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Shortcut Trick

Memorize common percentage to fraction conversions for faster calculation.

12 ¹/₂% = 1/8 and 66 ²/₃% = 2/3

⇒ Expression = (1/8) × 560 + (2/3) × 240

⇒ Expression = 70 + 160 = 230

∴ The correct answer is 230 km.

Alternate Method

Given:

12 ¹/₂% of 560 km + 66 ²/₃% of 240 km

Calculation:

⇒ 12 ¹/₂% = 25/2 % = (25/2) × (1/100) = 25/200 = 1/8

⇒ 66 ²/₃% = 200/3 % = (200/3) × (1/100) = 200/300 = 2/3

Substituting the fractional values into the original expression:

⇒ (1/8 × 560) + (2/3 × 240)

⇒ 70 + 160

⇒ 230

∴ The correct answer is 230 km.

Additional Information

Common Percentage-Fraction Equivalents

Memorizing standard equivalents saves calculation time: 16 ²/₃% = 1/6, 14 ²/₇% = 1/7, 33 ¹/₃% = 1/3, and 37 ¹/₂% = 3/8.

BODMAS Rule

Always follow the order of operations: Brackets, Orders, Division/Multiplication (left to right), Addition/Subtraction (left to right). The word 'of' indicates multiplication and is solved before division.

Percentage Reversibility

X% of Y is always equal to Y% of X. This property can be used interchangeably to make calculations easier when numbers are reversed.

Shortcut Trick

Daily commission from shirts = 3 × (2% of 500) = 3 × 10 = ₹30

Daily commission from ties = 4 × (12% of 150) = 4 × 18 = ₹72

Total daily commission = 30 + 72 = ₹102

Total commission earned in 30 days = 102 × 30 = ₹3060

∴ The correct answer is ₹3060.

Alternate Method

Given:

Price of a shirt = ₹500, Commission on a shirt = 2%

Price of a tie = ₹150, Commission on a tie = 12%

Items sold per day = 3 shirts and 4 ties

Total days = 30

Shirts Price: ₹500 | Comm: 2% Comm/item: ₹10 Daily (x3): ₹30

Ties Price: ₹150 | Comm: 12% Comm/item: ₹18 Daily (x4): ₹72

Total in 30 Days = ₹3060

Calculations:

⇒ Commission earned on one shirt = 2% of 500 = (2 ÷ 100) × 500 = ₹10

⇒ Commission earned on 3 shirts per day = 3 × 10 = ₹30

⇒ Commission earned on one tie = 12% of 150 = (12 ÷ 100) × 150 = ₹18

⇒ Commission earned on 4 ties per day = 4 × 18 = ₹72

⇒ Total commission earned per day = 30 + 72 = ₹102

⇒ Total commission earned in 30 days = 102 × 30 = ₹3060

∴ The correct answer is ₹3060.

Additional Information

Percentage Application

Percentage implies 'out of 100'. Thus, x% of y is calculated as (x ÷ 100) × y. In practical applications like commissions, discounts, or taxes, the percentage is applied directly to the base value.

Commission and Base Amount

Commission is generally paid as a percentage of the total sales value. If multiple items are sold with varying commission rates, they must be calculated individually before summation.

Unitary Method

Finding the value for a single unit and then multiplying it by the total number of units required is called the unitary method. It is highly effective for scaling daily quantities to monthly or annual values.

Shortcut Trick

Let the Cost Price (CP) of 100 units of apples be ₹100.

Selling Price (SP) at a claimed 5% loss = ₹95.

Actual weight used is 20% less, so the seller gives only 80 units (True CP = ₹80).

Profit earned by the seller = SP − CP = 95 − 80 = ₹15.

Total Profit % = (15 ÷ 80) × 100 = 18.75%.

∴ The correct answer is 18.75%.

Alternate Method

Given:

Loss percentage on Cost Price = 5%

False weight used = 20% less than actual weight

Formula Used:

Profit % = (Profit ÷ Cost Price) × 100

Actual CP800 gm = ₹800Selling Price1000 gm claimed = ₹950Profit = ₹150Profit % = (150 ÷ 800) × 100 = 18.75%

Calculations:

⇒ Let the cost price of 1000 gm of apples be ₹1000.

⇒ Selling price at a 5% loss = 1000 − (5% of 1000) = 1000 − 50 = ₹950.

⇒ The seller uses a false weight that is 20% less than the actual weight.

⇒ Actual weight given by the seller = 1000 − (20% of 1000) = 1000 − 200 = 800 gm.

