Discount

Discount is a reduction given on the marked price (or list price) of an article. It is a common practice in retail to attract customers. Understanding how to calculate and manage discounts is crucial for both buyers and sellers.

Types of Discounts

There are primarily two types of discounts:

• Trade Discount: This is a discount given by a wholesaler to a retailer on the list price of goods. It's usually a percentage of the list price and is often not recorded in the books of accounts.
• Cash Discount: This is a discount given by a seller to a buyer for prompt payment of an invoice. It is usually a small percentage and is recorded in the books of accounts.

Key Terms Related to Discount

• Marked Price (MP) / List Price (LP): The price printed on the article or listed in a catalogue. This is the price from which discounts are calculated.
• Selling Price (SP): The price at which the article is actually sold after deducting the discount.
• Discount Amount: The actual amount reduced from the marked price.
• Discount Percentage: The discount expressed as a percentage of the marked price.

Calculating Discount

The fundamental formula for calculating the selling price when a discount is given is:

Selling Price (SP) = Marked Price (MP) - Discount Amount

If the discount is given as a percentage, say 'd%', then:

Discount Amount = (d/100) * MP

Therefore,

SP = MP - (d/100) * MP

This can be simplified to:

SP = MP * (1 - d/100)

Or,

SP = MP * ((100 - d) / 100)

Example 1: Single Discount

A shopkeeper marks an article at ₹500. He offers a discount of 20%. Find the selling price.

Marked Price (MP) = ₹500

Discount Percentage = 20%

Discount Amount = (20/100) * 500 = ₹100

Selling Price (SP) = MP - Discount Amount = 500 - 100 = ₹400

Alternatively, SP = 500 * ((100 - 20)/100) = 500 * (80/100) = ₹400

Successive Discounts

Often, a shopkeeper offers more than one discount on an item. These are called successive discounts. For example, a 10% discount followed by a 20% discount. It's important to note that successive discounts are not simply added together. They are applied one after another to the successively reduced prices.

If two successive discounts of d1% and d2% are given on an article, the final selling price is calculated as follows:

First Discount Amount = (d1/100) * MP

Price after first discount = MP - (d1/100) * MP = MP * ((100 - d1)/100)

Second Discount Amount = (d2/100) * [MP * ((100 - d1)/100)]

Final Selling Price (SP) = [MP * ((100 - d1)/100)] - (d2/100) * [MP * ((100 - d1)/100)]

This simplifies to:

SP = MP * ((100 - d1)/100) * ((100 - d2)/100)

Example 2: Successive Discounts

A shopkeeper marks an article at ₹1000. He offers two successive discounts of 10% and 20%. Find the selling price.

Marked Price (MP) = ₹1000

First Discount = 10%

Price after first discount = 1000 * ((100 - 10)/100) = 1000 * (90/100) = ₹900

Second Discount = 20% (applied on ₹900)

Selling Price (SP) = 900 * ((100 - 20)/100) = 900 * (80/100) = ₹720

Alternatively, using the formula:

SP = 1000 * ((100 - 10)/100) * ((100 - 20)/100)

SP = 1000 * (90/100) * (80/100)

SP = 1000 * 0.9 * 0.8 = 1000 * 0.72 = ₹720

Equivalent Single Discount

Sometimes, it is useful to find a single discount percentage that is equivalent to a series of successive discounts.

If the selling price after successive discounts d1%, d2%, d3%... is SP, and the marked price is MP, then:

SP = MP * ((100 - d1)/100) * ((100 - d2)/100) * ((100 - d3)/100) ...

Let the equivalent single discount be D%. Then,

SP = MP * ((100 - D)/100)

Equating the two expressions for SP:

MP * ((100 - D)/100) = MP * ((100 - d1)/100) * ((100 - d2)/100) * ...

(100 - D)/100 = ((100 - d1)/100) * ((100 - d2)/100) * ...

1 - D/100 = ((100 - d1)/100) * ((100 - d2)/100) * ...

D/100 = 1 - [((100 - d1)/100) * ((100 - d2)/100) * ...]

D = 100 * (1 - [((100 - d1)/100) * ((100 - d2)/100) * ...])

Example 3: Equivalent Single Discount

Find the single discount equivalent to two successive discounts of 10% and 20%.

Here, d1 = 10% and d2 = 20%.

Using the formula for SP:

SP = MP * ((100 - 10)/100) * ((100 - 20)/100)

SP = MP * (90/100) * (80/100)

SP = MP * (0.9 * 0.8)

SP = MP * 0.72

This means the final price is 72% of the marked price.

