Math 7 · 7.RP.A.3

Markup and margin

A seller buys something for one price and sells it for more. The extra is what percent? A percent of what it cost, or of what it sells for?

You need

Percent increase and decreaseMath 7 Finding what percent one number is of anotherMath 7 Equal shares and naming the wholeGrade 1

Where you are 3 in Percent in money situations

Fractions, ratio and percent3

Percent8

Percent in money situations3

Markup and margin

Discounts, sales tax, tips and commission Simple interest

When you reach for it

Anyone who sells things has to get them first. A shop buys its stock from a supplier, a craft seller buys clay or beads or yarn, a café buys milk and coffee, a school fundraiser stall buys snacks in bulk. What the seller pays for one item is its cost. When the item is bought ready-made from a supplier, in bulk, the cost is also called the wholesale price. What a customer pays for it is the selling price. The gap between the two is the profit on the item:

\text{profit} = \text{selling price} - \text{cost}

In this lesson profit means exactly that gap and nothing more: the price minus what the item itself cost. A shop’s rent, a café’s wages and a market stall’s fee are left out of it, so it is not the same as the money the seller ends up keeping.

A profit in dollars, like any change, says little on its own. Two dollars on a cup of coffee that cost 30 cents to make is a lot; two dollars on a bicycle that cost 300 dollars is almost nothing. So sellers state the profit as a percent. And here there is a real choice, because there are two natural wholes to measure it against.

The markup is the profit as a percent of the cost.
The margin, or profit margin, is the profit as a percent of the selling price.

The two numbers are different, and they are different for the reason that runs through every percent question: a percent means nothing until you have named the whole. Both are the question from finding what percent one number is of another, with the profit as the part. Markup takes the cost as the whole; margin takes the price.

The word markup is also used for the dollar amount itself, as in “a markup of 14 dollars”, and then the percent is called the markup rate. Margin is used both ways too. When a percent follows the word, the percent is meant.

This lesson is taught over two days. Part 1 is markup: finding a price, a markup rate or a cost. Part 2 is margin, how it differs from markup, how to change one into the other, and how to set a price that gives a chosen margin.

The procedure

Part 1: markup

Markup is a percent increase. The cost is where the amount starts, the selling price is where it ends, and the change is measured against the start:

\text{markup rate} = \frac{\text{selling price} - \text{cost}}{\text{cost}}

Any two of cost, selling price and markup rate fix the third, so there are three questions to ask.

Cost and markup rate known: find the price. Teodoro’s shop buys backpacks wholesale for 35 dollars each and marks them up 40\%. The markup in dollars is 40\% of the cost, because the cost is the whole:

0.40 \times 35 = 14

The selling price is the cost with the markup added on: 35 + 14 = 49 dollars.

Figure 1The backpack’s cost of 35 dollars is the whole, 100\%. The markup of 14 dollars is the extra piece, and the selling price of 49 dollars sits at 140\% of the cost.

The figure shows the same thing on a double number line. The cost is 100\%, the markup is the extra 40\%, so the price sits at 140\% of the cost. That gives a one-step way to the price:

1.40 \times 35 = 49

It is allowed because \text{cost} + 0.40 \times \text{cost} = (1 + 0.40) \times \text{cost}, which is the distributive property read from right to left. The number 1 + 0.40 = 1.40 is the multiplier for a 40\% increase. In general, with the markup rate r written as a decimal,

\text{selling price} = (1 + r) \times \text{cost}

Cost and price known: find the markup rate. Subtract to get the profit, then divide by the cost, because the cost is the whole. For the backpack, 49 - 35 = 14 and \displaystyle \frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.

Price and markup rate known: find the cost. The price is (1 + r) times the cost, and dividing undoes multiplying. If all you knew was that a backpack sells for 49 dollars after a 40\% markup, then 49 is 140\% of the cost, and

\text{cost} = \frac{49}{1.40} = 35

This is finding the whole from a part and a percent, with the price as the part. The 40\% cannot be taken of the 49 dollars: it is a percent of the cost, and the cost is the number you are looking for.

