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Commission, tips and markups

Ratio, rates and money · level 5, lesson 4 of 7

A question to start

A clothing shop buys a sweater from the factory for 30 dollars and sells it for 50 dollars. The shop keeps 20 dollars on each sweater.

The owner says: "We add 67% to our cost." The shop assistant says: "No, 40% of every sale is ours." Who is right?

They both are. The 20 dollars is about 67% of the cost (30 dollars) and exactly 40% of the selling price (50 dollars). Shops, waiters and salespeople all use percents every day, and the most important question is always the same one you met in the last lesson: percent of what?

The idea

This lesson is about three everyday uses of "percent of an amount". Each one has its own words, but the math is the same: find a percent of some base amount, then add it or keep it.

Tips

A tip (or gratuity) is extra money you give for a service, such as a meal in a restaurant, a haircut or a taxi ride. In Canada a tip is usually worked out as a percent of the bill, and at a sit-down restaurant 15% to 20% is common. (Customs change over time and between places and countries, so check what is usual where you are. In some countries tipping is not expected at all.)

Tips are usually worked out on the bill before sales tax, though many people simply tip on the total.

Mental tipping. You rarely need a calculator.

  • 10%: move the decimal point one place left. 10% of 64 dollars is 6.40 dollars.
  • 5%: half of 10%. 5% of 64 dollars is 3.20 dollars.
  • 15% \(= 10\% + 5\%\): \(6.40 + 3.20 = 9.60\) dollars.
  • 20%: double the 10%: \(2 \times 6.40 = 12.80\) dollars.
  • 18%: 20% minus 2% (and 2% is a fifth of 10%): \(12.80 - 1.28 = 11.52\) dollars.

Why this works. Percent of an amount is directly proportional to the percent: 15% is 1.5 times 10%, so 15% of the bill is 1.5 times 10% of the bill. That is the direct proportion from lesson 2 in disguise.

The total. Bill plus a 15% tip is 115% of the bill: \(1.15 \times 64 = 73.60\) dollars. Again the one-step way, with a multiplier.

Markups

A shop buys things at a cost price and sells them at a higher selling price. The difference is the markup (in dollars). Shops usually set the selling price by choosing a markup percent of the cost:

\[\text{selling price} = \text{cost} + \text{markup} = \text{cost} \times \left(1 + \frac{\text{markup percent}}{100}\right)\]

A 40% markup on a 25 dollar item: markup \(= 0.40 \times 25 = 10\) dollars, so the selling price is 35 dollars. In one step, \(1.40 \times 25 = 35\).

The markup is not all profit. Out of it the shop pays rent, wages, electricity and everything else. What is left after all that is the profit.

Markup and margin

Shops also talk about margin: the same dollar amount, but as a percent of the selling price instead of the cost.

Formula Sweater (cost 30, price 50)
Markup percent \(\dfrac{\text{price} - \text{cost}}{\text{cost}}\) \(\dfrac{20}{30} \approx 66.7\%\)
Margin percent \(\dfrac{\text{price} - \text{cost}}{\text{price}}\) \(\dfrac{20}{50} = 40\%\)

The top (20 dollars) is the same. Only the bottom changes. Since the price is bigger than the cost, the margin percent is always smaller than the markup percent. This is exactly the "40 to 50 is up 25%, but 50 to 40 is down 20%" effect from the last lesson.

A margin can never reach 100% (that would mean the item cost nothing), but a markup can be any size: a café might sell a coffee costing it 1 dollar for 4 dollars, a 300% markup and a 75% margin.

Going backwards: from price to cost

If you know the selling price and the markup percent, divide by the multiplier. A shirt sells for 84 dollars after a 40% markup:

\[\text{cost} \times 1.40 = 84 \quad\Longrightarrow\quad \text{cost} = 84 \div 1.40 = 60 \text{ dollars}.\]

Why not just take 40% off 84? Because the 40% was 40% of the cost, not of 84. Taking 40% off 84 gives 50.40 dollars, which is wrong: \(50.40 \times 1.40 = 70.56\), not 84. You will practise this kind of reverse percent in level 6.

Commission

A commission is pay worked out as a percent of what someone sells. Salespeople for cars, houses, furniture and insurance are often paid this way. The percent is called the commission rate.

\[\text{commission} = \text{commission rate} \times \text{sales}\]

Common set-ups:

  • Straight commission: all pay is commission. Sell nothing, earn nothing.
  • Salary plus commission: a fixed base salary plus a commission on top. Steadier pay.
  • Tiered (or graduated) commission: a low rate on the first part of the sales and a higher rate on sales above a target, to reward selling more.

