3rd Grade math - Rounding?

[Russ Filtz]

Was checking my 3rd graders math homework and spotted an error (at least in every math course I remember from my engineering degree). They are teaching to round UP every time you have a "5"! So a 4.5 rounded would be 5, and 45 rounded would be 50. Absolutely wrong from an engineering standpoint. We are taught to round to the EVEN number. So 4.5 would be 4, not 5. That way "statistically" speaking, you aren't always rounding in one direction thereby creating even more errors. Of course, even us engineers take liberties sometimes with the concept and round to the odd number on a "5". IF it makes the calculation more conservative.

Not a huge deal I guess. Maybe it's easier to teach the concept of rounding first, and then teach it the right way later for 3rd grade brains? Any teachers out there that know the theory? Drives me insane thought, thinking it's easier to just teach the "right" way in the first place.

[David Epperson]

During my engineering studies I never heard of "rounding to the even" number. Always to the next whole significant digit. 0.45 would round to either 0 or 0.5 never to 0.4.

[Rob Russell]

What David said.

[Greg Peterson]

I'm no math whiz, but I did stay in a Holiday Express last night. I vote for round to the nearest significant digit.

[glenn bradley]

What David said again. Round to the nearest larger whole number.

[Greg Cuetara]

It all depends on how much precision and accuracy you want. If the directions state to round 4.5 to the nearest whole number then that would be 5. If you had 4.4 then the nearest whole number would be 4. In engineering if you round it should always be up...if you come out with needing 4.1 bolts then you have to round up to 5 because there is no way that you can get .1 bolts and 4 bolts will not satisfy your requirements.

[Russ Filtz]

Not talking sig figs, but after review, it seems like rounding up all the time IS the method being taught now. In my method 4.5 rounded to integers would be 4. 5.5 would be 6. 6.5 would still be 6, etc. 1.45 rounded to tenths would be 1.4, not 1.5 as being taught now.

Whoever taught me the round to even method stated it would tend to even out cumulative errors in a chain of calculations. Google searches show the round up with "5" approach, but no real reasoning behind it other than convention.

[Cliff Rohrabacher]

When doing quick calc's in my head I round up and down and it all sort of evens out. The answer I get isn't accurate but it's plenty close for horseshoes.

[Mike Henderson]

I was taught that a "5" in the rounding position caused the adjacent digit to go up by one. So 4.5 rounds to 5, 45 rounds to 50, etc.

Anything less than 5 in the rounding digit does not cause the adjacent digit to go up by one. So 4.4 is rounded to 4.0 and 44 is rounded to 40.

I guess you can think of it that there are five digits that cause rounding up and five that do not. The five that do not are 0, 1, 2, 3, and 4. The five that do cause rounding up are 5, 6, 7, 8, and 9.

So there are equal probabilities of going either way, which makes it fair (or makes it even out).

Mike

[Russ Filtz]

Although rounding up may be convention now. May not have always been the case! Found this on various websites.

"1. Some statisticians prefer to round 5 to the nearest even number. As a result, about half of the time 5 will be rounded up, and about half of the time it will be rounded down. In this way, 26.5 rounded to the nearest even number would be 26—it would be rounded down. And, 77.5 rounded to the nearest even number would be 78—it would be rounded up."

http://en.wikipedia.org/wiki/Rounding#Round-to-even_method
"The Round-to-even method has been the ASTM (E-29) standard since 1940. The origin of the terms unbiased rounding and statistician's rounding are fairly self-explanatory. In the 1906 4th edition of Probability and Theory of Errors [1] Robert Woodward called this "the computer's rule" indicating that it was then in common use by human computers who calculated mathematical tables. Churchill Eisenhart's 1947 paper "Effects of Rounding or Grouping Data" (in Selected Techniques of Statistical Analysis, McGrawHill, 1947, Eisenhart, Hastay, and Wallis, editors) indicated that the practice was already "well established" in data analysis."

[Chris Kennedy]

I remember learning the idea of rounding to the nearest even on a 5 when I was taking high school chemistry and physics, but in my math classes, we were always supposed to round up.