Part 2: margin

The margin uses the same profit and measures it against the selling price instead:

\text{margin} = \frac{\text{selling price} - \text{cost}}{\text{selling price}}

For the backpack the profit is still 14 dollars, but the whole is now 49:

\frac{14}{49} = \frac{2}{7} \approx 0.286 = 28.6\%

So the backpack has a markup of 40\% and a margin of about 28.6\%. Neither is wrong. They are answers to two different questions: “how much more than the cost is the price?” and “how much of the price is profit?”

Why the margin is smaller. When the seller makes a profit, the price is the cost plus something, so the price is the bigger number. The part is the same, the whole is bigger, and a part of a bigger whole is a smaller fraction of it. Seen another way, markup is the percent increase going up from cost to price, and margin is the percent decrease going down from price to cost. The same change is always a smaller percent of the larger amount.

What the margin tells you. The margin reads straight off the money taken in. A margin of 28.6\% on backpacks means that of every dollar a customer pays for one, about 28.6 cents is profit and the other 71.4 cents pays back what the backpack cost. The two shares make the whole price, so they add to 100\%.

Changing one into the other. Both percents compare the same three amounts, so either one fixes the other, whatever the cost happens to be. Doubling both the cost and the price doubles the profit too, and the two fractions do not change. So it is enough to imagine a cost of 100 dollars. A 40\% markup makes the price 140 and the profit 40, so the margin is \displaystyle \frac{40}{140} \approx 28.6\%, as before. In letters, take a cost of 1 and a markup rate r: the price is 1 + r and the profit is r. Or take a price of 1 and a margin m: the profit is m and the cost is the rest, 1 - m. Dividing the profit by the right whole in each case gives

\text{margin} = \frac{r}{1 + r} \qquad\qquad \text{markup} = \frac{m}{1 - m}

Setting a price for a chosen margin. If the margin is to be m, the profit is that share of the price and the cost is the rest, 1 - m of the price. The cost is known, so the price is the whole in a whole-from-a-part question:

\text{selling price} = \frac{\text{cost}}{1 - m}

To give the backpack a 30\% margin, the cost must be 70\% of the price, and the price is \displaystyle \frac{35}{0.70} = 50 dollars. Check: the profit is 15, and \displaystyle \frac{15}{50} = 30\%.

Worked examples

The plain case, a cost and a markup rate giving a price, is the backpack worked above. Examples 1 to 3 go with Part 1. Examples 4 to 7 go with Part 2.

Example 1

A markup above 100\%

A café works out that the milk, cocoa and paper cup for one hot chocolate cost 0.80 dollars. It sells the hot chocolate for 3.20 dollars. Find the markup rate.

Solution

The profit on one cup is 3.20 - 0.80 = 2.40 dollars. The markup measures it against the cost:

\frac{2.40}{0.80} = 3 = 300\%

The markup is 300\%. A markup can be far more than 100\%: here the profit is three times what the ingredients cost. The price itself is \displaystyle \frac{3.20}{0.80} = 4 times the cost, that is 400\% of it, which is 100\% for the cost plus 300\% for the markup. The markup is the extra above the cost, so it is the 300\%, not the 400\%.

Example 2

Buying by the pack, selling one at a time

Vikram runs a school fundraiser stall. He pays 9.60 dollars for a pack of 24 juice boxes and sells each box for 0.60 dollars. Find the markup rate.

Solution

The price is for one box and the cost is for a pack, so the two cannot be compared yet. A markup compares the cost and the price of the same thing. The cost of one box is

\frac{9.60}{24} = 0.40 \text{ dollars}

The profit on a box is 0.60 - 0.40 = 0.20 dollars, and the markup rate is

\frac{0.20}{0.40} = \frac{1}{2} = 50\%

The whole pack gives the same answer, as it must. Sold box by box it brings in 24 \times 0.60 = 14.40 dollars, a profit of 14.40 - 9.60 = 4.80 on a cost of 9.60, and \displaystyle \frac{4.80}{9.60} = 50\%. What matters is that the cost and the price describe the same amount of juice.