So, the equivalent single discount is 100% - 72% = 28%.

Using the formula for D:

D = 100 * (1 - [(100 - 10)/100] * [(100 - 20)/100])

D = 100 * (1 - [90/100] * [80/100])

D = 100 * (1 - 0.9 * 0.8)

D = 100 * (1 - 0.72)

D = 100 * 0.28 = 28%

Discount vs. Profit/Loss

It's important to distinguish between discount and profit/loss. The discount is calculated on the Marked Price (MP), while profit or loss is calculated on the Cost Price (CP).

Profit = SP - CP

Loss = CP - SP

Profit Percentage = (Profit / CP) * 100

Loss Percentage = (Loss / CP) * 100

A shopkeeper marks up his goods to make a profit. He then offers a discount to attract customers, but he still aims to sell above his cost price to make a profit.

We can relate MP, SP, CP, profit, loss, and discount using the following relationships:

SP = MP * (1 - d/100)

SP = CP * (1 + p/100) (if there is a profit p%)

SP = CP * (1 - l/100) (if there is a loss l%)

Example 4: Discount and Profit

A shopkeeper marks an article at 40% above its cost price and then offers a discount of 20% on the marked price. What is his gain or loss percentage?

Let the Cost Price (CP) = ₹100.

The article is marked up by 40%.

Marked Price (MP) = 100 * (1 + 40/100) = 100 * (140/100) = ₹140.

A discount of 20% is offered on the MP.

Selling Price (SP) = MP * (1 - 20/100) = 140 * (80/100) = 140 * 0.8 = ₹112.

Now, compare SP with CP.

Profit = SP - CP = 112 - 100 = ₹12.

Profit Percentage = (Profit / CP) * 100 = (12 / 100) * 100 = 12%.

The shopkeeper makes a 12% profit.

Types of Discounts in Practice

• Percentage Discount: e.g., 20% off.
• Flat Discount: A fixed amount off, e.g., ₹500 off.
• Buy One Get One Free (BOGO): Effectively a 50% discount on two items.
• Buy X Get Y Free: e.g., Buy 3 Get 1 Free. This is equivalent to a discount of (Y / (X+Y)) * 100 %. For Buy 3 Get 1 Free, discount = (1 / (3+1)) * 100 = 25%.

Example 5: Buy X Get Y Free

A shop offers a "Buy 2, Get 1 Free" deal on shirts. If each shirt costs ₹600, what is the effective discount percentage?

Here, X = 2 (shirts bought), Y = 1 (shirt free).

Total number of shirts received = X + Y = 2 + 1 = 3.

Cost of 2 shirts = 2 * ₹600 = ₹1200. This is the effective selling price for 3 shirts.

The total value of 3 shirts without the offer would be 3 * ₹600 = ₹1800. This is the effective marked price.

Discount Amount = Total Value - Effective SP = 1800 - 1200 = ₹600.

Effective Discount Percentage = (Discount Amount / Total Value) * 100

= (600 / 1800) * 100 = (1/3) * 100 = 33.33%.

Using the shortcut: Discount % = (Y / (X+Y)) * 100 = (1 / (2+1)) * 100 = (1/3) * 100 = 33.33%.

Key Takeaways for Discount Problems

• Always identify the Marked Price (MP) and Selling Price (SP).
• Discount is always calculated on MP.
• Successive discounts are applied sequentially, not added.
• Profit/Loss is always calculated on Cost Price (CP).
• Use the formula CP * (100 + Profit%) = MP * (100 - Discount%) for combined problems.

Partnership Business

A partnership is a business structure where two or more individuals agree to share in the profits or losses of a business. This agreement can be formal (written partnership deed) or informal. In this type of business, partners invest capital, time, and expertise. The profits are then distributed among the partners based on their agreement.

Key Concepts in Partnership Problems

The core of partnership problems in quantitative aptitude revolves around the distribution of profits based on the capital invested and the duration for which the capital was invested.

• Capital (C): The amount of money or assets invested by a partner in the business.
• Time (T): The duration for which the capital was invested.
• Profit (P): The earnings of the business.

The Fundamental Principle: Ratio of Investment and Profit

The profit earned by each partner is directly proportional to the product of their capital and the time period for which it was invested. This product is often referred to as the 'investment' or 'share'.