I can tell you mathematically speaking that the idea of rounding up _measurements_ to the nearest even is flawed. When you are rounding, you are losing a degree of accuracy/precision (I always forget which one is which), or more to the point, you are saying that you don't have that degree. So, when you round 45.4 to 45, you are really saying that the measured quantity is 45 within a margin of error -- you cannot say with certainty what the next digit is.

So, in particular, you round 45.0, 45.1, 45.2, 45.3, and 45.4 to 45, which is actually 45 plus some unknown error, so in particular, you cannot say it is 45.0. The point being, you are actually rounding 45.0 to 45 (with a margin of error). You round up 45.5, 45.6, 45.7, 45.8, 45.9 to 46 for the same reason. Hence, you round down on 5 digits and you round up on 5 digits and round the same number of times up as down.

The thing is that scientists and engineers may not view rounding 45.0 to 45 as actual rounding (it depends on whether you think it is 45 or 45 plus a margin of error), which is where the idea to round to the nearest even came from.

Cheers,

Chris

[Kevin L. Pauba]

I also have an engineering degree (mechanical) and we were taught like you were, round to the nearest even number. And I too get frustrated by some of the ways my kids have been taught but, so far, they're doing OK. I figure if one of the remaining not in college decide on the engineering path (unlikely, :( ), they'll relearn it.

[Russ Filtz]

IYou round up 45.5, 45.6, 45.7, 45.8, 45.9 to 46 for the same reason. Hence, you round down on 5 digits and you round up on 5 digits and round the same number of times up as down.

The thing is that scientists and engineers may not view rounding 45.0 to 45 as actual rounding (it depends on whether you think it is 45 or 45 plus a margin of error), which is where the idea to round to the nearest even came from.

Cheers,

Chris

I think that whole round down 5, round up 5 argument is flawed. why would you include 45.0 and NOT 46.0? Therefore you're now rounding down 5 numbers and up SIX numbers if you include 46.0! Sounds like a made up argument to legitimize the rounding method.

[Tom Veatch]

why would you include 45.0 and NOT 46.0?

Because you've already counted the "round on zero" sample with "45.0". Including "46.0" would effectively be counting the same sample twice and would skew the statistics.

As far as "round to even" goes, put another mark in the "That's the way I learned it." column.

[Ben Rafael]

For finances, Mike's way is the standard since things will average out over time. It is the standard in, AFAIK, the financial industry. I believe that method is the way computers round off also.
Mathematicians, generally, use the rounding method called "whatever makes my life easier", that is, when mathematicians actually compute figures. Most mathematics is theoretical and doesn't involve a lot of computations. The computations come in to play when engineers and programmers apply what the mathematicians have produced.

[David Freed]

So, in particular, you round 45.0, 45.1, 45.2, 45.3, and 45.4 to 45, which is actually 45 plus some unknown error, so in particular, you cannot say it is 45.0. The point being, you are actually rounding 45.0 to 45 (with a margin of error). You round up 45.5, 45.6, 45.7, 45.8, 45.9 to 46 for the same reason. Hence, you round down on 5 digits and you round up on 5 digits and round the same number of times up as down.

Cheers,

Chris

That is how I was taught to round numbers almost 40 years ago.

[John Dykes]

If only Richard Pryor were alive... he would know.

- jbd in Denver

[Gregory Lyons]

I've been working in engineering for 22 years (not a degreed engineer) and have never heard of this. A quick informal survey of some of the PEs in my office didn't turn up anyone who was taught the 'round-to-even' method.

While I agree that the round-to-even method is statistically more accurate when the end goal is a function of a set of rounded numbers, you're applying this accuracy to what is ultimately an estimation technique.

Also, the accuracy of a function applied to the set of rounded numbers is only acheivable if there is an equal distribution of odd and even integer components. If the set is biased towards even, the result will err on the low side, odd and the error occurs on the high side.

Again, rounding is all about estimation; speed, not accuracy. Like the old adage says: 'Measure with a micrometer, mark with chalk, cut with an axe.

FWIW, Excel uses the round up at 5 method. I'm curious what my credit union's software uses, maybe I should try to renegotiate my mortgage.