Notice that salary plus commission is not proportional to sales: it is "a fixed part plus a rate", the "neither" case from lesson 2. Straight commission is proportional.

Worked examples

1. A tip. A family's restaurant bill is 64 dollars before tax. They tip 15%. How much is the tip?

  • 10% is 6.40 dollars, 5% is 3.20 dollars, so 15% is 9.60 dollars.

2. An 18% tip. The bill is 85 dollars. Find an 18% tip.

  • \(0.18 \times 85 = 15.30\) dollars.
  • Mental check: 10% is 8.50, 20% is 17.00, and 2% is 1.70, so \(17.00 - 1.70 = 15.30\). ✓

3. A markup. A shop buys a lamp for 25 dollars and marks it up 40%. Selling price?

  • \(1.40 \times 25 =\) 35 dollars.

4. Markup or margin? The sweater costs 30 dollars and sells for 50. Find the markup percent and the margin percent.

  • Markup: \(\dfrac{20}{30} \approx 0.667\), about 66.7%.
  • Margin: \(\dfrac{20}{50} = 0.40\), 40%.

5. Salary plus commission. Maya earns a base of 2000 dollars a month plus 5% of her sales. In March she sells 36 000 dollars of furniture.

  • Commission: \(0.05 \times 36\,000 = 1800\) dollars.
  • Total pay: \(2000 + 1800 =\) 3800 dollars.

6. Tiered commission. A salesperson earns 3% on the first 10 000 dollars of sales each month and 6% on sales above that. This month she sells 25 000 dollars.

  • First 10 000: \(0.03 \times 10\,000 = 300\) dollars.
  • The rest is \(25\,000 - 10\,000 = 15\,000\): \(0.06 \times 15\,000 = 900\) dollars.
  • Total commission: 1200 dollars. (Not 6% of everything, which would be 1500.)

7. Backwards from the price. A shirt sells for 84 dollars after a 40% markup. What did the shop pay?

  • \(84 \div 1.40 =\) 60 dollars. Check: \(1.40 \times 60 = 84\). ✓

Try it

  1. Find a 15% tip on a 48 dollar bill, without a calculator.
  2. Find a 20% tip on a 37.50 dollar bill.
  3. A shop buys mugs for 18 dollars each and marks them up 50%. What is the selling price?
  4. A book costs a shop 12 dollars and sells for 20 dollars. Find the markup percent and the margin percent.
  5. A real-estate agent's assistant earns 4% commission on sales of 12 500 dollars. How much commission?
  6. Leo earns a base of 1500 dollars a month plus 6% of his sales. One month his total pay is 2700 dollars. What were his sales?
  7. A jacket sells for 91 dollars after a 30% markup. What was the cost?
  8. Four friends share a restaurant bill of 96 dollars before tax. They add 13% HST (Ontario's rate; check the current rate) and a 15% tip worked out on the bill before tax. They split the total equally. How much does each pay?
Answers
  1. 10% is 4.80 and 5% is 2.40, so 15% is \(4.80 + 2.40 = 7.20\) dollars.
  2. 10% is 3.75, so 20% is \(2 \times 3.75 = 7.50\) dollars.
  3. \(1.50 \times 18 = 27\) dollars.
  4. Markup in dollars: \(20 - 12 = 8\). Markup percent: \(\dfrac{8}{12} \approx 66.7\%\). Margin percent: \(\dfrac{8}{20} = 40\%\).
  5. \(0.04 \times 12\,500 = 500\) dollars.
  6. Commission \(= 2700 - 1500 = 1200\) dollars. So \(0.06 \times \text{sales} = 1200\), and sales \(= 1200 \div 0.06 = 20\,000\) dollars.
  7. \(\text{cost} \times 1.30 = 91\), so cost \(= 91 \div 1.30 = 70\) dollars. Check: 30% of 70 is 21, and \(70 + 21 = 91\). ✓
  8. Tax: \(0.13 \times 96 = 12.48\) dollars. Tip: \(0.15 \times 96 = 14.40\) dollars. Total: \(96 + 12.48 + 14.40 = 122.88\) dollars. Each pays \(122.88 \div 4 = 30.72\) dollars. (One step: tax and tip are both percents of 96, so the total is \(1.28 \times 96 = 122.88\).)

Common mistake

Using the wrong base. Markup percent is of the cost; margin percent is of the selling price; a tip is usually of the bill before tax. Mixing them up changes the answer. Always ask "percent of what?"

Taking the percent off the new price to go backwards. If the price is 84 dollars after a 40% markup, the cost is \(84 \div 1.40 = 60\), not \(84 - 0.40 \times 84 = 50.40\). Divide by the multiplier.

Where it leads