Marked price and markup

A shop­keeper does not usu­ally print his cost price on an arti­cle. He adds some­thing to it first, and prints the big­ger price. That printed price is the marked price, and the amount he adds is called the markup. Then, at sale time, he may give a dis­count on the marked price. In this les­son you will work with markup and dis­count together, and solve some harder profit prob­lems.

Markup#

Markup is the amount added to the cost price to get the marked price.

$$\text{M.P.} = \text{C.P.} + \text{markup}$$

The markup per­cent­age is worked out on the cost price:

$$\displaystyle \text{markup}\% = \left(\frac{\text{markup}}{\text{C.P.}} \times 100\right)\%$$

So if the markup is $m\%$, then

$$\displaystyle \text{M.P.} = \left(\frac{100 + m}{100}\right)\text{C.P.}$$

Exam­ple 1#

The cost price of an arti­cle is ₹300 and the markup is 20%. What is the marked price?

$$\displaystyle \begin{aligned} \text{M.P.} &= 300 + 20\% \text{ of } 300 \\ &= 300 + \frac{20}{100} \times 300 \\ &= 300 + 60 = 360 \end{aligned}$$

The marked price is ₹360.

Exam­ple 2#

The marked price of an arti­cle is ₹450 and the markup is 12.5%. What is the cost price?

Let the cost price be $x$. Then $x$ plus 12.5% of $x$ is 450.

$$\displaystyle \begin{aligned} x + \frac{12.5}{100}x &= 450 \\ x + \frac{125}{1000}x &= 450 \\ \frac{1125}{1000}x &= 450 \\ x &= \frac{450 \times 1000}{1125} = 400 \end{aligned}$$

The cost price is ₹400. Check: 12.5% of 400 is 50, and $400 + 50 = 450$.

Markup and dis­count together#

Many shops mark the price up and then give a dis­count. Work in two steps: first from C.P. to M.P., then from M.P. to S.P.

Exam­ple 3#

A shop­keeper buys a lamp for ₹800. He marks it 25% above the cost price, and then gives a 10% dis­count. Find his profit per cent.

$$\displaystyle \text{M.P.} = \frac{125}{100} \times 800 = 1000$$

$$\displaystyle \text{S.P.} = \frac{90}{100} \times 1000 = 900$$

Profit $= 900 - 800 = 100$ rupees.

$$\displaystyle \text{profit}\% = \frac{100}{800} \times 100 = 12.5\%$$

The markup was 25% and the dis­count was 10%, but the profit is not 15%. The dis­count is worked out on the marked price, which is a big­ger amount than the cost price.

Com­par­ing many items#

Exam­ple 4: 10 sold for the cost of 12#

The sell­ing price of 10 tele­vi­sion sets is the same as the cost price of 12 sets. Find the profit or loss per cent.

$$\displaystyle 10 \times \text{S.P.} = 12 \times \text{C.P.} \quad \text{so} \quad \frac{\text{S.P.}}{\text{C.P.}} = \frac{12}{10} = \frac{6}{5}$$

The ratio is more than 1, so S.P. is more than C.P. There is a profit.

Take S.P. $= 6k$ and C.P. $= 5k$.

$$\displaystyle \text{profit}\% = \frac{6k - 5k}{5k} \times 100 = \frac{1}{5} \times 100 = 20\%$$

Work­ing back from the sell­ing price#

Exam­ple 5: A con­ces­sion on the label price#

A man got a 30% con­ces­sion on the label price of an arti­cle. He then sold it for ₹8750, mak­ing a 25% profit on the price he paid. What was the label price?

A con­ces­sion is the same as a dis­count.

Let the label price (M.P.) be $x$. The man paid 30% less than that:

$$\displaystyle \text{C.P.} = x - \frac{30}{100}x = \frac{70}{100}x = \frac{7x}{10}$$

He sold it at a 25% profit for ₹8750:

$$\displaystyle \begin{aligned} 8750 &= \frac{100 + 25}{100} \times \frac{7x}{10} \\ 8750 &= \frac{125 \times 7x}{1000} \\ x &= \frac{8750 \times 1000}{125 \times 7} = 10000 \end{aligned}$$

The label price was ₹10,000. He paid $\displaystyle \frac{7}{10} \times 10000 = 7000$ rupees for it.

Check: 25% profit on ₹7000 is ₹1750, and $7000 + 1750 = 8750$. Cor­rect.

Buy­ing and sell­ing in dif­fer­ent lots#

Exam­ple 6: 11 for ₹100, 10 for ₹110#

I bought 11 books for ₹100 and sold 10 books for ₹110. What is my profit per cent on each book?

First method. Find the price of one book.

C.P. of 1 book $\displaystyle = \frac{100}{11}$ rupees. S.P. of 1 book $\displaystyle = \frac{110}{10} = 11$ rupees.

$$\displaystyle \begin{aligned} \text{profit}\% &= \frac{11 - \frac{100}{11}}{\frac{100}{11}} \times 100 \\ &= \frac{\frac{121 - 100}{11}}{\frac{100}{11}} \times 100 \\ &= \frac{21}{100} \times 100 = 21\% \end{aligned}$$

Sec­ond method. Use a num­ber of books that both lots go into. 110 is a mul­ti­ple of both 11 and 10.

C.P. of 11 books is ₹100, so C.P. of 110 books is ₹1000.

S.P. of 10 books is ₹110, so S.P. of 110 books is $110 \times 11 = 1210$ rupees.

Profit on 110 books $= 1210 - 1000 = 210$ rupees.

$$\displaystyle \text{profit}\% = \frac{210}{1000} \times 100 = 21\%$$

Both meth­ods give 21%.

Prac­tice#

1. An arti­cle costs ₹1500 and is marked up by 30%. Find the marked price.
2. An arti­cle has a marked price of ₹1320 after a markup of 10%. Find the cost price.
3. A radio costs a shop ₹1600. It is marked ₹2000 and sold at a 15% dis­count. Find the profit per cent.
4. A shop­keeper marks his goods 40% above cost price and gives a 25% dis­count. Find his profit per cent.
5. The sell­ing price of 15 arti­cles equals the cost price of 18 arti­cles. Find the profit per cent.
6. A boy buys 8 pens for ₹60 and sells 6 pens for ₹54. Find his profit per cent.
7. Ravi got a 20% dis­count on the label price of a phone. He sold it for ₹7200 and made a 20% profit on what he paid. Find the label price.
8. An arti­cle marked ₹1250 is sold for ₹1000. Find the dis­count per cent.

Answers#