~g

[Ken Garlock]

Rounding to an even number just because it is an even number is just stupid.:mad:

Rounding to me and how I used it when programming in COBOL consisted of first specifying 'n' the number of digits you wanted, two or twenty, and where the decimal point was located. Then operations that produced n+1 digits were rounded based upon the "five rule". IF the n+1 digit was less than 5, it was discarded, if it was 5 or more the nth number was increased by one and n+1 was discarded. Any number that would fit in the specified n digits was left alone and right filled with zeros as required. Even or odd numbers had NO bearing on the calculation.

No wonder high school kids don't know Jack about arithmetic.:rolleyes:

[Jeff Bratt]

The "rounding to even" methodology is the most accurate way of rounding numbers - this has been well known for years (or centuries). Still, the methods that are taught and the algorithms implemented in computer programs vary as to what type of rounding is actually used. Plus the subject gets even more complicated when negative numbers are included.

It is easy to demonstrate that you get more accurate results when using the "rounding to even" method. For example (I used a spreadsheet) take a set of numbers 0.0, 0.1, 0.2, 0.3 ... 30,7,30.8, 30.9. There are 310 numbers here - their sum is 4789.5, the average value is 15.45. These results are exact - the "original" data.

Now compare the same set of 310 values after all the individual numbers have been rounded to integers. The results are:
After rounding up - sum 4805, avg 15.500
After "rounding to even" - sum 4789, avg 15.448...

The "rounding to even" results are much closer to the original values. All the difference comes from the way the values in the x.5 column are treated. In the original data these values sum to 480.5. Rounding up results - sum = 496, rounding to even results - sum = 480. Real world situations like this can result from digital readouts (like my calipers when set to mm) that only read whole numbers or increments of .5.

The tendency for sets of numbers to increase after rounding is why the "rounding to even" rule was formulated. It is unfortunate that there is still disagreement on this subject. Still, the intricacies of these differences are going to be lost on third graders. And most of the rest of us will end up with whatever rounding the particular computer program we're using actually implements.

Still, there are situations - like buying lumber - where always rounding up (or adding 20%) might be best. In situations where having a little extra is always a good idea, mathematical accuracy is not the goal - and the "rounding to even" method is for achieving more accuracy.

[Tom Veatch]

Rounding to an even number just because it is an even number is just stupid.:mad:
...

Well, I wouldn't go quite that far. Nor would I characterize it as being done "just because it is an even number".

The idea is to minimize the error in the sum of a series of rounded numbers, and would be just as effective if it was "round to odd" rather than "round to even". As mentioned in another post, minimum error would occur when numbers in the series were evenly distributed between even and odd "Nth" digits whose "N+1" digit was 5. Any other distribution would increase the rounding error - but not as much as if the convention were to always truncate (round down) or to always round up when the next digit is 5.

Nor do I see where selection of any one of the 4 possible rules for rounding on 5 ("round up", "round down", "round to even", "round to odd") would detract from the understanding of arithmetic any more than any of the other 3. It's just a convention/rule intended to remove the ambiguity from an indeterminate case.

Of course, the case is not indeterminate unless the digits following the 5 are repeating zeros. Otherwise, the case is one of "round up" since the value being rounded is >5. And that's the way I learned the rounding convention - greater than 5 round up, less than 5 round down, exactly 5 (5 followed by repeating zeros, a rare case) round to even.

All in all, IMHO, a tempest in a teapot.

[Travis Lavallee]

To look at this you have to think why you are doing the rounding in the first place:

Are you trying to analyze a lot of data and you want the statistics to remain accurate?

Are you trying to calculate some safety factor where and underestimation could be 'disastrous', as in too few bdft for a project?

Or are you trying to teach a student a mathematical concept and just need a specific rule to deal with the one case that can lead to ambiguity?

[David Freed]

There are 310 numbers here.

I come up with 400, not 310.


Now compare the same set of 310 values after all the individual numbers have been rounded to integers. The results are:
After rounding up - sum 4805, avg 15.500
After "rounding to even" - sum 4789, avg 15.448...

I didn't take the time to add the numbers, so I don't know how far off the answer is, but you can't get an odd numbered sum by adding even numbers.

[Eric Larsen]

I was taught to never round any numbers, 34.231543392906598382021 years ago.... ;)

Whoops, in the time I added the smiley, it went up to 34.23154339290658382194 years.

[Tom Veatch]

...you can't get an odd numbered sum by adding even numbers.

