Compared to What?

Compared to What? — The Most Misused Number in Modern Life

Raise a price by 20 percent, then lower it by 20 percent, and you do not end up where you started. Mark a coat down 25 percent, then take another 25 percent off, and the total discount is not 50 percent.

Why not?

Because almost every percentage quietly depends on something it may never tell you: the number it is being compared with.

In 1909, Boston retailer Edward Filene created a pricing system that made that comparison unusually clear. In the basement of his store at Washington and Summer Streets, every markdown was measured against the item’s original price. The tag showed both that price and the date the item first went on sale.

After 12 days, the price dropped by 25 percent. After 18 days, it was 50 percent off. After 24 days, it was 75 percent off. After 30 days, the item was donated to charity.

There was no haggling and no manager deciding the discount. Just a paper tag, a calendar, and one fixed starting point.

It became one of the clearest percentage systems in retail history—and the perfect way to learn the question that makes every percentage meaningful:

Compared to what?

IN THIS EPISODE

– Why “25 percent off, then another 25 percent off” is not the same as 50 percent off
– The $17 difference on a $100 coat that makes the math easy to see
– The difference between relative change and absolute change
– How a headline can be completely true and still tell you very little
– Why doubling a tiny risk can still leave the risk tiny
– Why a 40 percent loss requires about a 67 percent gain to recover
– The single mental habit that makes percent change easier to understand

THE HABIT LEARNED: NAME THE BASELINE

Whenever you hear that something went up, went down, became cheaper, grew, shrank, doubled, or was cut in half, do not begin with a formula.

Begin with a question:

Percent of what?

Find the number the percentage is being measured against. Then check whether that starting number stayed the same—or quietly changed along the way.

That is the skill.

Chapters: 

00:18 Introduction

00:35 What Percent Change Really Means 

1:08 Inside Filene’s Basement 

03:48 The Markdown System That Changed Retail 

05:20 How Stores Use Percentages Today 

06:26 Every Percentage Needs a Baseline 

06:53 The Habit That Makes Percentages Easier 

08:16 Why Going Up and Coming Down Are Different 

09:31 Percentages in the Headlines 

12:03 The End of Filene’s Basement 

12:49 The Big Idea: Compared to What? 

13:44 Closing

PRACTICE

Several short practice problems are available on the website. They take only a few minutes and will help make the idea stick:

storyproblem.org

THE SERIES

Story Problem builds understanding one question at a time.

Episode 1 asked, “Part of what?” through fractions and the human heart.

Episode 2 asked, “In step with what?” through ratios, dosage, and the history of medicine.

Episode 3 continues the idea with the question behind every percentage change:

“Compared to what?”

Each episode stands on its own, but together they build a way of thinking about numbers.

LISTEN AND FOLLOW

Website and practice: storyproblem.org
Apple Podcasts: podcasts.apple.com/us/podcast/story-problem/id6792299369
Spotify: open.spotify.com/show/033RDVfodnsN8CAUQOwKfa
YouTube

ABOUT THE SHOW

Story Problem teaches real math and science through true stories—for parents helping with homework, adult learners returning to subjects they once found difficult, and anyone who was ever told they were not a “math person.”

You were.

Hosted by Behi Rabbani.

Topics: percent change, percentage increase and decrease, relative versus absolute change, relative risk, mental math, math for parents, Filene’s Basement history, calculating discounts, and SAT math skills