Profit Share of a Partner ∝ Capital Invested × Time Period

If partners A, B, and C invest capitals C_A, C_B, and C_C for time periods T_A, T_B, and T_C respectively, then the ratio of their profits (P_A : P_B : P_C) is given by:

P_A : P_B : P_C = (C_A × T_A) : (C_B × T_B) : (C_C × T_C)

Example 1: Simple Partnership

A and B started a business by investing ₹20,000 and ₹30,000 respectively. After one year, they made a profit of ₹25,000. What is the share of each partner?

Capital of A (C_A) = ₹20,000

Capital of B (C_B) = ₹30,000

Time period for both is the same (T_A = T_B = 1 year).

Ratio of Profits = (C_A × T_A) : (C_B × T_B)

Ratio = (20000 × 1) : (30000 × 1)

Ratio = 20000 : 30000

Simplifying the ratio by dividing by 10000:

Ratio = 2 : 3

Total profit = ₹25,000.

Total parts in the ratio = 2 + 3 = 5.

A's share = (2 / 5) * 25000 = ₹10,000.

B's share = (3 / 5) * 25000 = ₹15,000.

Example 2: Different Time Periods

A invested ₹50,000 for 6 months and B invested ₹60,000 for 5 months. They earned a total profit of ₹72,000. Find the profit share of each.

A's Investment = 50,000 for 6 months.

B's Investment = 60,000 for 5 months.

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

Ratio = (50000 × 6) : (60000 × 5)

Ratio = 300000 : 300000

Ratio = 1 : 1

Total profit = ₹72,000.

Total parts = 1 + 1 = 2.

A's share = (1 / 2) * 72000 = ₹36,000.

B's share = (1 / 2) * 72000 = ₹36,000.

Types of Partnerships

Partnerships can be classified based on the duration of the partnership and the nature of capital investment.

• Simple Partnership: All partners invest their capital for the same duration. In this case, the profit ratio is simply the ratio of their capitals.

  Profit Ratio = C_A : C_B : C_C

• Compound Partnership: Partners invest their capital for different durations. This is the case we've discussed where profit ratio = (C_A × T_A) : (C_B × T_B) : (C_C × T_C).

Introduction of New Partner or Retirement of a Partner

When a new partner joins or an existing partner leaves, the profit-sharing ratio for the period after the change needs to be considered. The calculation of profit share for the period before the change remains based on the old ratios and investments.

Example 3: New Partner Entry

A and B entered into a partnership investing ₹40,000 and ₹60,000 respectively. After 4 months, C joined them with an investment of ₹80,000. If the total profit at the end of the year was ₹45,000, find the share of each partner.

A invested ₹40,000 for 12 months (full year).

B invested ₹60,000 for 12 months (full year).

C invested ₹80,000 for 8 months (since he joined after 4 months, the remaining period is 12 - 4 = 8 months).

Ratio of Profit Shares = (A's Capital × A's Time) : (B's Capital × B's Time) : (C's Capital × C's Time)

Ratio = (40000 × 12) : (60000 × 12) : (80000 × 8)

Ratio = 480000 : 720000 : 640000

Simplify by dividing by 10000:

Ratio = 48 : 72 : 64

Simplify further by dividing by their Greatest Common Divisor (GCD), which is 8:

Ratio = 6 : 9 : 8

Total profit = ₹45,000.

Total parts = 6 + 9 + 8 = 23.

A's share = (6 / 23) * 45000

B's share = (9 / 23) * 45000

C's share = (8 / 23) * 45000

(Note: This example might result in non-integer shares if the total profit isn't divisible by 23. In exams, values are usually chosen to give round figures). Let's assume total profit was ₹46,000 for cleaner calculation.

If Total Profit = ₹46,000:

A's share = (6 / 23) * 46000 = 6 * 2000 = ₹12,000.

B's share = (9 / 23) * 46000 = 9 * 2000 = ₹18,000.

C's share = (8 / 23) * 46000 = 8 * 2000 = ₹16,000.

Check: 12000 + 18000 + 16000 = 46000. Correct.

Working Partners and Sleeping Partners

In some partnerships, one or more partners may actively manage the business (working partners), while others may only contribute capital (sleeping partners). The partnership deed usually specifies:

• Salary for Working Partners: Working partners might receive a fixed salary in addition to their profit share. This salary is usually deducted from the total profit before distributing the remaining profit in the ratio of capital or agreed ratio.
• Interest on Capital: Partners might receive interest on the capital they have invested. This interest is also deducted from the total profit before distribution.
• Interest on Drawings: If partners withdraw money (drawings), they might be charged interest on it. This interest is added to the profit.