⇒ Actual cost price incurred by the seller (for 800 gm) = ₹800.

⇒ Profit earned = Selling Price − Actual Cost Price

⇒ Profit = 950 − 800 = ₹150.

⇒ Total Profit % = (Profit ÷ Cost Price) × 100

⇒ Total Profit % = (150 ÷ 800) × 100 = 18.75%.

∴ The correct answer is 18.75%.

Additional Information

Dishonest Dealer (Selling at Cost Price)

If a trader professes to sell his goods at cost price but uses false weights, then his Profit % = [Error ÷ (True Value − Error)] × 100.

Successive Percentage Method

The overall profit can be evaluated using successive percentages. Giving 20% less weight means a gain of (20 ÷ 80) × 100 = 25%. Combined with a 5% loss on price, the Net % = 25 − 5 − (25 × 5)÷100 = 20 − 1.25 = 18.75%.

Ratio Method for Dishonest Dealer

Write the price ratio and quantity ratio separately. Price ratio (SP:CP) = 95:100. Quantity given ratio (SP:CP) = 100:80. Net Ratio = (95 × 100) : (100 × 80) = 95 : 80, which results in a profit of 15 on a base of 80.

Shortcut Trick

If the profit at one selling price equals the loss at another, the Cost Price is exactly halfway between them.

CP = (SP₁ + SP₂) ÷ 2

CP = (2200 + 1800) ÷ 2 = ₹2000

New SP = ₹2050 ⇒ Profit = 2050 − 2000 = ₹50

Profit % = (50 ÷ 2000) × 100 = 2.5%

∴ The correct answer is Profit of 2.5%.

Alternate Method

Given:

Selling Price 1 (SP₁) = ₹2200

Selling Price 2 (SP₂) = ₹1800

Profit at SP₁ = Loss at SP₂

Formula Used:

Profit = SP − CP

Loss = CP − SP

Profit % = (Profit ÷ CP) × 100

₹1800 CP = ₹2000 ₹2200

New SP = ₹2050

Loss (L)

Profit (P = L)

Profit = ₹50

Calculations:

Let the cost price of the item be CP.

⇒ Profit at SP₁ = Loss at SP₂

⇒ 2200 − CP = CP − 1800

⇒ 2CP = 2200 + 1800

⇒ 2CP = 4000

⇒ CP = ₹2000

Now, the item is sold for a new SP = ₹2050.

⇒ Profit = New SP − CP = 2050 − 2000 = ₹50

⇒ Profit % = (50 ÷ 2000) × 100

⇒ Profit % = 2.5%

∴ The correct answer is Profit of 2.5%.

Additional Information

Marked Price and Successive Discount

Successive discounts of a% and b% are equivalent to a single net discount of (a + b − (a × b) ÷ 100)%.

Relation between CP and MP

The exact ratio of Cost Price to Marked Price is directly proportional to the discount and profit given: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Dishonest Dealer

If a dealer claims to sell goods at cost price but uses a false weight, the actual profit percentage is (Error ÷ False Weight) × 100.

Shortcut Trick

Cost Price (CP) to Selling Price (SP1) ratio at 20% profit = 100 : 120 = 5 : 6.

Assuming CP = 5 units, then initial SP1 = P = 6 units.

Given the festive Marked Price (MP) = 1.5P ⇒ MP = 1.5 × 6 = 9 units.

A discount of 10% is given, so New SP = 9 × 0.9 = 8.1 units.

Profit % = [(New SP − CP) ÷ CP] × 100 = [(8.1 − 5) ÷ 5] × 100 = 62%.

∴ The correct answer is 62% (or 0.62).

Alternate Method

Given:

Initial Selling Price = ₹P

Initial Profit = 20%

New Marked Price (MP) = 1.5P

Festive Discount = 10%

Formula Used:

CP = SP × [100 ÷ (100 + Profit%)]

New SP = MP × [100 − Discount%] ÷ 100

Profit % = [(New SP − CP) ÷ CP] × 100

CP 5P/6 SP1 P SP2 1.35P MP 1.5P +20% -10% New Profit = 62%

Calculations:

⇒ CP = P × (100 ÷ 120) = 5P/6

⇒ New SP = 1.5P × (90 ÷ 100) = 1.35P = 27P/20

⇒ Profit = New SP − CP = (27P/20) − (5P/6)

⇒ LCM of 20 and 6 is 60.

