48÷2(9+3) = ?

[grundle]

Texas Instruments TI-85 says:

48÷2(9+3) = 2

But Texas Instruments TI-86 says:

48÷2(9+3) = 288

[magslinger]

Or design a space telescope.

[re_nortex]

Well, I’m absolutely certain the correct answer is between 1 and 300.

Not if you're using octal. It's 440!

I am certain that when I awake on the morrow, this thread will have hit somewhere >= 600 (decimal notation) posts. :-)

[SeaHawkFan]

48 divided by 2(9+3)= 2
48 divided by 2 x (9+3)= 288.

Just follow the example at purplemath; it’s clear.

[SeaHawkFan]

Look at the purplemath example again, and follow the example and read the explanation. It’s clear.

[RBranha]

Nice example, except it’s given only to show an ambiguity which arises when people don’t follow the standard order of operations. A previous post indicated that the mathematicians and/or engineers at TI do not give implicit multiplication higher precedence.

“Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear.”

The original post in the thread asks why 2 calculators give different results. From Dr. Math on mathforum.org:

“In my research for another Dr. Math “patient,” I found that some calculators have experimented with this rule [implied multiplication]. Calculators have somewhat different needs than mathematicians, since they have to take input linearly, one character after another, so they are forced to make a decision about it. On the TI Web site I learned that they deliberately put this “feature” into the TI 82, and then took it out of the TI 83, probably because they decided it was not a standard rule and would confuse people.”

Summary: implied multiplication is not a standard rule, and the programming of the TI calculators was consequently changed.

[aruanan]

Why did you perform the operation between the 2 and the parenthesis before the division? They have the same mathematical precedence, and left-to-right processing says divide first.

The left to right argument is a superficial misapplication of directionality in English to orders of operation in math. The properties of multiplication and division take precedence over mere left to right notation. Beyond this, to clearly demonstrate intent of operations, one uses parentheses. Here is the only circumstance under which 288 obtains: (48÷2)(9+3). It is only the placement of the parentheses that says, "two is the divisor of 48, not a multiplicand of the sum of 9 and 3. In 48÷2(9+3), the use of parentheses makes 2 and the sum of 9 and 3 to be multiplicands and subject to properties between multiplicands. The only correct answer is that which is also consistent with properties of multiplication and division, that is, 2.

Please note that in the symbol ÷, the dot on the top represents the dividend (in this problem the 48) and the dot below, the divisor (in this problem, everything else to the right and before the equality sign). There is commutativity between multiplicands but not between a dividend and a divisor. So 2(9+3) is the same thing as (9+3)2 but 4÷2 is not the same thing as 2÷4.

In this problem as written, 48÷2(9+3), there is one dividend (48) and two multiplicands, 2 and (9+3). The operation between like terms, the multiplicands, is carried out first. Thus, the multiplication between 2 and the sum of 9 and 3. Or, because of the distributive property, 2(9+3) could have been written as (18+6). So the problem as written, 48÷2(9+3), could also be written as 48÷(18+6).

If the problem had contained more than two multiplicands, the answer would be even more obvious. For example, 48÷2(2+2)(5-2)= 48÷2*4*3, the multiplication between all the multiplicands must take place first because of laws of association (a(bc) = (ab)c) and commutation (abc = cba) that exist between multiplicands but not between a dividend and a single divisor. Thus, the 48 will always be divided by the product of 2 and 4 and 3.

If I, as the one writing the problem had meant for the 2 to be the divisor of 48, I would have had to indicate this by the use of parentheses so that 2 would not appear to be a multiplicand of 4 and 3. If I failed to do this, I couldn't maintain that the correct answer is 288 because of a "left to right" approach. "The quotient of 48 divided by 2 then multiplied by 12" is not the same left to right reading as "The quotient of 48 divided by the product of 2 and 4 and 3."

In the instance of this problem, 48÷2(9+3), the operation between the multiplicands must be completed first otherwise commutation around the operand, allowed for in multiplication, would result in two different divisors, 2 or 12, which would result in two different answers, 288 = [(48÷2)(12)] or 8 = [(48÷12)(2)].

Remember this also:

PEMDAS. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

48÷2(9+3) =

Parentheses: 48÷2(12) =
Powers: 48÷2(12) =
Multiplication: 48÷24 =
Division: 2

[grateful]

The only case in which you can justify the answer of 2 is if you argue implied multiplication has higher precedence over explicit multiplication. By normal order of operations, the answer is 288.

http://knowyourmeme.com/memes/48293

I could go on and on citing sources, but I don't think it'll convince any 2ers.