True, but that series contains all the odd integers from 1 to 29, so it's not all even numbers. In fact, assuming that even/odd is determined by the units digit of the number, there a 160 even numbers, 0.0, ...0.9, 2.0..., 30.0...30.9, and 150 odd numbers, 1.0..., 3.0...,29.0 ... 29.9, I'd expect the result to be odd. Using Gauss' Equation, the sum is as previously stated 4789.5.

For those who might not know, and the (probably) smaller number of those who might be interested, Gauss' Equation for the sum of a sequence of integers is simply the average value multiplied by the total number of integers; for a sequence beginning at 0 or 1, sum = (N)*(N+1)/2 where N is the largest integer in the sequence. More useless knowledge remembered from years ago. Now if I could just remember where I left that *(^)*# tape measure.

[Chris Kennedy]

Now compare the same set of 310 values after all the individual numbers have been rounded to integers. The results are:
After rounding up - sum 4805, avg 15.500
After "rounding to even" - sum 4789, avg 15.448...


I'm sorry to say this, but you have made a classic mistake. You never round during the computation -- only at the end, if (and this is a big IF) you are concerned with the accuracy/precision of the computation -- which is what you are doing if you proceed to take an average. If you want to get the average, you would average the original data and then round.

This actually goes back to several comments made so far, and probably the purpose of the original 3rd grade problem that spawned all of this. If you are estimating for the purpose of speeding up the computation and aiming to get a "ball park" figure, then seriously, it really doesn't matter. If you want to know, approximately, the sum/difference/product/quotient, whether you round 5 up or down has little bearing on the situation. After all, it's an estimation. Any person in their right mind would add a fudge factor to an estimation.

Also, as several posters have pointed out, there are many situations where you have to round up, just from a logical standpoint. The number of board feet and bolts/screws aside, think about the House of Representatives. They take the population (from the census), divide by 435, and figure how many people per representative. The fact that it doesn't divide evenly and the decision as to how to deal with that fact leads to paradoxes.

As I mentioned in an earlier post, the round to the nearest even is a measurement protocol, where you cannot be certain as to the various digits beyond the ability of your measurement device. If you know those digits, very simply put -- you don't round. There isn't any need to.

If you don't know those digits with certainty, then the underlying principle is that computation cannot add precision/accuracy and you aim to round up as often as you round down. The notion that you round to the nearest even on the 5 is based on the assumption that if you get a 0 in the rounding place, you aren't rounding. From a purely mathematical standpoint, that is incorrect.

And I am sorry, but I cannot resist this. Back in the chain of responses, it was stated:


Mathematicians, generally, use the rounding method called "whatever makes my life easier", that is, when mathematicians actually compute figures.
Seriously?:D Because that is what we mathematicians are known for -- making life easier. Why, just this morning my calculus students accused me of just that.:D

Quite to the opposite, I'm afraid.

Cheers,

Chris

[Jeff Bratt]

On a spreadsheet there are 31 rows with 10 numbers each
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9
...
30.0, 30.1, 30.2, 30.3, 30.4, 30.5, 30.6, 30.7, 30.8, 30.9

That is obviously 310 numbers. This set is not particularly special - it's just a completely even distribution of numbers to be rounded that happen to easily fit on the screen... basically the same results would be obtained from a large enough set of random numbers.

[David Freed]

On a spreadsheet there are 31 rows with 10 numbers each
0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9
...
30.0, 30.1, 30.2, 30.3, 30.4, 30.5, 30.6, 30.7, 30.8, 30.9

That is obviously 310 numbers. This set is not particularly special - it's just a completely even distribution of numbers to be rounded that happen to easily fit on the screen... basically the same results would be obtained from a large enough set of random numbers.

I agree. I was looking crosseyed a few minutes ago and thought it ended in 39.9


True, but that series contains all the odd integers from 1 to 29, so it's not all even numbers.

If you are rounding each one to an even number (which I don't necessarily agree with), then they are all even.

I took the time to add the numbers after rounding to even, and I came up with 4800 for a total.

[Tom Veatch]

...If you are rounding each one to an even number (which I don't necessarily agree with), then they are all even....

