In this lesson, we are going to solve different varieties of profit and loss questions having different levels of difficulty. This lesson only covers the practice of profit and loss problems with their solutions and it does not explain the basic concepts and derivation of formulas. The purpose of this tutorial is to master and apply already learned concepts.

Before heading onto examples, let’s first see some important formulas that are commonly used while solving profit and loss questions. Note that some formulas are derived from basic formulas, they are written here so that we can quickly solve the question and do not indulge in their derivation.

## Basic concepts and formulas

Let profit be P, loss be L, selling price be SP and cost Price be CP.

- P = SP – CP, the selling price is greater than the cost price when there is a profit.
- L = CP – SP, the selling price is less than the cost price when there is a loss.
- Profit Percentage = (P/CP) × 100
- Loss Percentage = (L/CP) × 100

Selling price when there is profit

- SP = { (100 + Profit %) / 100 } × CP

Selling Price when there is loss

- SP = { (100 – Loss %) / 100 } × CP

Cost price when there is profit

- CP = { 100 / (100 + Profit %) } × SP

Cost price when there is loss

- CP = { 100 / (100 – Loss %) } × SP
- Selling Price = Marked price – Discount
- Discount Percentage = (Discount / Marked price) × 100

If you don’t know the shortcut tricks to find percentage click on Percentage tricks and shortcuts.

Following Image is a summary of all profit and loss formulas. You can download it.

Now, let’s solve some examples with the help of above profit and loss formulas.

## Profit and loss questions

Here, important profit and loss problems are compiled these are commonly asked questions in any aptitude competition exam.

### Simple question to find gain/loss

**Problem 1**

Person A buys a product for Rs. 6000 and sells it to person B for Rs. 7000. Person B sells the same product for Rs. 7500 to person C. Find the profit gained by A and B.

**Solution**

For person A

Cost price = CP = 6000

Selling price = SP = 7000

Profit gained by A = SP – CP

Profit = 7000 – 6000

Profit = 1000

**Person A earned a profit of Rs. 1000.**

Similarly, for person B

Cost price = CP = 7000

Selling price = SP = 7500

Profit gained by B = SP – CP

= 7500 – 7000 = 500

**Person B earned a profit of Rs. 500.**

### Questions to find percentage gain or loss

**Problem 2**

Consider a shopkeeper who has bought a table for Rs. 2000 and sold it at the price of 1400. What is the percentage of loss?

**Solution**

Cost Price = CP = 2000

Selling Price = SP = 1400

Loss Percentage = (L / CP) × 100

We know that L = CP – SP

L = 2000 – 1400

L = 600

Putting the value of L in the loss percentage equation. We get,

Loss Percentage = (600 / 2000) × 100

Loss Percentage = 30%

**Shopkeeper suffered a loss of 30%.**

**Problem 3**

Suppose Ali sells two articles for Rs. 1200 each. He makes a profit of 10% on one article and a loss of 5% on the other. Find out the percentage gain or loss?

**Solution**

Selling Price = SP = 1200

Profit % = P = 10%

Loss % = L = 5%

Finding the cost price for first article where profit has occurred. Using formula

CP = { 100 / (100 + Profit %) } × SP

= { 100 / (100 + 10)} × 1200

= (100 / 110) × 1200

= 1090.908

Finding the cost price for second article where loss has occurred. Using formula

CP = { 100 / (100 – Loss %) } × SP

= { 100 / (100 – 5)} × 1200

= (100 / 95) × 1200

= 1263.1572

Total selling price = 1200 + 1200 = 2400

Total cost price = 1090.908 + 1263.1572 = 2354.0652

Since total cost price is less than the selling price so profit has occurred.

Finding the profit percentage = { (SP – CP) / CP } × 100

% gain = {(2400 – 2354.0652) / 2354.0652} × 100

% gain = 1.95

**Alternate Approach**

We can solve these types of problems in which we have to find the percentage gain or loss when we sell two things at the same price, but we get profit in one case and loss in the other, we use formula

**Percentage gain or loss **

**= [100(Profit % – Loss %) – 2 * Profit % * Loss %] / [(100 + Pofit %) + (100 – Loss %)]**

Putting values

% gain or loss = [100(10 – 5) – 2(5)(10)] / [(100+10) + (100 – 5)]

= [100(5) – 100] / (110 + 95)

= (400) / (205)

= (400) / (205)

= 1.95

If the value of % gain or loss is positive, then the profit has occurred. Otherwise, loss has occurred.

**A profit of 1.95% is obtained.**

**Problem 4**

Consider a student at bake sale who sells two cupcakes for Rs. 200 each. He gains a profit of 15% on one cupcake and a loss of 15% on the other. Find the percentage loss or gain?