Transcript

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SPEAKER_00 0:00

Hello, thank you for joining us again. I'm Behe Rabani, and this is Story Problem, the show where real math and real science hide inside true stories. Today we're going to talk about one of the more misedused numbers in modern life, the percent change. My hope is by the end you won't hear something like up 40% or cuts your risk in half without asking the one question that makes it mean something. There's a thread running through this show, and today it finishes a thought. In episode one, it was part of what? A fraction only makes sense when you know the whole. In episode two, it was compared with what? The ratio only makes sense when you know two quantities being compared. And today we take the question one step further, compared with what starting point, because that's where percentages, even very simple ones, trip people up every day. And the lesson's going to come from us from a rather strange place, a basement in a department store in Boston. Picture a staircase in downtown Boston, at the corner of Washington and Summer Streets. It's a sweltering day in 1909. You walk down, not up, down below the street into a low-ceiling room packed shoulder to shoulder with people. This is Filene's basement, and a man named Edward Filene has just done something a little strange with it. Upstairs, Filene's is a proper department store, polished counters, careful salespeople. Down here, it is chaos. Overstock, odd sizes, last season's coats, whole lots bought cheap from stores that went under. And every single item has a tag. But the tag doesn't just show a price, it shows a price and a date. The day that that item first landed on the floor. People learn to read those dates the way a hunter reads tracks. There are stories, versions of the same story, repeated again and again, of a woman finding a coat she loves, checking the date on the tag, and deciding the price has not dropped far enough. So she hides it behind a radiator, uh, beneath a pile of clothes, somewhere on another shopper might miss it. Then she comes back a week later, hoping it's still there because she knows the price is about to fall again. This is not being foolish. She is calculating, and the store created the system that taught her how. Here's how the store worked, and it ran on a calendar with almost no human decision making. When an item hit the floor, the clock started. After 12 selling days, if it hadn't sold, the price automatically dropped by 25%. After 18 days, 50% off. After 24 days, 75% off. And if after 30 days it still hadn't sold, it was given to charity. No haggling, no manager, no exceptions. The tag and the calendar did the work. And it was ruthlessly effective. Something like 90% of everything in that basement sold in the first 12 days before the first markdown ever hit. The bargain hunters and the impatient shoppers pushed against each other perfectly. Wait too long and it's gone. Buy too early and you overpaid. Phylene had turned pricing into a game with a ticking clock, and the earnings from that basement were strong enough that they helped carry the entire Phylene's company through the Great Depression. But the history is not really why we're here. We're here for one small detail in how those markdowns worked, a detail that made the percentage clear, useful, and honest. And it is the same detail that's missing from many of the percentages we hear today. Every one of those markdowns, 25%, 50%, 75% off, was taken from the original price, the first price, the one printed on the tag the day the coat arrived, not off last week's reduced price, off the original every time. That may sound like a minor detail, but it changes everything. Let's use a $100 coat to see why, because with round numbers, the math is easy to follow. With filene system, the original price never changed as the starting point. So if the coat began at $100, $5% off meant $25 came off, leaving a price of $75. 50% came off, leaving $50. And $75% off meant $75 came off, leaving $25. Every markdown was measured against the same original $100 price. The percentage on the tag told you exactly how much had been taken off the price you first saw. Now watch what happens if you do it the way almost every modern store does it, where each discount comes off the new, already lowered price. Same coat, same three cuts of 25%, but stacked on top of each other. $100, take 25% off $75. Now take 25% off of that, not off 100, off 75, and you get $56.25. Take another 25% off of that, and you land at about $42.19. Three cuts of 25% each. Your gut says that that's 75% off. It is not. Never trust a numerator until you know its denominator. The same rule applies to percentages. Never trust a percent until you know what it's being measured against. 