Not all are rounded to even numbers. Only those that end with 5 are "rounded to even". All the others round to the nearest integer whether even or odd. 31.5 rounds to 32, but 31.4 rounds to 31. So, even after rounding, there's still a mix of even and odd.

But the original summation was on the unrounded numbers, so whatever rounding convention was used, the original sum wouldn't be affected.

[Ben Rafael]

I
And I am sorry, but I cannot resist this. Back in the chain of responses, it was stated:
Seriously?:D Because that is what we mathematicians are known for -- making life easier. Why, just this morning my calculus students accused me of just that.:D

Quite to the opposite, I'm afraid.

Cheers,

Chris

I didn't mean mathematicians make life easier for others; They try to make life easier for themselves. :) I spent too many years as a math student to believe otherwise.

[Jeff Bratt]

I'm sorry to say this, but you have made a classic mistake. You never round during the computation -- only at the end, if (and this is a big IF) you are concerned with the accuracy/precision of the computation -- which is what you are doing if you proceed to take an average. If you want to get the average, you would average the original data and then round.

Since the question is about the rounding process itself, and how that should be accomplished, the simplistic admonition to "round at the end" doesn't apply here. Rounding involves making a change to the original number - sometimes very little, sometimes more. To determine what changes result from a particular rounding process, one should look at a distribution of numbers subjected to that rounding.


As I mentioned in an earlier post, the round to the nearest even is a measurement protocol, where you cannot be certain as to the various digits beyond the ability of your measurement device. If you know those digits, very simply put -- you don't round. There isn't any need to.

If you don't know those digits with certainty, then the underlying principle is that computation cannot add precision/accuracy and you aim to round up as often as you round down. The notion that you round to the nearest even on the 5 is based on the assumption that if you get a 0 in the rounding place, you aren't rounding. From a purely mathematical standpoint, that is incorrect.

Rounding is necessary in cases other than just measuring. Rounding can be required after a division - like computing repeating fractions such as 1/3, or calculating with irrational numbers like pi or phi. Converting between base 10 and base 2 - going on inside our computers - can also require rounding.

The rule of "rounding to even" is not based on the notion you stated. A more complete statement of how rounding affects numbers is more like:
(Place means the digit being rounded away)
The value of numbers where place=0 is not changed by rounding.
Of the numbers that do have their value changed by rounding, those with place=1,2,3,4 are balanced out by those with place=6,7,8,9. (There is no difference between both rounding schemes on how to treat all these numbers so far.) The only numbers remaining have place=5. If you always round these up, then on average, rounding will increase the value of the set of numbers being rounded. This is why the rule is to "round to even", half of the time the place=5 numbers are rounded up, and half of the time rounded down. Thus the average value of the set remains unchanged. (Or almost unchanged - this is a statistical value after all.)

None of this applies to making a quick and dirty estimate, making sure you really have enough, or figuring out how to manage 2.3 children. What this does demonstrate is why "rounding to even" is mathematically the most accurate rounding scheme.

[David Freed]

Not all are rounded to even numbers. Only those that end with 5 are "rounded to even". All the others round to the nearest integer whether even or odd. 31.5 rounds to 32, but 31.4 rounds to 31. So, even after rounding, there's still a mix of even and odd.

But the original summation was on the unrounded numbers, so whatever rounding convention was used, the original sum wouldn't be affected.

Well, if that is the case, you are probably right. I am going to write this problem off as a case of; I am an old dog, and this is a new trick.

I only round numbers if I am trying to get a quick answer when adding in my head. When I am doing that, I don't necessarily follow any particular rules; BECAUSE IT DOESN'T MATTER IF I AM OFF BY 1 OR 1/10.

Good luck to the rest of you. Maybe you can get it figured out before the world implodes or explodes or whatever is supposed to happen since they smashed or created an atom the other day. Maybe you can tell; I'm real worried about that too.

[Russ Filtz]

I've been working in engineering for 22 years (not a degreed engineer) and have never heard of this. A quick informal survey of some of the PEs in my office didn't turn up anyone who was taught the 'round-to-even' method.
~g

OK, now we have to recall all the PE licenses of your firm!! :D (I am a degreed & licensed PE, BTW)

[Bill Eicher]

The purpose of teaching elementary students rounding is an attempt to develop skills to enable estimating and mental math. Those abilities allow students to determine if the answer obtained by another method is reasonable. As a special education teacher for fourth and fifth graders I believe the rule for rounding 5 and larger up is simplified and close enough for estimation at the elementary level. Those who need more sophisticated rounding methods can learn them when the need arises.

