Percent traps

The coat that came off by 60, and the sign that said 70

A coat is marked half off. At the register the shop takes another 20 percent off the sale price. Add the two signs together and you get 70 percent. The receipt says otherwise.

Half the price survives the first cut. Four fifths of that half survives the second. You pay 0.5 times 0.8, which is 0.4 of the sticker, so 60 percent has come off and the last ten points never existed. That is Half off, then 20 percent more off, and it takes about fifteen seconds once you know where to look.

Nearly every percent trap runs on the same machinery. A percent is a share of a base. Move the base between two steps and one word ends up describing two different amounts of money.

Why does the same 20 percent give back less than it took?

Three mechanisms cover most of the damage, and all three are about the base.

The base moves between the steps. A jacket at 100 rises 20 percent to 120, then falls 20 percent and lands on 96. The rise was worth 20 and the fall was worth 24, because the fall was charged on the larger number. Two moves of equal size leave you 4 percent down.

Losses and recoveries are not symmetric. A fund that drops 50 percent has to rise 100 percent to get level. Halving takes 100 down to 50. Climbing from 50 back to 100 is a gain of 50 on a base of 50. The same money, twice the percent. Down 50 percent, back to even is the two line version of every bad quarter you have read about.

Points and percent are different units. A mayor polls 40 percent in June and 44 in July. The gap is four points. Four on a base of 40 is a tenth, so support rose 10 percent. Both figures are correct and they describe different things, which is why From 40 percent to 44 percent catches people who read polls for a living.

How do you take a percent without a calculator?

Five moves cover almost everything that lands on a receipt.

Find one percent first, then copy it. Move the decimal point two places left and you have one percent. Three percent of 250 dollars: one percent is 2.50, three copies come to 7.50. Every number stays small enough to hold in your head.

Take ten percent, then build from it. Move the point one place for ten percent, halve that for five percent, add the pieces you need. Fifteen percent of 2,400 is 240 plus 120, so 360. Twenty percent is ten percent doubled.

Chain what survives instead of adding the cuts. Two discounts of 50 and 20 leave 0.5 times 0.8. Two rises of 10 and 10 give 1.1 times 1.1. Multiply the survivors, read the answer off the end, and the order of the steps stops mattering.

Flip the percent to the easy side. Eighteen percent of 50 is the same product as 50 percent of 18, so take half of 18 and answer 9. A percent is a multiplication, and multiplication ignores the order of its parts.

To undo a discount, divide by what survived. Boots ring up at 96 after 20 percent off. Eighty percent of the tag survived, so the tag was 96 divided by 0.8, which is 120. Adding 20 percent back gives 115.20 and answers a different question. That is Ninety six dollars after 20 percent off.

Which mistakes show up most often

Adding two discounts. The second cut is taken from the reduced price, so it is always worth less than its headline. Fifty and twenty make sixty, never seventy.

Adding a percent back to reverse it. Reversing a percent is a division. Adding 20 percent to 96 uses 96 as the base, and the discount used 120.

Averaging two percentages of different sized groups. Ten students averaging 90 and thirty averaging 70 do not average 80 across the room. The forty students hold 3,000 points between them, so the average is 75. Weight each group by its size before you divide.

Assuming the order of two multipliers matters. A 45 dollar lamp gets 20 percent off and the state adds 8 percent sales tax. Taxing before the discount and taxing after it both multiply 45 by 0.8 and by 1.08, so both land on 38.88. Taxing the smaller number feels cheaper, and the saving is real on that one step, after which the discount applies to a larger figure and hands it straight back. Tax first, or discount first is worth answering before you argue about it at a till.

Treating percent and price as interchangeable. A shampoo bottle offering 20 percent more free divides the price per ounce by 1.2, a cut of about 16.7 percent. The bottle beside it at 20 percent off multiplies the price by 0.8, a full fifth. Two labels carrying the same number, two different answers, which is the whole of 20 percent more free, or 20 percent off.

What Berkeley found when it split the numbers

In 1973 the graduate school at Berkeley admitted 44 percent of 8,442 male applicants and 35 percent of 4,321 women. The gap looked plain enough to worry the university, so statisticians went through it department by department.

Department by department, the figures leaned slightly toward women. Women had applied in larger numbers to the departments that turned nearly everyone away, and the pooled rate carried that choice rather than the decisions of any admissions committee. The finding was published in Science in 1975 and is now the standard example of Simpson's paradox.

The mechanism is the same one that runs through this whole topic. A pooled percentage is a weighted average, and the weights are doing work you cannot see from the total. Every part of a group can lean one way while the group leans the other. Berkeley, 1973, and a bias that moved puts the question to you before it gives you the answer.

Where percent traps actually reach you

Sale tags carry them first. A second markdown, a coupon on top of a promotion, a price that says 96 with the original hidden. Then loans: a fee of 15 dollars on 100 borrowed for two weeks is 15 percent for two weeks, and two weeks repeats 26 times in a year.

Investment statements are next. Drawdowns and recoveries are quoted in the same unit and measured against different bases, so a fund can post a 50 percent loss and a 60 percent gain and still be under water. Polls, market share and conversion rates round it out, where the difference between points and percent decides whether a change reads as tiny or large.

None of this needs long arithmetic. It needs one habit: before you accept a percent, ask what it is a percent of. Almost every question in this topic is answered by that sentence.

Work through them one at a time

The feed above runs on the same questions described here, one to a card. You type an answer, the card tells you whether it holds, and a line underneath explains the move that gets there quickly. There is no timer and no score, so a question you miss costs you nothing except the moment where it clicks.

If you would rather read first, the blog covers the same ground at longer length, starting with the viral 8 divided by 2(2+2) argument. For the money side of the same habit, Everyday money takes the receipts, the tips and the loan offers.