**Calculating Markups and Markdowns :**

A markup is one kind of percent increase. We can use a bar model to represent the retail price of an item, that is, the total price including the markup.

An example of a percent decrease is a discount, or markdown. A price after a markdown may be called a sale price. We can also use a bar model to represent the price of an item including the markdown.

**Mark up ----> Increasing **

To get profit in a business, a trader increases the cost price and sells the product. This increment in price is called as "Mark up"

This "Mark up can either be in percent or in dollars.

**Mark Down ----> Decreasing**

To increase the sale, stores will decrease the price of a product by giving offer or discount. This offer or discount is called as "Mark down".

This mark down can either be in percent or in dollars.

**Hint 1 :**

Cost price and marked up percentage are given.

Cost price = C.P, Marked up percentage = m%

Then,

Selling price (S.P) = (100 + m)% ⋅ C.P

**Hint 2 :**

List price and marked down percentage are given.

List price = L.P, Marked down percentage = m%

Then,

Selling price (S.P) = (100 - m)% ⋅ L.P

**Hint 3 :**

List price price and marked down value (in dollars ) are given.

List price = L.P, Marked down value = $m

Then,

mark down rate = (m / L.P) ⋅ 100%

**Hint 4 :**

Cost price and marked up value are given

Cost price = C.P, Marked up value = $m

Then,

mark up rate = (m / C.P) ⋅ 100%

**Hint 5 :**

Cost price and selling price are given.

Cost price = C.P, Selling price = S.P and S.P > C.P

So, Gain = S.P - C.P

Then,

mark up rate = (Gain / C.P) ⋅ 100%

**Hint 6 :**

Selling price and profit percentage are given.

How to find cost price ?

Use hint 1 and solve for C.P

**Hint 7 :**

Selling price and loss percentage are given.

How to find cost price ?

Use hint 2 and solve for C.P

**Hint 8 :**

Marked price : It is the price before discount given.

Selling price = Marked price - Discount value

**Hint 9 :**

Marked price = M.P, Discount percentage = D%

Then, the discount value is

= D% ⋅ M.P

Selling price is

= (100 - D)% ⋅ M.P

**Hint 10 :**

Marked price (M.P) and discount value are given.

Then the discount percentage is

= (Discount value / M.P) ⋅ 100%

**Hint 11 :**

Retailer using false weight :

A trader cheats his customer to make a profit by stating that he sells at cost price. But he gives his customer less than 1000 grams (false weight) for every 1 kilogram.

Then, the profit percentage is

= (Cheated value / False weight) ⋅ 100 %

Here,

Cheated value = Original weight - False weight

**Hint 12 :**

Two articles are sold at the same price. But, one is sold at a profit of p% and other one is sold at a loss of p%.

Then, the net result of the transaction is loss.

The loss percentage is

= (p^{2} / 100)%

**Hint 13 :**

The cost price of two articles is same. But, one is sold at a profit of p% and other one is sold at a loss of p%.

Then, the net result of the transaction is no profit and no loss.

**Example 1 : **

To make a profit, stores mark up the prices on the items they sell. A sports store buys skateboards from a supplier for s dollars. What is the retail price for skateboards that the manager buys for $35 and $56 after a 42% markup ?

**Solution : **

**Step 1 : **

Use a bar model.

Draw a bar for the cost of the skateboard S.

Then draw a bar that shows the markup: 42% of S, or 0.42S.

These bars together represent the cost plus the markup.

That is

S + 0.42S

**Step 2 :**

Retail price = Original cost + Markup

= S + 0.42S

= 1S + 0.42S

= 1.42S

**Step 3 : **

Use the expression to find the retail price of each skateboard.

S = $35 ----> Retail price = 1.42($35) = $49.70

S = $56 ----> Retail price = 1.42($56) = $79.52

**Example 2 : **

A discount store marks down all of its holiday merchandise by 20% off the regular selling price. Find the discounted price of decorations that regularly sell for $16 and $23.

**Solution : **

**Step 1 : **

Use a bar model.

Draw a bar for the regular price P.

Then draw a bar that shows the discount: 20% of P, or 0.2P.

The difference between these two bars represents the price minus the discount.

That is,

P - 0.2P

**Step 2 :**

Sale price = Original price - Markdown

= p - 0.2p

= 1p - 0.2p

= 0.8p

**Step 3 :**

Use the expression to find the sale price of each decoration.

p = $16 ---> Sale price = 0.8($16) = $12.80

p = $23 ---> Sale price = 0.8($23) = $18.40

**Example 3 : **

On selling 20 units of an item, the profit is equal to cost price of 5 units. Find the mark mark up rate.

**Solution :**

Let m be the cost price of one unit.

Then, we have

Cost price of 5 units = 5m

Cost price of 20 units = 20m

**Given :** On selling 20 units of an item, the profit is equal to cost price of 5 units.

Then, we have

Profit on selling 20 units = C.P of 5 units = 5m

Mark up rate = (profit / cost) ⋅ 100%

Mark up rate = (5m / 20m) ⋅ 100%

Mark up rate = (1 / 4) ⋅ 100%

Mark up rate = 25%

**Example 4 :**

Difference between the cost price of two products is $10. Difference between the selling price is $20. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.

**Solution :**

Let x and y be the cost prices of two products.

Then, we have

x - y = 10 -----(1)

Let us assume that x is sold at 20% profit.

Then, the selling price of x is

= 120% ⋅ x

= 1.2x

Let us assume that y is sold at 20% loss.

Then, the selling price of y is

= 80% ⋅ y

= 0.8y

**Given :** Difference between the selling price is $20

1.2x - 0.8y = 20

Multiply each side by 10.

12x - 8y = 200

Divide each side by 4.

3x - 2y = 50 -----(2)

Solving (1) and (2), we get

x = 30

y = 20

So, the cost prices of the two products are $30 and $20.

**Example 5 : **

A trader marks his goods 20% above the cost price and allows a discount of 10% for cash. Find the mark up rate.

**Solution :**

Let the cost price be $100.

Then, marked price is $120.

Let x be the selling price.

From the above picture, we get

90% of (M.P) = x

(0.9) ⋅ 120 = x

108 = x

Therefore, the selling price is $108.

Cost price = $100

Selling Price = $108

Mark up rate = 8%

After having gone through the stuff given above, we hope that the students would have understood, how to solve mark up and mark down word problems.

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