Bill

[Mike Henderson]

Just FYI - there's a good article on Wikipedia on rounding (http://en.wikipedia.org/wiki/Rounding). Many of you may have already found it.

Mike

[Justin Leiwig]

Seriously?

Is this the only forum where someone can ask about 3rd grade rounding and have a debate between engineering and math people?

I think I want my 5 minutes back that I spent reading this thread! I almost learned something and I try to never do that before 10 am! :D

[jeremy levine]

Just FYI - there's a good article on Wikipedia on rounding (http://en.wikipedia.org/wiki/Rounding). Many of you may have already found it.

Mike
And if you want to see what a pain it is for those of us in the Computer field
check out http://en.wikipedia.org/wiki/IEEE_754 which includes a small note on

"Round to Nearest – rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which occurs 50% of the time (in IEEE 754r (http://en.wikipedia.org/wiki/IEEE_754r) this mode is called roundTiesToEven to distinguish it from another round-to-nearest mode)"

[Ben Rafael]

The purpose of teaching elementary students rounding is an attempt to develop skills to enable estimating and mental math. Those abilities allow students to determine if the answer obtained by another method is reasonable. As a special education teacher for fourth and fifth graders I believe the rule for rounding 5 and larger up is simplified and close enough for estimation at the elementary level. Those who need more sophisticated rounding methods can learn them when the need arises.

Bill
Bingo!
How and why you round is dependent upon your goal.

[John Schreiber]

. . . I think I want my 5 minutes back that I spent reading this thread! . . .
Not me, this is the Fun Stuff.

I've been a "round the 5 to an even" man since high school chemistry and it has served me well in statistics. Jeff Bratt's argument above demonstrates it clearly to me. When I saw Chris Kennedy's response and saw that he teaches calculus, I thought I might be confused, but now I'm pretty sure Jeff's got it right.

As many have pointed out, the purposes of the rounding should determine the method used.

My favorite historical example of an error in rounding is human body temperature. When expressed in Celsius it's 37. With a normal understanding of significant figures, that means that normal is between 36.5 and 37.5. When expressed in Fahrenheit, people call it 98.6 which implies that normal is between 98.55 and 98.65. That means people often think that their body temperature is abnormal.

My preference would be to say that a healthy person has a temperature under the tongue between x and y z% of the time. But I'm a statistician.

Now, anybody want to talk about getting rid of the cent and the nickle?

[Ben Rafael]

John,
What people think is "normal" and what statisticians think is "normal" is not the same thing.
Normally, normal people do not know what you think normal is.
Few people know the difference between average, mean, and median but will use those words interchangeably.
And dont get me started on mode.:)

Also, count me in for getting rid of pennies and nickels...at least pennies.

[Russ Filtz]

John,

Few people know the difference between average, mean, and median but will use those words interchangeably.
And dont get me started on mode.:)

Esp dangerous when people start talking "average" salaries and the like. Very easy to think your "mean" is the best way to go, when median (or maybe even mode) gives a better view of what the "average" Joe makes. It's those darn super rich that skew everything!

psst. What's the sigma on your mean! :p

[Ben Rafael]

And a nice use of "average":

"Stop to think about how stupid the average person is and then realize that 1/2 the population is even more stupid than that"-George Carlin

[John Schreiber]

And dont get me started on mode.:)
Mode?

The popular way? Like with ice cream on it?

A la mode.

Apple pie with ice cream has the highest frequency of dessert choices (I just made that up).

[Ben Rafael]

What is the average dessert choice?
And is it normal?

[Jeff Bratt]

Pennies - actual physical pieces of metal - have long outlived their usefulness. There's no reason to ban "electronic pennies" in financial calculations, but if you have to pay cash it would be very easy to just round to the nearest nickel - there's no ambiguous halfway value. Many retail places already have a little penny stash that accomplishes this very thing. Plus the "fears" that everything will get more a little expensive (no more $1.99 type pricing) are unfounded, these prices would just become $1.95 instead.