⇒ Profit = (81P − 50P) ÷ 60 = 31P/60

⇒ Profit % = [(31P/60) ÷ (5P/6)] × 100

⇒ Profit % = (31/60) × (6/5) × 100

⇒ Profit % = (31/50) × 100 = 62%

⇒ Expressed as a decimal fraction, it is 0.62.

∴ The correct answer is 62% (or 0.62).

Additional Information

Successive Percentage Change

When a value undergoes two successive changes of a% and b%, the net percentage change is given by: Net Change = a + b + (a × b) ÷ 100.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). The Selling Price is derived as: SP = MP × [100 − Discount%] ÷ 100.

Relationship between CP, MP, Profit and Discount

A direct relationship ratio is given by: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Using the effective successive percentage change formula:

Net Profit % = M − D − (M × D) / 100

Where Markup (M) = 25% and Discount (D) = 10%

⇒ Profit % = 25 − 10 − (25 × 10) / 100

⇒ Profit % = 15 − 2.5 = 12.5%

∴ The correct answer is 12.50%.

Alternate Method

Given:

Markup = 25% above Cost Price (CP)

Discount = 10% on Marked Price (MP)

CP = 100

SP = 112.5

MP = 125

+25% Markup

−10%

+12.5% Profit

Calculations:

⇒ Let the Cost Price (CP) = 100

⇒ Marked Price (MP) = CP + 25% of CP = 100 + 25 = 125

⇒ Discount = 10% of MP = (10 / 100) × 125 = 12.5

⇒ Selling Price (SP) = MP − Discount = 125 − 12.5 = 112.5

⇒ Profit = SP − CP = 112.5 − 100 = 12.5

⇒ Profit Percentage = (Profit / CP) × 100 = (12.5 / 100) × 100 = 12.5%

∴ The correct answer is 12.50%.

Additional Information

Marked Price & Discount

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

Successive Percentage Rule

When a value is increased by A% and then decreased by B%, the effective change percentage is given by the formula: A − B − (A × B) / 100.

Profit & Loss Basics

Profit or Loss percentage is always calculated on the original Cost Price (CP) unless specifically stated otherwise.

Shortcut Trick

Let CP = 100%. Then Marked Price (MP) = 160% (as it is 60% above CP).

Selling Price (SP) after 25% discount = 160% − (25% of 160%) = 120%.

Actual Final SP = 100% + 14% profit = 114%.

Difference in SPs = 120% − 114% = 6%, which corresponds to the ₹150 cash discount.

⇒ 6% of CP = 150 ⇒ CP = 150 × 100 ÷ 6 = ₹2500.

∴ The correct answer is ₹2500.

Alternate Method

Given:

Markup = 60% above Cost Price

Discount = 25%

Additional Cash Discount = ₹150

Overall Profit = 14%

Formula Used:

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

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

CP = 100%

+60%

MP = 160%

-25%

SP = 120%

+14% Profit

Final SP = 114%

• ₹150

120% - 114% = 6% → ₹150   ∴ 100% = ₹2500

Calculations:

⇒ Let the Cost Price (CP) be 100x.

⇒ Marked Price (MP) = 100x + (60% of 100x) = 160x.

⇒ Selling Price before cash discount = 160x − (25% of 160x) = 160x − 40x = 120x.

⇒ Final Selling Price = 120x − 150.

⇒ Based on the profit percentage, Final SP = CP + (14% of CP) = 114x.

⇒ Equating both values of Final SP:

⇒ 120x − 150 = 114x

⇒ 120x − 114x = 150

⇒ 6x = 150

⇒ x = 25

⇒ Cost Price = 100x = 100 × 25 = ₹2500.

∴ The correct answer is ₹2500.

Additional Information

Marked Price & Discount

Discount is exclusively given on the Marked Price (MP), never on Cost Price. It is simply a reduction in price directly impacting the Selling Price. Formula: SP = MP × (100 − Discount%) ÷ 100.

Direct Formula for CP and MP

When Profit% and Discount% are known, the direct ratio between Cost Price and Marked Price is efficiently calculated as CP / MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

When multiple flat discounts x% and y% are given sequentially, the single equivalent discount percentage is evaluated using: (x + y − x × y ÷ 100)%.

Shortcut Trick

First, find the Cost Price (CP) of the mixture from its Selling Price (SP) and Gain %.

CP of mixture = SP ÷ (1 + Gain%) = 43.70 ÷ 1.15 = ₹38/kg

Using Alligation, Ratio of quantities = (38 − 32) : (45 − 38) = 6 : 7

Since 7 parts correspond to 25 kg, 6 parts will correspond to: (25 ÷ 7) × 6 = 150 ÷ 7 ≈ 21.43 kg

∴ The correct answer is 21 kg.

