Q₁ + CP₂ × Q₂) ÷ (Q₁ + Q₂)

Gain % = [(SP − CP) ÷ CP] × 100

Calculations:

⇒ Average CP = (80 × 3 + 100 × 1) ÷ (3 + 1)

⇒ Average CP = (240 + 100) ÷ 4

⇒ Average CP = 340 ÷ 4 = ₹85 per kg

⇒ Since SP (₹91.80) is greater than CP (₹85), there is a gain.

⇒ Gain = SP − CP = 91.80 − 85.00 = ₹6.80

⇒ Gain % = (6.80 ÷ 85) × 100

⇒ Gain % = 680 ÷ 85 = 8%

∴ The correct answer is 8% Gain.

Additional Information

Rule of Alligation

When two items of different prices are mixed, the ratio of their quantities is inversely proportional to the differences of their prices from the mean price. Ratio = (Price of dearer − Mean Price) : (Mean Price − Price of cheaper).

Weighted Average

Instead of the Alligation cross method, you can directly use the weighted average formula: A_w = (w₁x₁ + w₂x₂) ÷ (w₁ + w₂), where w represents ratio parts and x represents the price values.

Profit and Loss Basics

Profit occurs when the Selling Price (SP) is strictly greater than the Cost Price (CP). Profit percentage is always calculated on the base of Cost Price unless explicitly stated otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Shortcut Trick

Profit ratio = (Investment of A × Time of A) : (Investment of B × Time of B)

Profit ratio = (45000 × 8) : (60000 × 6)

Profit ratio = 360000 : 360000 = 1 : 1

Total profit = ₹36,000, so A's share = 1 / 2 × 36000

A's share = ₹18,000

∴ The correct answer is ₹18,000.

Alternate Method

Given:

Investment of A = ₹45,000

Time period of A = 8 months

Investment of B = ₹60,000

Time period of B = 6 months

Total Profit = ₹36,000

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Partner A ₹45,000 × 8 mo Profit: 360000

Partner B ₹60,000 × 6 mo Profit: 360000

Ratio = 1 : 1

Calculations:

⇒ Profit share of A : Profit share of B = (45000 × 8) : (60000 × 6)

⇒ Ratio = 360000 : 360000

⇒ Simplified Ratio = 1 : 1

⇒ Total profit ratio parts = 1 + 1 = 2 parts

⇒ Given, 2 parts = ₹36,000

⇒ 1 part = 36000 ÷ 2 = ₹18,000

⇒ A's share = 1 part = ₹18,000

∴ The correct answer is ₹18,000.

Additional Information

Working vs Sleeping Partner

A working partner receives a fixed salary or a specific percentage of the profit for managing the business. The remaining profit is distributed among all partners based on their capital and time invested. A sleeping partner only receives their share based on their capital investment.

Joining Mid-way in Business

If a partner joins the business x months after it starts and the total accounting period is 1 year (12 months), their effective investment duration for calculating the profit ratio is taken as (12 − x) months.

Multiple Partners

For a business with multiple partners, the profit ratio is always the compound ratio of their investments and respective time periods. Profit Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Shortcut Trick

Weight of removed student = Old Average + (Remaining Students × Drop in Average)

⇒ Weight = 60 + (29 × 0.5)

⇒ Weight = 60 + 14.5 = 74.5

∴ The correct answer is 74.5 kg.

Alternate Method

Given:

Average weight of 30 students = 60 kg

Average weight of remaining 29 students = 59.5 kg

Total Weight of 30 Students = 1800 kg

29 Students = 1725.5 kg

1 Left = ? kg

Formula Used:

Total Weight = Average × Number of Students

Weight of removed student = Total initial weight − Total new weight

Calculation:

⇒ Total weight of 30 students = 30 × 60 = 1800 kg

⇒ Total weight of 29 students = 29 × 59.5 = 1725.5 kg

⇒ Weight of removed student = 1800 − 1725.5

⇒ Weight of removed student = 74.5 kg

∴ The correct answer is 74.5 kg.

Additional Information

Average Change on Removal

If removing an entity decreases the group's average, the value of the removed entity must be greater than the initial average. The direct logic formula is: Removed Value = Old Average + (Remaining Items × Decrease in Average).

Average Change on Addition

If a new entity joins and increases the average, its value is greater than the initial average. The direct logic formula is: Added Value = Old Average + (New Total Items × Increase in Average).

Combined Average

When combining two groups of sizes n₁ and n₂ having averages A₁ and A₂ respectively, the total combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Net change in total sum = Correct value − Incorrect value = 156 − 165 = −9 cm.