Example 3

Working back from the price to the cost

Ulrike’s shop marks every item up 60\%. A table lamp is priced at 72 dollars. What did the shop pay for it?

Solution

The 60\% is a percent of the cost, and the cost is what is unknown. So the price is 160\% of the cost:

1.60 \times \text{cost} = 72

Dividing both sides by 1.60,

\text{cost} = \frac{72}{1.60} = 45 \text{ dollars}

Check by going forwards: 60\% of 45 is 0.60 \times 45 = 27, and 45 + 27 = 72.

Taking 60\% of the 72 dollars would give 43.20, and 72 - 43.20 = 28.80 dollars. That would be the cost if the 60\% were a share of the price, and then 28.80 marked up 60\% would be only 1.60 \times 28.80 = 46.08 dollars, not 72. The percent belongs to its own whole, which here is the cost.

Example 4

One profit, two percents

Wilhelmina makes mugs and sells them at a weekend craft market. The clay, the glaze and her share of the kiln cost 10 dollars a mug, and she sells each mug for 16 dollars. Find the markup and the margin.

Solution

The profit on a mug is 16 - 10 = 6 dollars. Against the cost, the markup is

\frac{6}{10} = 60\%

Against the price, the margin is

\frac{6}{16} = \frac{3}{8} = 0.375 = 37.5\%

Figure 2The same profit of 6 dollars, three pieces, measured against two different wholes. Against the cost it is 3 pieces out of 5; against the price it is 3 pieces out of 8.

The figure cuts everything into 2-dollar pieces. The cost is 5 pieces, the profit 3 and the price 8. The markup asks what fraction 3 pieces are of 5, and the margin what fraction they are of 8. Same part, bigger whole, smaller percent. The margin also reads off the takings: of each 16 dollars a customer hands over, 6 dollars, 37.5\%, is profit.

Example 5

Turning a markup into a margin, and back

(a) An item is marked up 25\%. What is its margin?

(b) An item is sold for twice what it cost. What are its markup and its margin?

Solution

The cost does not matter, since scaling the cost and the price together leaves both fractions alone. So pick a convenient one.

(a) Imagine a cost of 100 dollars. A 25\% markup makes the price 125 and the profit 25. The margin is

\frac{25}{125} = \frac{1}{5} = 20\%

The formula agrees: \displaystyle \frac{r}{1 + r} = \frac{0.25}{1.25} = 0.20.

(b) A cost of 100 and a price of 200 give a profit of 100. The markup is \displaystyle \frac{100}{100} = 100\% and the margin is \displaystyle \frac{100}{200} = 50\%. Doubling the cost is a 100\% markup but only a 50\% margin.

(c) Now the margin is given, so imagine a price of 100 dollars. The profit is 75 and the cost is the rest, 25. The markup is

\frac{75}{25} = 3 = 300\%

This is the café’s hot chocolate from Example 1: its 300\% markup is a margin of \displaystyle \frac{2.40}{3.20} = 75\%. The formula agrees again: \displaystyle \frac{m}{1 - m} = \frac{0.75}{0.25} = 3.

Example 6

Pricing for a margin

Xochitl runs a café. A mango lassi costs her 1.20 dollars in yoghurt, mango and ice, and she wants a margin of 70\% on it. What should she charge?

Solution

A 70\% margin means 70\% of the price is profit, so the other 30\% of the price is the cost:

0.30 \times \text{price} = 1.20

Dividing both sides by 0.30,

\text{price} = \frac{1.20}{0.30} = 4 \text{ dollars}

Check: the profit is 4 - 1.20 = 2.80, and \displaystyle \frac{2.80}{4} = 0.70 = 70\%. As a markup that same profit is \displaystyle \frac{2.80}{1.20} = \frac{7}{3} = 233\tfrac{1}{3}\%.

Adding 70\% of the cost would not do it. That gives 1.20 + 0.84 = 2.04 dollars, a 70\% markup, whose margin is only \displaystyle \frac{0.84}{2.04} \approx 41.2\%. A margin is set by dividing the cost by the share of the price it is to fill.