**Solution**

Selling Price = SP = 200

Profit % = P = 15%

Loss % = L = 15%

This is the same type of problem that was solved above. However, here profit percentage and loss percentage are the same.

Percentage gain or loss = [100(P – L) – 2PL] / [(100 + P) + (100 – L)]

let P = L = x

The above reduces to

% gain or loss = -x^{2} / 100

Here a negative sign shows a loss

% loss= (15)^{2} / 100

= 225 / 100 = 2.25

**A loss of 2.25% is observed.**

**Problem 5**

Ali bought a phone and paid 10% less than its original price. He sold it to another person at a profit of 40% of the cost price. What was the percentage of profit on the original price?

**Solution**

let x be the original price of the phone.

Cost price = x – (10 / 100)x

= x – 0.1x

= CP = 0.9x

Finding selling price

SP = { (100 + P) / 100 } × CP

= { (100 + 40) / 100 } × 0.9x

= (140 / 100) × 0.9x

SP = 1.4 × 0.9x

SP = 1.26x

Finding percentage of profit on the original price

Profit Percentage = (P/CP) × 100

= { (SP- CP) / CP) } × 100

= { (1.26x- x) / x) } × 100

= { (0.26x) / x) } × 100

= 0.26 × 100

Profit Percentage = 26%

**Profit Percentage of 26% was received on the original price of the phone.**

### How to find cost price or selling price

**Problem 6**

A seller could have made a profit of 10% on a product instead of a 15% loss if he would have charged 1200 extra. What is the cost price?

**Solution**

Let cost price = CP = x

Let the initial selling price = y

Percentage loss = 15%

Percentage gain = 10%

A loss of 15% has occurred at the initial selling price.

Using selling price formula for loss

SP = { (100 – Loss %) / 100 } × CP

y = { (100 – 15) / 100 } × x

= (85 / 100) × CP

y = 0.85x

If the selling price is y + 1200 then the profit of 10% is obtained.

Using selling price formula for profit

SP = { (100 + Profit %) / 100 } × CP

y + 1200 = { (100 + 10) / 100 } × x

substituting the value of y from the above equation

0.85x + 1200 = (110 / 100) × x

0.85x + 1200 = 1.1x

1.1x – 0.85x = 1200

0.25x = 1200

x = 1200 / 0.25

x = 4800

**The cost price is Rs. 4800.**

### Find the quantity or number of items

**Problem 7**

The cost of 8 pens is the same as the selling price of n pens. If there is a loss of 20%, what is the value of n?

**Solution**

Let the cost of 1 pen be x.

Then, the cost of 8 pens will be 8x.

Let the selling price of 1 pen be y.

Then, the selling price of n pens will be ny.

We know from the question that the cost of 8 pens is equal to the selling price of n pens. i.e.

8x = ny → equation (a)

A loss of 20% has occurred. Using the formula

Loss Percentage = (L/CP) × 100

Loss Percentage = { (CP – SP) / CP } × 100

20 = { (x – y) / x } × 100

20 / 100 = (x – y) / x

0.2x = x – y

y = x – 0.2x

y = 0.8x → equation (b)

From equation (a), we have,

8x = ny

or x = (ny) / 8

Putting the value of x in equation (b), we get,

y = 0.8 × { (ny) / 8 }

1 = 0.8n / 8

1 = 0.1n

n = 10

**The value of n is 10.**

### Profit and loss questions involving markup price and discount

**Problem 8**

If a shopkeeper marks up his price by 40% and then allows a discount of 20%, what is his percentage profit or loss?

**Solution**

Marked price is the price that is printed on the item to be sold. It is also known as listed price.

A discount is a deduction in the marked price.

The selling price is the price at which the item is sold to the customer.

Therefore,

**Selling Price = Marked price – Discount**

Moreover,

**Discount Percentage = (Discount / Marked price) × 100**

Let the cost price be x. Then, the marked price will be

Marked price = x + (40 / 100)x

Discount Percentage = 20%. Discount can be found as

Discount Percentage = (Discount / Marked price) × 100

20 = [Discount / { x + (40 / 100)x }] × 100

20 / 100 = Discount / { x + (40 / 100)x }

Discount = (20 / 100) × { x + (40 / 100)x }

Now,

Selling Price = Marked price – Discount

Putting the values

Selling Price = x + (40 / 100)x – (20 / 100) × { x + (40 / 100)x }

Selling Price = x + 0.4x – 0.2x – 0.08x

Selling Price = 1.12x

It is obvious from the above equation that selling price is greater than cost i.e. profit will occur. Finding the profit percentage,

Profit Percentage = { (SP – CP) / CP } × 100

= { (1.12x – x) / x } × 100

= { (0.12x) / x } × 100

= 0.12 × 100

Profit Percentage = 12%

**Seller has gained 12% of profit.**

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