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How to solve $a÷b(c+d)$?

For example, $2÷4(8+16)$.

Is it $($$\frac{2}{4}$$)(8+16)$ = $\frac{1}{2}$(8+16) = $\frac{1}{2}$(24) = 12?

or $\frac{2}{4(8+16)}$ = $\frac{2}{4*24}$ = $\frac{2}{96}$ = $\frac{1}{48}$?

Which one is the right one?

User
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3 Answers3

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I'd argue that a better solution is to ask the person who wrote this expression down to re-write it in a more clear manner.

The purpose of notation (and implicitly, the order of operations rules people talk about) is not to torture and confuse students, but to communicate information successfully from one person to another. Expressions like $$ a \div b (c + d) $$ are technically unambiguous if we have all agreed on the order of operations beforehand. However, they aren't very clear (and hence are in practice ambiguous) and so a better solution is to either add more brackets before you write it down, or to write it as either $$ \frac{a}{b(c + d)} $$ or $$ \frac{a}{b}(c + d) $$ depending on which of those two is actually meant.

Written mathematics shouldn't be a guessing game for the reader. If they don't know what you meant due to potentially unclear notation, this isn't their fault.

Simon Rose
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  • The exercise could be to use order of operations, so asking for clarification defeats the purpose. That might not be the case, but we do welcome questions of all levels. – Em. Feb 17 '16 at 10:16
  • Fair! But I do see these things being passed around often with many, many people arguing over the "right" answer, so just in case it is that, I felt I should add this perspective in. – Simon Rose Feb 17 '16 at 10:57
  • I think it's also important to note that the order of operations is not something handed down from on high in the form of stone tablets. They are a convention to communicate information unambiguously, and no more. – Simon Rose Feb 17 '16 at 10:57
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They way it is written, assuming the usual order of operations, it should be $$2/4(8+16) = 2/4(24) = .5(24) = 12$$ since you do division OR multiplication from left to right, after parentheses and exponents.

Em.
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Never forget your order of operations:

Parentheses
Exponentiation
Multiplication and Division
Addition and Subtraction

For the "and" lines: these operations have the same level of priority. Now you go left to right.

For example: $35 \cdot 2 / 7 \cdot 6 = 70 / 7 \cdot 6 = 10 \cdot 6 = 60$

Newb
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