Example 7

How much to sell to raise a target

Vikram’s fundraiser stall buys snack bars for 0.75 dollars each and sells them for 1.25 dollars.

(a) Find the markup and the margin.

(b) The stall wants to make 150 dollars of profit. How much must it take in sales, how many bars is that, and how much will it spend buying them?

Solution

(a) The profit on a bar is 1.25 - 0.75 = 0.50 dollars. The markup is \displaystyle \frac{0.50}{0.75} = \frac{2}{3} = 66\tfrac{2}{3}\%, and the margin is \displaystyle \frac{0.50}{1.25} = \frac{2}{5} = 40\%.

(b) The margin is the useful number for takings: 40\% of every dollar taken is profit. So 150 dollars of profit is 40\% of the takings, and

\text{takings} = \frac{150}{0.40} = 375 \text{ dollars}

At 1.25 dollars a bar that is \displaystyle \frac{375}{1.25} = 300 bars, and buying them costs 300 \times 0.75 = 225 dollars.

The markup is the useful number for what to spend: the profit is \displaystyle \frac{2}{3} of the spending, so the spending is \displaystyle 150 \div \frac{2}{3} = 225 dollars, the same answer. Check: 375 - 225 = 150, and 300 bars at 0.50 dollars profit each is also 150 dollars.

Boundaries

The cost and the price must be for the same thing. One item against one item, or a whole batch against the same batch, as in Example 2. A pack’s cost against one box’s price compares nothing.

Each percent belongs to its own whole. A markup is a percent of the cost and a margin a percent of the price. A markup rate cannot be used as a share of the price, or a margin as a share of the cost, without changing it first by \displaystyle \frac{r}{1 + r} or \displaystyle \frac{m}{1 - m}. Examples 3 and 6 show what goes wrong otherwise.

A markup rate needs a cost above zero. The markup rate divides by the cost. Snacks donated to a fundraiser stall cost it nothing, so every cent taken is profit: the margin is 100\%, and the markup rate does not exist, because division by zero is undefined.

A margin stays below 100\%; a markup has no ceiling. While the cost is above zero, the profit is less than the whole price, so the margin is under 100\%. The markup can be any size, as Example 1’s 300\% shows. The formula \displaystyle \frac{m}{1 - m} breaks down at m = 1 for the same reason.

Selling below cost makes both negative. Then the profit is a loss, a negative number, and both percents are negative. Since the price is then the smaller amount, the margin is the larger of the two in size.

Profit here means price minus the item’s cost, and nothing else. As set out at the start, a stall’s fee, a shop’s rent and a café’s wages are not in it. A seller can have a healthy margin on every item and still lose money overall if those other costs are larger than the profit on everything sold.

Prices round to the cent. When a markup gives a fraction of a cent, the price is rounded, and the markup actually charged is then very slightly different from the rate.

Where this goes next

The one-step price, (1 + r) \times \text{cost}, is the multiplier for a percent change at work. That lesson turns every percent increase or decrease into a single multiplication and runs it backwards by division.

The same shops that mark goods up also mark them down. A discount is a percent taken off the price, not added to the cost. A markup followed by a discount is two changes made one after the other, each against its own whole, which is the subject of successive percent changes.

Margin and markup are also a first case of a question that runs all through statistics and finance: two honest numbers describing the same facts can differ because they were measured against different wholes. Whenever a percent is quoted, the first question to ask is “of what?”

Exercises

Part 1: finding the selling price

A shop buys phone cases wholesale for 8 dollars each and marks them up 75\%. Find the markup in dollars and the selling price.
Saskia makes beaded bracelets for a craft market. The beads and wire for one bracelet cost 4.50 dollars, and she marks them up 120\%. Find the selling price.
A café’s pot of tea uses 0.43 dollars of tea and milk, and the café marks it up 375\%. Find the selling price to the nearest cent.