25% off, but 25% off of which price? The original price, the most recent price. This is not a technicality. It can be the difference between paying $75 and paying $58. So here's the one habit I want you to take from this episode. Whenever you hear that something rose or fell by a certain percent, don't start with the formula. Start by finding the baseline, the number the percentage is being measured against. Ask percent of what, compared with which number. And if there's more than one change, check whether that starting point stayed the same or changed along the way. It's the same habit we practiced in episode two. There, the clue was the word per, miles per hour, milligrams per kilogram. The word tells you what one quantity is being measured against. This time, listen for the word that signal a change. Up, down, off, more, less, doubled, or halved. Each one tells you that something changed, but the change means nothing until you know where it started. More than what? Less than what? Down from what? Before you think about the percentage, find the number that came before it. If you want to practice this the way it shows up in real life, I've put a few problems on the website, storyproblem.org. You can visit the website to also look at prior episodes, listen to prior episodes, or try other practice problems in the study hall section. But stay with me because the baseline has one more trick to show you, and it's the one that costs people real money. Now let's look at the same problem from the other direction. If something rises by a certain percent and then falls by that same percent, it does not return to where it started. It feels as though the two changes should cancel each other out, but they do not, because the second percentage is being taken away from a different number. Say a $100 item gets marked up 20%. Now it's 120. Then next week it gets marked down 20%. Back to 100? No, because the second 20% comes off 120, not off 100. 20% of 120 is $24. 120 minus 24 is 96. You're $4 down than where you started. From two moves that sounded like they erased each other. The words are the same and the percentage is the same, but the starting point is different, so the amount of money is different. That's what Finlean system made clear and what modern sales science can make easy to miss. The next time you see an extra 30% off an already reduced price, you'll know that that 30% is being taken off the reduced price, not the original price. Now let's move beyond the basement and into the headlines, because the baseline does not just determine what you pay for a coat. It shapes how you understand the numbers you hear in the news, and in many cases, how you understand what is happening in the world. Imagine a headline, new report. This doubles your risk of a rare disease. Doubles. That sounds like a five-alarm fire. But double means nothing until you ask, doubled from what? If the risk was one in a million and it doubled, it's now two in a million. Twice as big and still almost nothing. Doubled is a relative change. It tells you the ratio. What it hides is the absolute number, the actual size, and you almost always need the absolute number to know whether to care. Here's a simple and slightly ridiculous example. Suppose one lottery ticket gives you one in a 300 million chance of winning. Buy a second ticket, and your chance doubles to two in 300 million. That really is a 100% increase, but your chance of winning is still almost zero. The word doubled tells you how much the number changed. It does not tell you whether the number was large or small to begin with. Both statements are true. Your odds doubled and your odds remain vanishingly small. But only the second tells you what those odds actually mean. And the baseline can work against you too. Suppose something loses 40% of its value. How much does it need to gain to get back where it started? Your first instinct may be 40%, but after the loss, you started from a smaller number. Start with $100. 40% loss leaves you with $60. To return from 60 back to 100, you need to gain $40, and 40 is about 67% of 60. So a 40% loss requires roughly a 67% gain to recover. In the same way, a 30% loss requires about a 43% gain to recover. A drop and an equal rise are not mirror images because they are measured from different starting points. Once again, the baseline changed. So what happened to the system? The automatic markdown continued for nearly a century at the original downtown crossing store, the basement between Washington and Summer Streets. In January 2004, the store slowed the schedule slightly, extending the time between markdowns to 14 days. Then in 2007, the flagship basement closed. The filing's basement name continued elsewhere for a time, but the original system, the dated tag, the fixed schedule, the calendar itself determining the price, ended in the very basement where it had begun. And here's the part I find genuinely lovely. In 1909, a retailer using nothing more than a paper tag and a calendar created one of the clearest percentages in retail history. He was not trying to teach anyone mathematics. He simply made one decision and stayed faithful to it. Every markdown would be measured against the original price, and it lasted for over a century. So whenever the percentage changed, the starting point did not. The tag told you exactly what the discount was being measured against. And that kind of clarity is rarer than it should be, not only in stores, but in nearly every part of life where percentages are used. You've been asking that question your whole life, compared with what? All I'm suggesting is that you keep asking it, even when the number appears in a headline dressed in authority and sounding completely certain of itself. So this week, name the baseline. Ask for the absolute number, not just the percent. And when anybody, a sign, a headline, a salesman, a study hands you a change without telling you where it changed from, ask them the oldest question in the basement compared to what.org. They take only about several minutes, and hopefully they'll make this stick. And if this made one old idea feel new, follow the show wherever you're listening, or suggest it and recommend it to a friend. See you then.