Example 4: Salary and Interest on Capital

A and B entered into a partnership investing ₹80,000 and ₹1,20,000 respectively. They decided that:

• Interest on capital will be given at 10% per annum.
• A, being a working partner, will receive a salary of ₹12,000 per annum.
• The remaining profit will be shared in the ratio of their capitals.

The profit at the end of the year was ₹60,000. Find the profit share of each.

A's Capital = ₹80,000

B's Capital = ₹1,20,000

Profit = ₹60,000

Step 1: Calculate Interest on Capital

Interest for A = 10% of 80,000 = (10/100) * 80000 = ₹8,000.

Interest for B = 10% of 1,20,000 = (10/100) * 120000 = ₹12,000.

Total Interest = 8,000 + 12,000 = ₹20,000.

Step 2: Add Salary for Working Partner

A's Salary = ₹12,000.

Step 3: Calculate the total amount to be deducted/paid before profit sharing

Total Deductions = Total Interest + A's Salary = 20,000 + 12,000 = ₹32,000.

Step 4: Calculate the Remaining Profit

Remaining Profit = Total Profit - Total Deductions = 60,000 - 32,000 = ₹28,000.

Step 5: Determine the Profit Sharing Ratio

Ratio of Capitals = 80000 : 120000 = 8 : 12 = 2 : 3.

Step 6: Distribute the Remaining Profit

A's share of remaining profit = (2 / (2+3)) * 28000 = (2/5) * 28000 = 2 * 5600 = ₹11,200.

B's share of remaining profit = (3 / (2+3)) * 28000 = (3/5) * 28000 = 3 * 5600 = ₹16,800.

Step 7: Calculate Total Earnings for Each Partner

A's Total Earnings = Interest on Capital + Salary + Share of Remaining Profit

A's Total Earnings = 8,000 + 12,000 + 11,200 = ₹31,200.

B's Total Earnings = Interest on Capital + Share of Remaining Profit

B's Total Earnings = 12,000 + 16,800 = ₹28,800.

Check: Total Earnings = 31,200 + 28,800 = ₹60,000. Correct.

Important Considerations

• Partnership Deed: A written agreement is highly recommended to avoid disputes. It should cover capital contribution, profit/loss sharing ratio, salaries, interest rates, admission/retirement procedures, etc.
• Joint Liability: In a general partnership, partners have unlimited liability, meaning their personal assets can be used to pay off business debts.
• Continuity: The business may dissolve upon the death, insolvency, or retirement of a partner unless the partnership deed states otherwise.

Mixture and Allegation

The concepts of Mixture and Allegation are used to solve problems involving the mixing of two or more ingredients or quantities of different values or characteristics. Allegation is a rule used to find the ratio in which two or more ingredients must be mixed to produce a mixture of a desired mean value.

Key Concepts

• Mixture: A combination of two or more substances.
• Allegation: A method to determine the ratio of quantities of two ingredients when they are mixed to obtain a mixture of a specified value.
• Mean Value: The desired value or characteristic of the final mixture.

The Rule of Allegation (for two ingredients)

Suppose we want to mix two ingredients, Ingredient 1 (cheaper) and Ingredient 2 (dearer), to obtain a mixture with a Mean Value (MV). Let the value (price per unit) of Ingredient 1 be C (Cost) and the value of Ingredient 2 be D (Dear). The desired Mean Value is MV.

According to the rule of allegation, the ratio in which the two ingredients should be mixed is found as follows:

(Quantity of Cheaper Ingredient) / (Quantity of Dearer Ingredient) = (D - MV) / (MV - C)

This can be visually represented using a diagram:

Cheaper (C) Dearer (D)
(MV - C) represents the difference between the Mean Value and the Cheaper value.
(D - MV) represents the difference between the Dearer value and the Mean Value.

The ratio of the quantities is obtained by cross-subtracting:

C D - MV
\ / MV
/ \ D MV - C

So, the ratio of the quantity of the cheaper ingredient to the quantity of the dearer ingredient is (D - MV) : (MV - C).

Example 1: Simple Mixture (Price)

In what ratio should rice costing ₹20 per kg be mixed with rice costing ₹30 per kg so that the mixture costs ₹25 per kg?

Cheaper Ingredient (C) = ₹20/kg

Dearer Ingredient (D) = ₹30/kg

Mean Value (MV) = ₹25/kg

Using the allegation rule:

Ratio = (D - MV) : (MV - C)

Ratio = (30 - 25) : (25 - 20)

Ratio = 5 : 5

Ratio = 1 : 1

So, the rice should be mixed in the ratio 1:1.

