An embarrassingly easy math question: How to calculate a 15% difference?

Factual Questions
1

You may be wondering, okay, why are you asking a question if you know it’s easy? Well, I’ll tell you. It’s because I’ve been proofreading documents produced by a private investment bank for the past 10 hours and my brain has been turned to mush by having to read and edit the nearly incomprehensible language known as Bankerese, not to mention having to double-check multiple pie charts including several that should add up to 100%, but don’t, because my clients screwed up. (And they’re the financial experts!)

Having lost what remains of my sentience, I turn to you with a nagging issue. I’m staring in a mute haze at a graph consisting of two numbers, one of which is very obviously missing a digit. Now as a proofreader, I can just add the note “Missing a digit” in the margin and I’m sure that would be fine, but I’d rather be able to give them the correct figure. But I cannot fathom what formula to use. And I know it shouldn’t be this hard!

Here are the knowns:

a) First number: $2,981,075

b) Second number: $2,413,39 (this is the one missing a digit, as you can see)

c) The graph claims that there’s a 15% differential between these two numbers.

My first instinct was to calculate 15% of $2,981,075, which is $447,161.25. Then I subtracted that from $2,981,075. But it got me $2,533,913.75. Clearly my instinct was way wrong.

So please, please tell me: what’s the right formula for working this out? And better yet, what’s that stupid mofo of a missing digit supposed to be? Pretty please? I am barely able to keep my eyes open to finish editing the regular text, much less add math into the, uh, equation.

Many thanks if anyone read this far.

2

Your math approach was right. Whoever claimed there was a 15% differential was wrong. It’s a difference of about 19%.

3

Anyway, to answer your question, 15% of 2,981,075 is 447,161.25. You had the answer correct the first time.

4

So far as I can tell, you’re doing the right thing - there’s just more than one error is the data. No matter what the last digit of $2,413,39_ should be, it’s about 19% less than $2,981,075, not 15%.

5

Your approach of calculating 15% of the original and subtracting was correct - to save a step, you could also have just calculated 85% of the original (2981075*.85), the result would be the same.

6

There are, of course, (at least) two ways to compute the difference between two numbers A and B. |A-B|/A and |A-B|/B. But ding it the other way makes it worse for if you put the smaller number in the denominator, you’ll get a percentage difference even bigger than the 19% you got.

7

Percentage differences between two numbers can always be taken two different ways, so the other way is: 115% * x = $2,981,075.00

That gives x = 2,592,239.13

So it definitely doesn’t work out any way I can see. Write ‘MATH FAIL’ in the margin. The bold red letters are essential. :wink:

8

While you’re at it cross out the entire graph as well and write GRAPH FAIL. Putting two bars next to each other gives zero additional information, especially if the difference is stated in the text anyway.

9

Oh my God. I can’t even… Thank you, thank you! At least there are some pistons firing in my mental engine after all.

In fairness to the capitalist overlords I’ve been freelancing for, the chart does have labels. The figures represent (try to contain your excitement over this…) the difference between taking a lump sum payout of an IRA that you’ve inherited from an older deceased loved one, versus a “Stretch IRA,” which is this deal where you (as the young heir) get the IRA dividends according to your own life expectancy (rather than the deceased’s). Or something to that effect, I barely understand it after reading this document, but definitely not able to explain it.

So in this hypothetical, the chart assumes that the IRA was worth $1 million when the original owner either died or first set it up… I’m not quite sure. All I can tell you is that the smaller figure is the lump sum; the larger figure represents how much more awesome this Stretch IRA deal is, and why it’s better to make sure you designate someone instead of just “your estate” as a beneficiary.

I can’t even believe I’m jabbering about this crud. I’m a creative, damn it!

Anyway, does that somehow make this 15% thing make sense? I don’t see how it does–the way it’s labeled, there’s an arrow between the two differently sized number columns and in the middle it says “15% differential.” Doesn’t that imply what I’m saying–that quite simply, the larger figure is 15% bigger than the smaller one?

Damn, who even works on these documents? And this is not a rinky-dink operation, either. If I told you the name of this institution y’all would be hysterical. I’ll just say I now understand why it’s one of the most scandal-prone banks out there. Maybe it’s because these are the people doing their reporting to their clients.

(Oh, my favorite part of the chart? There’s this disclaimer: “Numbers have been rounded for convenience, are only estimates for illustrative purposes and should not be relied upon.” I’ll say!)

If the IRA stuff makes sense and somehow explains the weird numbers, please clue me in and I’ll happily take back my snark. If you saw the gibberish I’ve been editing you’d understand why I’m a mental blob of goo right now. On the plus side: doing this pays waaaay better than my fiction editing. But that’s much more enjoyable, and it’s right in my wheelhouse. Such are the choices one makes as a freelancer!

10

What is the definition of “15% difference”? 15% of which number? 115 is 15% more than 100, and 85 is 15% lower lhan 100. Both are “a 15% difference”. But 100 is not 15% larger than 85, and 100 is not 15% less than 115. So those are not a 15% difference.

11

As noted, it could be looked at two ways - is the larger supposed to be 15% more than the smaller, or is the smaller supposed to be 15% less than the larger (which is what you tried to calculate)?

The numbers work out differently - but in any case, neither works here.