Change in average = Net change ÷ Total boys = −9 ÷ 40 = −0.225 cm.

Correct average = Initial average + Change in average = 160 − 0.225 = 159.775 cm.

∴ The correct answer is 159.775 cm.

Alternate Method

159.775 cm Correct Avg

160 cm Initial Avg

Drop: -0.225 cm

Total sum drop = 9 cm for 40 boys

Given:

Total number of boys = 40

Initial average height = 160 cm

Incorrectly recorded height = 165 cm

Correct height = 156 cm

Formula Used:

Total Sum = Average × Number of Items

Correct Sum = Initial Sum − Incorrect Value + Correct Value

Calculations:

⇒ Initial sum of heights = 40 × 160 = 6400 cm

⇒ Correct sum of heights = 6400 − 165 + 156

⇒ Correct sum of heights = 6400 − 9 = 6391 cm

⇒ Accurate average height = 6391 ÷ 40

⇒ Accurate average height = 159.775 cm

∴ The correct answer is 159.775 cm.

Additional Information

Deviation Method in Averages

When an existing value is replaced, the net change in average is simply the difference between the new and old values divided by the total number of items.

Combined Average

If group A has n₁ items with average A₁ and group B has n₂ items with average A₂, the combined average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Adding or Removing an Item

If a new entity joins a group and the overall average increases, the value of the new entity is strictly greater than the original average. Conversely, if the average decreases, its value is less than the original average.

Shortcut Trick

Ratio of Amount to Principal = 6050 / 5000 = 121 / 100

For 2 years compound interest, the multiplying factor is the square root of the ratio.

⇒ Multiplying factor per year = √(121 / 100) = 11 / 10

⇒ Annual Interest Rate = (11 − 10) / 10 × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

0 Years 1 Year 2 Years P = ₹5000 A = ₹6050

Compound Interest (2 Years) Rate = R% p.a.

Given:

Principal (P) = ₹5000

Amount (A) = ₹6050

Time (T) = 2 years

Formula Used:

A = P × (1 + R / 100)^T

Calculation:

⇒ 6050 = 5000 × (1 + R / 100)²

⇒ 6050 / 5000 = (1 + R / 100)²

⇒ 121 / 100 = (1 + R / 100)²

⇒ (11 / 10)² = (1 + R / 100)²

Taking the square root on both sides:

⇒ 11 / 10 = 1 + R / 100

⇒ R / 100 = 11 / 10 − 1

⇒ R / 100 = 1 / 10

⇒ R = 10%

∴ The correct answer is 10%.

Additional Information

Compound Interest Formula

The amount after T years is calculated using A = P(1 + R/100)^T, where P is the principal and R is the rate of interest per annum.

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R/2) and the time duration doubles (2T). The formula becomes A = P(1 + R/200)^2T.

SI vs CI Difference

For a 2-year period at rate R%, the difference between Compound Interest and Simple Interest on the same principal is D = P(R/100)².

Effective Rate of Interest

For 2 years of compounding at an annual rate of R%, the total effective compound interest rate is (2R + R²/100)%.

Shortcut Trick

When profit is given in terms of Cost Price (CP), the total transaction amount (₹750) is not needed.

Base (Total Cost Price) = CP of 15 kg

Profit = CP of 5 kg

Profit % = (Profit ÷ Total CP) × 100

Profit % = (5 ÷ 15) × 100 = 33.33%

∴ The correct answer is 33.33%.

Alternate Method

Given:

Quantity sold = 15 kg

Selling Price (SP) of 15 kg = ₹750

Profit = Cost Price (CP) of 5 kg

Formula Used:

Profit = Total SP − Total CP

Profit % = (Profit ÷ Total CP) × 100

Total CP ₹562.5

Total SP ₹750

• Profit (₹187.5) (CP of 5 kg)

Calculations:

⇒ Let the CP of 1 kg of apples be x.

⇒ Total CP for 15 kg = 15 × x = 15x

⇒ Profit = CP of 5 kg = 5 × x = 5x

⇒ Total SP for 15 kg = Total CP + Profit = 15x + 5x = 20x

⇒ We are given that Total SP = ₹750

⇒ 20x = 750

⇒ x = 750 ÷ 20 = ₹37.5 (CP of 1 kg)

⇒ Total CP = 15 × 37.5 = ₹562.5

⇒ Profit = 5 × 37.5 = ₹187.5

⇒ Profit % = (187.5 ÷ 562.5) × 100

⇒ Profit % = (1 ÷ 3) × 100 = 33.33%

∴ The correct answer is 33.33%.