Alternate Method

Given:

CP of first type of sugar = ₹45/kg

CP of second type of sugar = ₹32/kg

Quantity of second type of sugar = 25 kg

SP of mixture = ₹43.70/kg

Profit on mixture = 15%

Formula Used:

CP = SP × [100 ÷ (100 + Gain%)]

Sugar 1 (₹45)Sugar 2 (₹32)₹3838 − 32 = 645 − 38 = 7Quantity Ratio = 6 : 7

Calculations:

⇒ Mean CP of mixture = 43.70 × [100 ÷ (100 + 15)]

⇒ Mean CP = (43.70 × 100) ÷ 115

⇒ Mean CP = 4370 ÷ 115 = ₹38/kg

⇒ Using the rule of alligation (as shown in the diagram), the ratio of the quantities is:

⇒ (Mean CP − CP₂) : (CP₁ − Mean CP) = (38 − 32) : (45 − 38) = 6 : 7

⇒ Let the required quantity of the first type of sugar be x kg.

⇒ x ÷ 25 = 6 ÷ 7

⇒ x = (25 × 6) ÷ 7

⇒ x = 150 ÷ 7 ≈ 21.43 kg

⇒ Therefore, approximately 21 kg of the first type of sugar must be mixed.

∴ The correct answer is 21 kg.

Additional Information

Rule of Alligation

It is a fast cross-calculation method used to find the ratio in which two ingredients at given unit prices must be mixed to produce a mixture of a desired target price. The target price must always lie exactly between the two given prices.

Mean Price Requirement

The central value placed in the alligation cross is called the Mean Price. It must always be the Cost Price (CP) of the mixture. Never place Selling Price (SP) in the alligation diagram.

Cost Price from Selling Price

When Selling Price (SP) and Gain Percentage (P%) are given, Cost Price (CP) is computed backwards using the relation: CP = [100 ÷ (100 + P%)] × SP.

Shortcut Trick

In a right circular cone, the height, base radius, and slant height always form a right-angled triangle.

Using the common Pythagorean triplet (7, 24, 25), where the two legs are 7 and 24, the hypotenuse is directly 25.

Since the given radius is 7 cm and the height is 24 cm, the slant height is directly 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Radius of the base (r) = 7 cm

Height of the cone (h) = 24 cm

Formula Used:

Slant height (l) = √(r² + h²)

h = 24 cm r = 7 cm l = ?

Calculations:

⇒ l = √(7² + 24²)

⇒ l = √(49 + 576)

⇒ l = √(625)

⇒ l = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Volume of a Cone

The volume is one-third of the volume of a cylinder with the same base and height: V = (1 ÷ 3) × π × r² × h.

Curved Surface Area (CSA)

The area of the curved surface alone is calculated using the slant height: CSA = π × r × l.

Total Surface Area (TSA)

The total surface area includes both the base area and the curved surface area: TSA = π × r × (r + l).

Shortcut Trick

Equivalent principal for 1 year = (5000 × 2) + (3000 × 4) = 10000 + 12000 = 22000

Total Simple Interest received = 2200

Rate = (Total Interest ÷ Equivalent Principal) × 100

Rate = (2200 ÷ 22000) × 100

Rate = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Principal lent to B = ₹5000, Time = 2 years

Principal lent to C = ₹3000, Time = 4 years

Total simple interest = ₹2200

Formula Used:

Simple Interest (SI) = Principal × Rate × Time

(where Rate is taken as a decimal fraction r)

Lent to B₹5000 × 2 yrsLent to C₹3000 × 4 yrsTotal SI = ₹2200

Calculations:

⇒ Interest from B = 5000 × r × 2 = 10000r

⇒ Interest from C = 3000 × r × 4 = 12000r

⇒ Total Interest = 10000r + 12000r

⇒ 22000r = 2200

⇒ r = (2200 ÷ 22000) × 100

⇒ r = 10%

∴ The correct answer is 0.1 per annum.

Additional Information

Simple Interest (SI) Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate in %, and T is Time in years.

Decimal vs Percentage Rate

When solving equations, the percentage rate R can be expressed directly as a decimal r (where r = R ÷ 100) making the SI calculation simply P × r × T.

Equivalent Principal

When multiple investments are made at the same simple interest rate, the overall principal equivalent to a 1-year term is the sum of the products of each Principal and its respective Time period.