2,981,075*.85 = 2533913.75 (the latter is 15% smaller than the former)
2,981,075/1.15 = 2592239.13 (the former is 15% larger than the latter)

12

*On edit… Oh, I see your point. Yes, you’re right, neither way works. But I do see there’s a difference. Anyway, I wrote the below before reading the above post. Not sure it matters but what the heck. *

Well, as I said, the chart has two vertical bars, a large one and a slightly smaller one. The large one says $2.9whatever million, the smaller one is $2.3whatever million. (I’m too tired to copy/paste the exact figures again.) Between the two, there’s this helpful indicator:

<------ 15% differential ------>

This document is all about convincing wealthy people to avoid various post-death financial penalties put upon your heirs due to a recent Supreme Court decision regarding IRAs and bankruptcy and yadda yadda kill me now.

They suggest that this horror can be avoided by setting up one of these Stretch IRA thingies. So I assume, and I grant you that it’s a big assumption, that since they’re trying to tell the reader that the bigger number (the $2.9M Stretch IRA result) is 15% more moolah for your heirs than if your heirs end up taking a lump sum payment of your IRA (the paltry $2.3M figure) once you’ve shuffled off the mortal coil.

Or not and I’m wrong. This doc is sooooooo not aimed at people like me, it’s for folks with financial advisors and trusts and whatnot who would actually put their money in a Private Bank like this in the first place.

… Which is why they’re paying an idiot pauper like me a ridiculous figure to proofread it. And the crazy part is, they love me! Go figure. I guess I represent the dumbest client they would be likely to have, so I’m able to point out where stuff is too woolly. (In my defense, I am a solid editor of text. Numbers, though… I think I must’ve been frightened by a protractor when I was a kid or something.)

13

The assumptions so far seem to agree that the number with a missing digit is missing it from the END ($2,413_). However, if we make it $2,_41339, look what happens when we fill that space with a “5.”

2,541,339 / 2,981,075 = 0.852490796105432 (that’s how many places the calculator on my phone goes out).

If we’re not fixated on the 15% being precise, that does work out to approximately a 15% differential.

14

This reminds me of a problem I used to have over here: VAT (sales tax) was set at 17½%. Now working that out in your head (10%+5%+2½%) was fairly easy, even for me. But when I tried to convince someone that selling something at 15% off was the same as not adding VAT in the first place, I failed.

15

When discussing % differences, I can think of at least three ways of doing it. You can use the first number as a base and see how far off the second number is from there. You can use the second number as a base and see how far off the first number is from there. Or, you can take the average of the two numbers and see how far off both numbers are from the average. IIRC, the third option is what we were taught to do in Physics class.

But when you’re reading someone else’s persuasion piece, you can usually bank on the idea that they will have chosen whichever method supports their position the strongest.

Suppose two different people conduct a census of a small town on two different days and get two different answers. The first answer is 450. The second answer is 600. You could say that 450 is 25% less than 600 and you’d be right. You could also say 600 is 33.3% more than 450 and you’d also be right. But it’s reasonable to imagine that the correct answer is actually 525 (halfway between 450 and 600), which means that the first measurement was 14.3% too low and the second was 14.3% too high, hence the two measurements are 28.6% apart from each other.

If your goal was to emphasize how different the two answers are, you’d probably quote the 33.3% figure. This is what is known as “lying with statistics”.

16

There are different ways of defining percent difference, but if the difference is small, then all of the different definitions will give you close to the same value. As a general rule, if you need more precision than that amount of variation, then you shouldn’t just be saying “percent difference” at all, but instead explicitly defining which one you mean, or using a different notation.

17

From a math perpective, it doesn’t matter what the 7th digit is, it’s not going to change the percentage of difference unless you are computing out to about 6 decimal places. For this reason, most of the documents I have seen like that simply round to the nearest 100.

18

Whew, thanks, all. The possibilities that they might have either exaggerated, messed up the math or left off an entirely different digit in some random place are all both likely and unsurprising.

This is a place where documents include a table with three figures: 1.1, 1.0 and 1.0, and the total was given as 3.4. They’re likely either rounding things way up or down, which is fine, but you have to include that rounding in both the total and the figures being added together, duh. (And of course you should be transparent and include a note indicating that figures are being rounded somewhere near the table.) Otherwise you just look like you suck at math. For a bank that’s kind of a bad impression to give.

Anyway, I sent the proofed document off with a note that recommended to double-check both figures, recalculate the percentage, or word it a different way. I really appreciate the assistance, everyone. I worked on that doc until 7AM today and that part was truly nagging at me.

I so wish I could somehow redact the file to remove the endless mentions of the client’s name, just so you could see just how jacked up this document is. I had to make so many comments/notes that it looks like a 1996 webpage that’s been hijacked by spam popup windows.

19

Is it possible the 15% difference is a difference in tax rate or after-tax yields or something, and not just a percentage difference between numbers? In other words, let’s say we’re taxing $10 million at 10% (1 million tax, 9 million after-tax) and comparing that to taxing it at 25% (2.5 million tax, 7.5 million after-tax). On a graph like that, one might label it “15% difference” and expect people to pick up from context that they mean [difference in tax rate] and not [difference between these two numbers].

Anyway, if you’re not able to follow the content well enough, it seems like your role as an editor is to point out that this appears to be an error that a subject matter expert needs to review.

I’d also add that confusion about the language of percentages that cause so many financial experts to talk about things in terms of “points.” A 15-point difference would make it pretty clear that I mean a change in percentage rate of 10% to 25%, and not just a change in the value of a number. Basis points are also frequently used, meaning hundredths of a percent. (So the difference between a 5.5% and a 5.6% return on investment is 10 basis points.) Maybe the bankerese can be made easier to read by adopting a convention like that.

20

You didn’t get this off of the Internet, did you?