Example 2: Mixture of Milk and Water

A shopkeeper mixes milk and water in the ratio 5:2. The cost of milk is ₹40 per litre. What is the cost price of the mixture per litre? (Assume water is free).

Here, the 'ingredients' are milk and water.

Cost of Milk (Dearer) = ₹40/litre

Cost of Water (Cheaper) = ₹0/litre (since it's free)

The ratio of mixing is given as 5:2 (Milk:Water).

Using the allegation rule, the ratio of Milk (Dearer) to Water (Cheaper) should be:

(Quantity of Water) / (Quantity of Milk) = (Cost of Milk - Mean Cost) / (Mean Cost - Cost of Water)

Ratio (Water : Milk) = (40 - MV) / (MV - 0)

We are given the ratio of Milk : Water = 5 : 2.

So, Ratio (Water : Milk) = 2 : 5.

Equating the ratios:

(40 - MV) / MV = 2 / 5

5 * (40 - MV) = 2 * MV

200 - 5*MV = 2*MV

200 = 7*MV

MV = 200 / 7 ≈ ₹28.57 per litre.

The cost price of the mixture is approximately ₹28.57 per litre.

Mixture Problems involving Profit/Loss

These problems often involve selling a mixture at a certain price and making a profit or loss. The key is to first find the cost price of the mixture using allegation and then use profit/loss formulas.

Example 3: Profit on Mixture

A merchant mixes two types of tea, one costing ₹40 per kg and another costing ₹55 per kg. He wants to sell the mixture at ₹50 per kg and make a profit of 10%. In what ratio should he mix them?

First, let's find the cost price (CP) of the mixture.

Selling Price (SP) = ₹50 per kg

Profit = 10%

We know SP = CP * (1 + Profit%/100)

50 = CP * (1 + 10/100)

50 = CP * (1 + 0.1)

50 = CP * 1.1

CP = 50 / 1.1 = 500 / 11 ≈ ₹45.45 per kg. This is the Mean Value (MV).

Now, apply the rule of allegation:

Cheaper Tea (C) = ₹40/kg

Dearer Tea (D) = ₹55/kg

Mean Value (MV) = 500/11 per kg

Ratio = (D - MV) : (MV - C)

Ratio = (55 - 500/11) : (500/11 - 40)

Calculate the terms:

55 - 500/11 = (55 * 11 - 500) / 11 = (605 - 500) / 11 = 105 / 11

500/11 - 40 = (500 - 40 * 11) / 11 = (500 - 440) / 11 = 60 / 11

Ratio = (105 / 11) : (60 / 11)

Ratio = 105 : 60

Simplify by dividing by GCD (which is 15):

Ratio = 7 : 4

The merchant should mix the teas in the ratio 7:4.

Allegation with More Than Two Ingredients

The rule of allegation can be extended to mixtures of three or more ingredients, but it becomes more complex. Usually, problems involve mixing two ingredients at a time, or a mixture with a third ingredient.

Example 4: Mixing a Mixture with another Ingredient

A shopkeeper has a mixture of milk and water in the ratio 7:3. He adds pure milk to this mixture in a quantity equal to the original mixture. Find the new ratio of milk and water.

Let the original mixture contain 7x litres of milk and 3x litres of water.

Total quantity of the original mixture = 7x + 3x = 10x litres.

The shopkeeper adds pure milk equal to the original mixture's quantity, which is 10x litres.

Amount of milk added = 10x litres.

Amount of water added = 0 litres.

New quantity of milk = Original milk + Added milk = 7x + 10x = 17x litres.

New quantity of water = Original water + Added water = 3x + 0 = 3x litres.

New ratio of milk and water = 17x : 3x = 17 : 3.

Key Applications of Mixture and Allegation

• Pricing: Mixing goods of different prices (e.g., tea, coffee, grains).
• Concentration: Mixing solutions of different concentrations (e.g., acid and water, alcohol and water).
• Proportions: Determining the ratio of components in a mixture.
• Profit/Loss Calculations: When selling mixtures.

Common Pitfalls

• Confusing Cost Price (CP) with Selling Price (SP) when profit/loss is involved. Always calculate the CP of the mixture first.
• Incorrectly applying the ratio. Ensure you know which quantity corresponds to which difference (e.g., (D-MV) relates to the cheaper ingredient's quantity).
• Not simplifying the final ratio.
• Ignoring the units (e.g., per kg, per litre).