[Lurkina.n.Learnin]

Interesting arguments here. Let me settle this. I just did this on my phone and it said the answer is 288. I’m not a math wiz but I know I’m right because my phone is a “smart” phone, at least that is what U S Cellular said when they sold it to me.

[knittnmom]

LOL! Perfect!

[aruanan]

48÷2(9+3)

#1 the () is the priority..
#2 next would be the other equation (48/2)..
#3 the () is multiplied times the answer of #2
#4 288


To begin with, 48÷2(9+3) is not an equation and 48/2 is not "the other equation."

Second, 2 would be the divisor of the dividend 48 only in this case: (48÷2)(9+2).

Third, as written, 2 is the first of two multiplicands; the sum of 9 and 3, or 12 is the second. The operation between multiplicands, like terms, must be completed first.

Fourth, the dividend 48 divided by the product (24) of two multiplicands (2 and 12) yields a quotient of 2.

[RBranha]

Perhaps you should publish a paper and explain to the people who wrote the arithmetic parsers for Google, Microsoft, HP, GNU, etc. why they are giving the wrong answer.

[IYAS9YAS]

Following your link, when you type the equation in their “Enter Problem” box 48/2(9+3) gave 288, with the solution being 48 over 2 first (24) then multiplied by 9+3 (12). So, it appears my answer of 2 before, is incorrect and should be 288. That’s using your quoted site.

[P.O.E.]

Question is, why do they show different answers.

Should be a warning about too much dependence on technology. (think Google filtering and “pushing” matches).

The way math is taught today answer should be 288, due to left-to-right precedence. But the real answer depends on what the original question is, and the intent of the calculation.

[InvisibleChurch]

72

[P.O.E.]

The answer 2 is obtained because the division sign is also a grouping symbol (like parentheses). So the two would be multiplied 12 before that product was divided in to 48.

[IYAS9YAS]

Following your link, when you type the equation in their “Enter Problem” box 48/2(9+3) gave 288, with the solution being 48 over 2 first (24) then multiplied by 9+3 (12). So, it appears my answer of 2 before, is incorrect and should be 288. That’s using your quoted site.

But, going back to the original equation as 48÷2(9+3) and it gives the answer of 2. So your sight treats the ÷ and the / differently. So, my take-away is my first guess was 2 and was right, based on the ÷. Using the / is a wrong assumption and gives 288.

[RBranha]

The calculators in question used /. See the pictures of the 2 calculators at the top of the thread. I can’t say for sure, but the link you’re looking at is almost certainly pre-parsing the problem text before evaluating it, and it’s “rewriting” the problem differently for the / and ÷ cases.

Google gives 288 for both / and ÷. Bing gives 288 for / but does not calculate with ÷. Matlab, C, C#, PHP, JavaScript, bc, Python, and the HP50g all say 288.

What does this mean? Math, like other languages, has evolved over the years. The implied multiplication question is still being asked, which is evidence of the continuing evolution. In the last 4-5 decades or so (and probably as a result of the development of computing,) implied multiplication has come to have the same precedence as explicit multiplication. This is evident in the fact that so many languages and computing platforms give 288 as the answer to the problem in question. Very few give an answer of 2. Even though some TI products produce 2, TI has changed their approach so that newer products say 288.

[spunkets]

"Did you ignore this part: “parentheses outrank division”?"

Parenthesis have no effect external to their boundary. In the woman's example "2(2)" all hte parenthesis do after the contents have been reduced to "2" is to distinguish 2(2) from 22. The example reduces to: 16*(1/2)*2+1=17. That's not 5 as the both of you contend, but you may continue to do so with your own special arbitrary, adhoc rules that are inconsistent with the standard rules of arithmetic, such as multiplicative commutivity and the concept of multiplicative inverse.

[CharacterCounts]

I tried several online algebra calculators.

With / I got 288, but when I used ÷, I got 2.

[RBranha]

I’d like to see the algorithm used by the online calculators to parse the problem text. To give different answers, they have to be parsing the text differently. Most likely they’re doing something like

48/2(9+3) => 48/2*(9+3)
and
48÷2(9+3) => 48/(2*(9+3))

The expressions after the => can be calculated directly in the languages used to build the sites.