Part 1: finding the markup rate

A school fundraiser stall buys bags of pretzels for 0.50 dollars each and sells them for 0.80 dollars. Find the markup rate.
A shop buys scarves for 18 dollars each and sells them for 27 dollars. Find the markup rate.
Saskia also knits hats. The yarn for one hat costs 6.40 dollars and she sells the hat for 20 dollars. Find the markup rate.
A fundraiser stall pays 12 dollars for a box of 30 fruit bars and sells them for 1 dollar each. Find the markup rate.
A café’s cup of filter coffee costs 0.30 dollars in coffee and cup, and sells for 2.70 dollars. Find the markup rate.

Part 1: working back to the cost

A shop marks everything up 25\%. A pair of headphones is priced at 60 dollars. What did the shop pay for it?
A café marks bottled drinks up 200\%. A bottle of sparkling water sells for 2.40 dollars. What did it cost the café?
After an 80\% markup a jacket is priced at 99 dollars. Find the cost and the markup in dollars.

Part 2: finding the margin

A shop buys notebooks for 3 dollars and sells them for 5 dollars. Find the markup and the margin.
A café’s sandwich uses 2.10 dollars of ingredients and sells for 6 dollars. Find the margin, and the markup to the nearest tenth of a percent.
Teodoro makes wooden toy trains. The wood and paint for one cost 9 dollars, and he sells it for 24 dollars. Find the margin.
A fundraiser stall buys packets of crisps for 0.35 dollars and sells them for 0.50 dollars. Find the margin, and the markup to the nearest tenth of a percent.

Part 2: changing between markup and margin

Find the margin that goes with each markup. (a) 50\% (b) 400\% (c) 150\%
Find the markup that goes with each margin. (a) 20\% (b) 60\% (c) 90\%

Part 2: pricing for a margin

Ulrike sells silver earrings at a craft market. The materials for a pair cost 5.40 dollars. What price gives a margin of 55\%?
A shop buys kettles for 26 dollars each. What price gives a margin of 35\%?
Xochitl wants every drink in her café to have a margin of at least 75\%. An iced tea costs her 0.90 dollars to make. What is the lowest price she can charge for it?
A fundraiser stall sells drinks at a margin of 45\% and wants to make 90 dollars of profit. How much must it take in sales, and how much will it have spent on the drinks?

Reasoning

Saskia and Teodoro each buy the same lamp for 30 dollars. Saskia marks it up 50\%; Teodoro prices it for a margin of 40\%. Find both prices. Explain why Teodoro’s price is higher even though 40 is less than 50.
Markups of 300\% or 800\% are possible, as Example 1 and Exercise 8 show. Explain why a margin can never reach 100\% as long as the item cost something.
Explain why, whenever an item sells for more than it cost, its markup is larger than its margin.
A fundraiser stall pays 0.50 dollars each for ice lollies. On a hot afternoon, with the lollies starting to melt, it sells the last of them for 0.40 dollars. Find the markup and the margin. Why are both negative, and which is larger in size?