Additional Information

Profit and Loss Base Rule

Profit or loss is always calculated on the Cost Price (CP) unless explicitly stated otherwise. Profit % = (Profit ÷ CP) × 100.

Dishonest Dealer Concept

When a dealer claims to sell at cost price but uses a false weight, the profit percentage is given by (Error ÷ (True Weight − Error)) × 100.

Same Selling Price with Same Profit/Loss %

If two articles are sold at the same price, one at a profit of x% and the other at a loss of x%, there is always an overall loss of (x² ÷ 100)%.

Shortcut Trick

Intermediate SP (before 20% second discount) = 8400 × (100 ÷ 80) = ₹10,500

First discount amount = Marked Price − Intermediate SP = 12000 − 10500 = ₹1,500

First discount % = (1500 ÷ 12000) × 100

First discount % = (1 ÷ 8) × 100 = 12.5%

∴ The correct answer is 12.5%.

Alternate Method

Given:

Marked Price (MP) = ₹12,000

Final Selling Price (SP) = ₹8,400

Second Discount (D₂) = 20%

Formula Used:

Final SP = MP × (1 − D₁ / 100) × (1 − D₂ / 100)

MP ₹12000

− D₁%

Intermediate ₹10500

− 20%

Final SP ₹8400

Successive Discount Process

Calculations:

⇒ Let the first discount be D₁%.

⇒ 8400 = 12000 × (1 − D₁ / 100) × (1 − 20 / 100)

⇒ 8400 = 12000 × (1 − D₁ / 100) × (80 / 100)

⇒ 8400 = 12000 × (1 − D₁ / 100) × 0.8

⇒ 8400 = 9600 × (1 − D₁ / 100)

⇒ 8400 ÷ 9600 = 1 − D₁ / 100

⇒ 7 / 8 = 1 − D₁ / 100

⇒ D₁ / 100 = 1 − 7 / 8

⇒ D₁ / 100 = 1 / 8

⇒ D₁ = 100 ÷ 8 = 12.5%

∴ The correct answer is 12.5%.

Additional Information

Successive Discount Formula

When two discounts a% and b% are given successively, the single equivalent discount is given by: (a + b − ab ÷ 100)%.

Marked Price (MP) Base

Any discount provided by a shopkeeper is fundamentally calculated on the Marked Price of the article, not the Cost Price. Selling Price (SP) = MP − Discount amount.

Multiplying Factor Method

For a standard discount of D%, the resultant value is found directly by multiplying the base with a factor of (1 − D ÷ 100). For continuous successive discounts, these multiplying factors are applied sequentially.

Shortcut Trick

Initial Ratio → Alcohol : Water = 7 : 4

Final Ratio → Alcohol : Water = 7 : 6

Since the alcohol quantity remains unchanged, its ratio parts (7) are identical.

Increase in water parts = 6 − 4 = 2 parts, which corresponds to the 12 litres added.

2 parts = 12 litres ⇒ 1 part = 6 litres.

Initial total quantity = (7 + 4) parts = 11 × 6 = 66 litres.

∴ The correct answer is 66 litres.

Alternate Method

Given:

Initial ratio of alcohol to water = 7 : 4

Amount of water added = 12 litres

Final ratio of alcohol to water = 7 : 6

Calculations:

Initial

Alcohol (7 parts)

Water (4)

Final

Alcohol (7 parts)

Water (4)

• 2

• 12 Litres

2 Parts = 12 L → 1 Part = 6 L Initial Total = 11 Parts = 66 Litres

⇒ Let the initial quantity of alcohol and water be 7x and 4x respectively.

⇒ Total initial mixture = 7x + 4x = 11x

⇒ After adding 12 litres of water, quantity of water = 4x + 12

⇒ According to the new ratio: 7x / (4x + 12) = 7 / 6

⇒ 7x × 6 = 7 × (4x + 12)

⇒ 42x = 28x + 84

⇒ 14x = 84

⇒ x = 6

⇒ Initial total quantity = 11x = 11 × 6 = 66 litres

∴ The correct answer is 66 litres.

Additional Information

Constant Quantity Principle

In mixture problems, when only one component is added or removed, the absolute amount of the other component stays constant. Matching the ratio values for this constant part is the fastest method to solve the equation.

Rule of Alligation

Used to find the ratio in which two ingredients at given concentrations must be mixed to produce a mixture at a desired concentration. Formula: (Quantity 1 / Quantity 2) = (Price 2 − Mean Price) / (Mean Price − Price 1).

Replacement Method

When a specific volume of a mixture is drained and replaced by one of its constituent liquids, the ratio of the remaining mixture right before the replacement occurs is identical to the original mixture.