Money, Cost and Profit

Money questions are percent questions with a dollar sign, and the difficulty is never the arithmetic. It is knowing which number is the base.

Total cost with more than one rate

The most common money shape is a bill assembled from parts.

A contractor charges $65 an hour for the first 8 hours and $95 an hour after that. What does a 13-hour job cost?

Two rates, two blocks of time. Do not average them.

• First 8 hours: 8 times 65 is 520.
• Remaining 5 hours: 5 times 95 is 475.
• Total: $995.

The wrong answer here is applying the higher rate to all 13 hours, giving 1,235, or applying the lower rate to all 13, giving 845. Both will be choices. Read where the rate changes.

Watch for a flat fee sitting alongside a rate. "A $40 call-out charge plus $28 an hour" means the 40 is added once, not per hour, and the per-hour version is always a choice.

Discounts

A discount is a percent of the original price.

A jacket listed at $120 is 35 percent off. What is the sale price?

Two routes, and the second is faster and safer.

• Long: 35 percent of 120 is 42, and 120 minus 42 is 78.
• Short: 35 percent off means you pay 65 percent, and 65 percent of 120 is $78.

Multiplying by what you pay beats computing the discount and subtracting. It is one operation instead of two, and it makes it impossible to hand in the discount amount as your answer, which is the most common error on this shape.

Two discounts in a row

An item is marked 20 percent off, then a further 10 percent off at the register. What is the total discount?

Not 30 percent. The second discount applies to the already-reduced price.

You pay 0.80 of the original, then 0.90 of that: 0.80 times 0.90 is 0.72. You pay 72 percent, so the total discount is 28 percent.

Successive percent changes multiply. This is true for two discounts, for two increases, and for an increase followed by a decrease. Thirty percent will be a choice every time.

An increase followed by the same percent decrease does not return you to where you started: up 10 percent then down 10 percent is 1.10 times 0.90, which is 0.99, a net one percent loss. That surprises people and it is a question.

Markup, profit and margin

These three get confused because they describe the same sale from different angles.

A shop buys a radio for $80 and sells it for $100.

• Profit is 100 minus 80, which is $20. Always a dollar amount, always revenue minus cost.
• Markup is the profit as a percent of what the shop paid: 20 over 80, which is 25 percent.
• Margin is the profit as a percent of what the shop charged: 20 over 100, which is 20 percent.

Same sale, two different percentages, both correct. The question tells you which base it wants, and the other one is a choice.

Markup is on cost. Margin is on price. If you remember only one thing from this section, remember which denominator each uses.

Going backward:

An item is marked up 40 percent from cost and sells for $126. What did it cost?

Selling price is 140 percent of cost, so cost is 126 divided by 1.40, which is $90.

Taking 40 percent off the 126 instead gives 75.60, which is wrong and is a choice. When a price has already been marked up, divide to get back. Do not subtract the same percent.

Pay, deductions and commission

Gross pay is what is earned before anything comes out; take-home (net) pay is what is left after deductions.

Gross pay is $2,200 with 18 percent deductions. Take-home is 82 percent of $2,200, which is $1,804.

Working backward is the harder version, and the base is the gross:

Take-home is $1,890 after 10 percent deductions. That $1,890 is 90 percent of the gross, so the gross is 1,890 / 0.90 = $2,100.

Commission is a percent of sales. 4 percent of $12,500 is $500. Base pay plus commission adds the two: $600 plus 8 percent of $9,500 ($760) is $1,360. A tiered commission applies each rate only to its own slice: 5 percent on the first $10,000 ($500) and 8 percent on the remaining $8,000 ($640) is $1,140.

Simple interest

Interest = principal x rate x time, with the rate as a decimal and the time in years.

$4,500 is borrowed at 6 percent simple interest for 3 years. What is the interest, and what is owed in total?

Interest: 4,500 times 0.06 times 3. Take 4,500 times 0.06 first, which is 270 a year, times 3 is $810.

Total owed: 4,500 plus 810 is $5,310.

Both numbers are choices, so read whether the question asks for the interest or the total.

Time must be in years. Eighteen months is 1.5 years; 9 months is 0.75 years. A question that gives months and is answered as though they were years is off by a factor of twelve, which is too big to miss if you check the size of your answer.

Compound interest, briefly

Compound interest earns interest on the interest. Each year, the rate applies to the new balance. $2,400 at 6 percent for 2 years grows to 2,400 x 1.06 x 1.06 = $2,696.64, which is $8.64 more than simple interest's $288. Only four questions ask it, all at the hard end.

Sales tax and tipping

Both are increases on a base, and both are one multiplication.

A 7 percent sales tax means you pay 107 percent: multiply by 1.07.

A purchase of $46.50 with 8 percent tax.

46.50 times 1.08. Take 46.50 plus 8 percent of 46.50, which is 3.72, giving $50.22.

The tax alone, 3.72, is a choice. Read whether the question wants the tax or the total.

What you can skip

Across the 242 questions on this topic:

• Investments. Stocks and depreciation come up once each, and dividends and exchange rates never.
• Compound interest beyond two years. The four compound questions are one or two periods, worked by multiplying by the growth factor each year.

Where people lose points

Answering the discount instead of the sale price, or the tax instead of the total. Both are one subtraction or addition from correct, and both are sitting in the choices.

Adding successive percent changes. They multiply.

Confusing markup with margin. Different denominators, different answers.

Subtracting a percent to undo it. To reverse a 40 percent markup, divide by 1.40. To reverse a 25 percent discount, divide by 0.75.

Applying one rate to all the hours when the problem states two.

Leaving time in months in a simple interest calculation.

Work one in under a minute

A shop buys a tool for $48 and marks it up 50 percent. It later sells at 20 percent off the marked price. What is the profit?

Marked price: 48 times 1.5 is 72.

Sale price: 72 times 0.80 is 57.60.

Profit: 57.60 minus 48 is $9.60.

Note the two different bases: the markup was on 48 and the discount was on 72. Running both off 48 gives 14.40, which is a choice and is wrong.

Where this leads

Every technique here is percent work with dollars attached, and the successive-change rule carries into any question where two changes stack.