Answers

Markup 6 dollars, price 14 dollars. 0.75 \times 8 = 6 and 8 + 6 = 14; in one step, 1.75 \times 8 = 14.
9.90 dollars. The markup is 1.20 \times 4.50 = 5.40, and 4.50 + 5.40 = 9.90; in one step, 2.20 \times 4.50 = 9.90.
2.04 dollars. The price is 475\% of the cost: 4.75 \times 0.43 = 2.0425, which rounds to 2.04.
60\%. The profit is 0.80 - 0.50 = 0.30, and \displaystyle \frac{0.30}{0.50} = 0.6.
50\%. The profit is 27 - 18 = 9, and \displaystyle \frac{9}{18} = \frac{1}{2}.
212.5\%. The profit is 20 - 6.40 = 13.60, and \displaystyle \frac{13.60}{6.40} = 2.125.
150\%. One bar costs \displaystyle \frac{12}{30} = 0.40 dollars, so the profit is 0.60, and \displaystyle \frac{0.60}{0.40} = 1.5.
800\%. The profit is 2.70 - 0.30 = 2.40, and \displaystyle \frac{2.40}{0.30} = 8.
48 dollars. The price is 125\% of the cost, so the cost is \displaystyle \frac{60}{1.25} = 48. Check: 0.25 \times 48 = 12 and 48 + 12 = 60.
0.80 dollars. A 200\% markup makes the price 300\% of the cost, so the cost is \displaystyle \frac{2.40}{3} = 0.80.
Cost 55 dollars, markup 44 dollars. The cost is \displaystyle \frac{99}{1.80} = 55, and 99 - 55 = 44. Check: 0.80 \times 55 = 44.
Markup 66\tfrac{2}{3}\%, margin 40\%. The profit is 2 dollars; \displaystyle \frac{2}{3} of the cost and \displaystyle \frac{2}{5} of the price.
Margin 65\%, markup about 185.7\%. The profit is 6 - 2.10 = 3.90; \displaystyle \frac{3.90}{6} = 0.65 and \displaystyle \frac{3.90}{2.10} = 1.857\ldots
62.5\%. The profit is 24 - 9 = 15, and \displaystyle \frac{15}{24} = \frac{5}{8} = 0.625.
Margin 30\%, markup about 42.9\%. The profit is 0.15; \displaystyle \frac{0.15}{0.50} = 0.3 and \displaystyle \frac{0.15}{0.35} = \frac{3}{7} \approx 0.4286.
(a) 33\tfrac{1}{3}\%: a cost of 100 gives a price of 150, and \displaystyle \frac{50}{150} = \frac{1}{3}. (b) 80\%: price 500, and \displaystyle \frac{400}{500} = \frac{4}{5}. (c) 60\%: price 250, and \displaystyle \frac{150}{250} = \frac{3}{5}.
(a) 25\%: a price of 100 leaves a cost of 80, and \displaystyle \frac{20}{80} = \frac{1}{4}. (b) 150\%: cost 40, and \displaystyle \frac{60}{40} = 1.5. (c) 900\%: cost 10, and \displaystyle \frac{90}{10} = 9.
12 dollars. The cost must be 45\% of the price, so the price is \displaystyle \frac{5.40}{0.45} = 12. Check: \displaystyle \frac{6.60}{12} = 0.55.
40 dollars. The cost must be 65\% of the price: \displaystyle \frac{26}{0.65} = 40. Check: \displaystyle \frac{14}{40} = 0.35.
3.60 dollars. At a 75\% margin the cost is 25\% of the price: \displaystyle \frac{0.90}{0.25} = 3.60. Any higher price leaves the cost a smaller share of it and so gives a larger margin.
Sales of 200 dollars, and 110 dollars spent. The profit is 45\% of the takings, so the takings are \displaystyle \frac{90}{0.45} = 200; the rest, 200 - 90 = 110, paid for the drinks.
Saskia 45 dollars, Teodoro 50 dollars. 1.50 \times 30 = 45 and \displaystyle \frac{30}{0.60} = 50. The two percents have different wholes. Teodoro’s 40\% margin is a markup of \displaystyle \frac{20}{30} = 66\tfrac{2}{3}\%, more than 50\%; put the other way, Saskia’s 50\% markup is a margin of only \displaystyle \frac{15}{45} = 33\tfrac{1}{3}\%, less than 40\%.
The margin is profit divided by price, and the price is the cost plus the profit. If the cost is above zero, the profit is less than the price, so the fraction is less than 1, that is under 100\%. A margin of exactly 100\% would need the whole price to be profit, which happens only when the cost is zero. A markup divides by the cost instead, and the price can be any number of times the cost, so the markup has no upper limit.
Both percents have the same part, the profit, which is above zero. The markup divides it by the cost and the margin by the price, and the price is the cost plus the profit, so it is the larger whole. The same positive amount divided by a larger number gives a smaller result, so the margin is smaller and the markup larger.
Markup -20\%, margin -25\%. The profit is 0.40 - 0.50 = -0.10 dollars, a loss, so both fractions have a negative top: \displaystyle \frac{-0.10}{0.50} = -0.2 and \displaystyle \frac{-0.10}{0.40} = -0.25. The margin is larger in size, because the same 10-cent loss is measured against the smaller amount, the price.